Angular Momentum in Quantum

Mechanics.

R. C. Johnson

March 10, 2015

1 Brief review of the language and concepts ofQuantum Mechanics.

We begin with a review of the basic concepts involved in the quantum mechanicaldescription of physical systems and the notation we will use in the lectures.

The notes in this Section are not intended to provide an introductory courseon Quantum Mechanics. They assume the reader has had at least a first courseat undergraduate physics level that covers some historical background and re-views the experimental evidence that leads to the necessity for the formalismdeveloped here as the most useful description we have of nature at the subatomiclevel.

Some familiarity with matrices, differential equations, complex numbers andvector is assumed. The basic concepts of Dirac’s vector space formulation ofnon-relativistic Quantum Mechanics are described in an informal, non-rigorousway. We believe that this formalism provides the best and most economicalframework for understanding of the quantum world in a notation that is of greatpracticality for the description of many-body systems. We believe that theseadvantages outweigh any difficulties that may arise because of the abstractionsinvolved. These difficulties are no worse then those involved in mastering theconcept of vectors in physical space.

1.1 The structure of theories in physics.

We first show how the way dynamical systems are described in quantum me-chanics fits into the same general scheme that we use in classical physics.

1.1.1 Newtownian particle mechanics.

1. Dynamical variables.Particle coordinates and momenta:

r1, p1, r2, p2, . . . (1)

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2. Definition of a state S at time t.A set of values of r1(t), p1(t), r2(t), p2(t), . . .3. Dynamical law describing the way the state changes with time.Newton’s Law of Motion:

dpi(t)

dt= F i(r1(t),p1(t), r2(t),p2(t), . . .),

dri(t)

dt=

pi(t)

mii = 1, 2, . . . . (2)

1.1.2 The Electromagnetic Field.

1. Dynamical variables.Electric and Magnetic fields at all points r:

E(r, t),B(r, t). (3)

2. Definition of a state S at time t.A set of values of E(r, t),B(r, t) for all r.3.Dynamical law describing the way the state changes with time.Maxwell’s equations:

1

c

∂E(r, t)

∂t= ∇∧B(r, t)− 4π

cJ(r, t),

1

c∇.E(r, t) = 4πρ(r, t),

1

c

∂B(r, t)

∂t= −∇ ∧E(r, t),

1

c∇.B(r, t) = 0. (4)

1.1.3 Quantum Mechanics of a spinless point particle.

1. Dynamical variables.The operators corresponding to the particle position coordinate and momen-

tum: r, p.2. Definition of a state S at time t.The ket vector | S, t〉 at time t.3. Dynamical law describing the way the state changes with time.The Schrodinger equation:

ıh∂

∂t| S, t〉 = H(r, p) | S, t〉, (5)

where H(r, p) is the Hamiltonian operator for the particle being considered.H(r, p) is constructed out of the operators r and p.

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1.1.4 Features of the 3 theories.

1. Note the occurrence of various key constants: mass, electric charge, Plank’sconstant.

2. In all cases the state S evolves in time according to differential equationsthat are first order in time derivatives, This means that they are all causal, i.e.,given a state S at time t the state at all other times is determined.

3. A key difference between Quantum Mechanics and the other theories isthe very different concept of ”state” and its connection with what we measurein the laboratory. In the world described by Newtonian Mechanics and Electro-magnetic Theory there is a one-to-one correspondence between the dynamicalvariables used in the theory and quantities that are believed to be measurablein the laboratory. In Quantum Mechanics the relationship is many-to-one. Onestate can correspond to many observed values of the dynamical variables andthe interpretation of the ket vector in terms of probability amplitudes reflectsthis fact.

In order to be able to use and interpret the Schrodinger equation (5) weclearly must understand the concept of ket vectors and operators.

1.2 The formalism of quantum mechanics.

1.2.1 Ket vectors.

We are used to describing an electric field in terms of a vector in 3-dimensionalspace, E. Note that the concept of a vector is a very abstract one. A vector is nota number, or even 3 numbers. In a particular coordinate system E is describedby its numerical components, E1, E2, E3, along 3 orthogonal directions in spacedenoted by the vectors of unit length e1, e2, e3. The vector E can then bewritten

E = E1e1 + E2e2 + E3e3. (6)

But if we change these axes the same vector E is described by 3 different num-bers. The vector description of the electric field collects into one concept theinfinite number of triplets of numbers that describe the same quantity in allpossible axes. Of course, all the triplets are related to each other by a simpleformula which defines what it means to be a vector. The use of vectors im-plies that the relations between the electric and magnetic fields embodied inMaxwell’s equations, eq.(4), are valid for any choice of coordinate axis in space.

In a similar way, there are many ways of describing a system in quantummechanics. All these descriptions are completely valid and related to each otherby linear equations that are analogous to the way the components that describeE and B in different coordinate systems are related. Dirac invented a languagefor writing down quantum mechanics that encapsulates these relationships.

A state in quantum mechanics is described by a vector in an infinite di-mensional complex space. Dirac called these vectors ket vectors and inventeda special notation for them: a vertical line, a label and angular bracket, >. A

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state S is described by a ket | S〉 labelled so that it is distinguished from a ketdescribing any another state. Different physical systems correspond to differentspaces. Different states of a particular physical system are described by differ-ent kets in the same space. Ket vectors in the same space can be added andmultiplied by complex numbers to give new kets.

Just like vectors in physical 3-dimensional space, ket vectors are abstractquantities. To make a connection with numbers we need axes in ket spacecalled basis kets, | u1〉, | u2〉, . . .. The infinite dimensions of Dirac’s ket spacesshows up as the need to use bases with an infinite number of basis vectors | ui〉.The complexity of the space means that the components of | S〉 are complexnumbers S1, S2, . . . in general. The fact that the | ui〉 form a basis means that,analogously to eq.(6), we can write

| S〉 = S1 | u1〉+ S2 | u2〉+ . . . . (7)

1.2.2 Physical interpretation of ket components.

In quantum mechanics the connection between the state of the system and theresults of measurements on the system is contained in the interpretation of thenumbers Si as probability amplitudes. The physical meaning of these ampli-tudes is particularly clear when the basis is orthonormal. We discuss below whatit means to be an orthonormall basis, but leaving the formal definition aside forthe moment, | Si |2 is the probability that measurements will give results con-sistent with being in the state described by the ket | ui〉. (See Appendix A forrevision notes on complex numbers.)

If we describe the same ket vector | S〉 using a different orthonormal basis thenumbers Si which will change to a new set of complex numbers that describethe probability amplitudes for a different set of observables corresponding tothe new basis states. If we want to ask a question about the probable resultsof particular measurements on a physical system we have to learn how thechoice of measurements picks out a particular basis. This link is guaranteedby choosing the | ui〉 to be eigenfunctions of one or more Hermitean operators.When we have a particular measurement in mind, ”All bases are equal, butsome are more equal than others,” to mis-quote ”1984”. To understand thesestatements we need to understand the connection between quantities observablein an experiment and operators and this in turn means we need to understandthe concept of an Hermitian operator.

The power of Dirac’s formulation in terms of ket vectors is revealed in theform of the Schrodinger equation given in eq.(5) in which there is no explicitreference to a particular basis. All bases have the same weight. To use thisformulation in terms of standard mathematical forms, such as differential oper-ators and matrices, we have to learn how to express kets and the Hamiltonianoperator H in different bases.

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1.2.3 Operators and observables.

In quantum mechanics every observable is represented by a Hermitian opera-tor. The only possible result of a measurement of an observable is one of theeigenvalues of the corresponding operator. If the system happens to have beenprepared in an eigenstate of an observable A corresponding to the operator Awith eigenvalue α1, then a measurement of A will result in the value α1 withcertainty. We label the corresponding eigenket with its eigenvalue α1. It satisfiesthe eigenvalue equation

A | α1〉 = α1 | α1〉. (8)

1.2.4 Inner product of kets.

The concept of an Hermitian operator arises when we have an inner productdefined between kets. This is analogous to the ”dot”product of ordinary vectors.The inner product of two kets is defined as a rule which assigns a complexnumber to any pair of kets. The inner product associated with the two kets | a〉and | b〉 is written 〈b | a〉.

To qualify as an inner product the rule that gives the complex number 〈b | a〉must satisfy have the following properties for all | a〉 and | b〉.

1.

〈b | a〉 = (〈a | b〉)∗. (9)

2. If α is an arbitrary complex number and | d〉 = α | a〉 then

〈b | d〉 = α〈b | a〉. (10)

3. If | a〉 =| e〉+ | f〉 then

〈b | a〉 = 〈b | e〉+ 〈b | f〉. (11)

Property (9) means that the symbol 〈b | a〉 is not symmetric in the labels aand b. Interchanging them is the same as complex conjugation.

Properties (10) and (11) means that if | g〉 = α | e〉+ β | f〉 then

〈b | g〉 = α〈b | e〉+ β〈b | f〉. (12)

We say that 〈b | a〉 is linear in | a〉. But (9) implies that

〈g | a〉 = α∗〈e | a〉+ β∗〈f | a〉. (13)

We say that 〈b | a〉 is antilinear in the ket on the left of the vertical line, | b〉.4.Property (9) implies that 〈a | a〉 is a real number. We also require that it

be positive or zero.

〈a | a〉 ≥ 0. (14)

The equality is satisfied only if | a〉 is the nul ket, | 0〉. The latter satisfies| a〉 =| a〉+ | 0〉 for all | a〉.

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It also follows from the defining properties that

〈0 | a〉 = 〈a | 0〉 = 〈0 | 0〉 = 0, (15)

for all | a〉.

1.2.5 Orthonormal sets of kets.

Two kets | a〉 and | b〉 are said to be orthogonal if their inner product vanishes,i.e.,

〈a | b〉 = 0. (16)

A set of kets | u1〉, | u2〉, . . . is said to be orthonormal if different kets in the setare orthogonal and each is normalized so that 〈ui | ui〉 = 1, i.e.,

〈ui | uj〉 = δi,j , (17)

where δi,j is the Kroneker delta symbol defined by

δi,j = 0, i 6= j,

= 1, i = j. (18)

1.2.6 Inner products in terms of ket components.

Having defined an inner product of kets and found an orthonormal basis we arenow in a position to actually evaluate an inner product of two arbitrary kets| S〉 and | S′〉 once we have an expression for them in terms of components andbasis kets as in eq.(7):

| S′〉 = S′1 | u1〉+ S′2 | u2〉+ . . . ,

| S〉 = S1 | u1〉+ S2 | u2〉+ . . . . (19)

Using the linearity property (12) we have

〈S′ | S〉 = S1〈S′ | u1〉+ S2〈S′ | u2〉+ . . . . (20)

But using (13) and (17)

〈S′ | u1〉 = (S′1)∗〈u1 | u1〉+ (S′2)∗〈u2 | u1〉+ . . .

= (S′1)∗, (21)

and

〈S′ | u2〉 = (S′1)∗〈u1 | u2〉+ (S′2)∗〈u2 | u2〉+ . . .

= (S′2)∗, (22)

and in general

〈S′ | ui〉 = (S′i)∗. (23)

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Using this result in (20) we have finally

〈S′ | S〉 = (S′1)∗S1 + (S′2)∗S2 + . . .

=

i=∞∑i=1

(S′i)∗Si. (24)

Note in particular that if the S and S′ are the same state then

〈S | S〉 = | S1 |2 + | S2 |2 + . . .

=

i=∞∑i=1

| Si |2 . (25)

Bearing in mind the interpretation of the components Si as probability ampli-tudes, we would like the components Si to satisfy

∑i=∞i=1 | Si |2= 1. Eq.(25) we

see that this is the same condition as requiring the ket | S〉 to be normalised sothat 〈S | S〉 = 1.

In an orthonormal basis eq.(23) gives us a useful formula for the componentsof an arbitrary ket in terms of an inner product:

S′i = 〈ui | S′〉. (26)

Eq.(19) can now be written

| S〉 = | u1〉〈u1 | S〉+ | u2〉〈u2 | S〉+ . . . ,

=∑i

| ui〉〈ui | S〉. (27)

Note that as an aid to memory we have extended are notation slightly by puttingthe comples numbers 〈ui | S〉 behind the kets | ui〉. By definition this meansexactly the same as 〈ui | S〉 | ui〉. We shall see below that using this conventiongives a simple way to express formulae for relations between kets into relationsbetween numbers.

1.2.7 Operators acting on kets.

In general we define an operator O as any unambiguous rule that when appliedto an arbitrary ket, gives another ket. Our notation for operators is to put a”hat” over a suitably chosen symbol, such as O. If according to the rule O itsaction on | S〉 is to produce | S′〉 we write

| S′〉 = O | S〉. (28)

In quantum mechanics we frequently deal with a sub-class of operators calledlinear operators. They satisfy

O(| a〉+ | b〉) = O | a〉+ O | b〉O(α× | a〉) = α× O | a〉, (29)

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where α is an arbitrary complex number.Unless specified otherwise, all operators referred to in these notes will be

linear operators1.An important example of a linear operator can be found in eq.(27). This

equation can be rewritten in a way that makes it look like a trivial identity ifwe agree to regard a quantity like | a〉〈b | as a linear operator defined by therule

(| a〉〈b |) | c〉 =| a〉(〈b | c〉). (30)

Eq.(27) can now be written

| S〉 =∑i

| ui〉〈ui | S〉

= (∑i

| ui〉〈ui |) | S〉. (31)

This reveals that this relation can be viewed as identifying the operator in roundbrackets as the unit operator:∑

i

| ui〉〈ui |= 1. (32)

This formula is often referred to as the completeness relation for basis kets.

1.2.8 Operators, kets and numbers.

If we take the inner product of the ket | b〉 defined in eq.(28) with the ket | c〉we write the resulting complex number 〈c | b〉 as

〈c | b〉 = 〈c | O | a〉. (33)

We often refer a number constructed in this way as a ”matrix element of theoperator O”. This reason for this terminology becomes clearer if we considerthe collection of numbers 〈ui | O | uj〉, where i, j run over all the members ofa set of basis kets. This ∞×∞ matrix plays a crucial role when, for example,we express the relation eq.(28) in terms of numbers.

Taking the inner product of both sides of eq.(28) with basis vector | ui〉 weget

〈ui | S′〉 = 〈ui | O | S〉. (34)

Using the formula (26) and the linearity of O we find that (34) can be written

S′i =∑j

〈ui | O | uj〉Sj . (35)

1The concept of an anti-linear operator occurs when the time reversal transformation isdiscussed. Instead of the second of properties (29), an anti-linear operator O satisfies

O(α× | a〉) = α∗ × O | a〉.

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This result has a useful interpretation in terms of standard operations on num-bers. We think of the numbers S1, S2, . . . , and S′1, S

′2, . . . , as two matrices each

with one column and an infinite number of rows. The formula (35) can now beunderstood as:

” Matrix S′ is the result of multiplying matrix S, according to the rules ofmatrix multiplication by the matrix O, where the element of O in the ith rowand the jth column is 〈ui | O | uj〉, i.e,

S′ = O × S.” (36)

In other words, the relation between kets and operators in eq.(28) is mappedprecisely onto a relation between matrices representing | S〉, | S′〉 and O ineq.(36).

1.2.9 Hermitian operators and Hermitian Matrices.

By definition, the Hermitian conjugate of an operator O is an operator O† thatsatisfies

〈a | O† | b〉 = 〈b | O | a〉∗, (37)

for ALL kets | a〉 and | b〉.If O† = O, i.e, if O† | a〉 = O | a〉 for all | a〉, we say that O is an Hermitian

operator.For an Hermitean operator eq.(37) tells us that

〈a | O | b〉 = 〈b | O | a〉∗. (38)

for all kets | a〉 and | b〉.PROBLEM. Use the definition (37) to show that

(AB)† = B†A†. (39)

We saw in the last subsection that an arbitrary linear operator O is repre-sented in an orthonormal basis by a matrix with components Oi,j given by

Oi,j = 〈ui | O | uj〉 (40)

If O is Hermitean, eq.(38) tells us that the matrix elements Oi,j must satisfy

Oi,j = O∗j,i. (41)

In other words, the matrix representing an Hermitian operator is the same asthe complex conjugate of the matrix obtained by interchanging its rows andcolumns. Such a matrix is known as an Hermitian matrix.

It is proved in text books that(1).The eigenvalues of an Hermitian operator are all real numbers.

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(2). Eigen kets corresponding to distinct eigenvalues of an Hermitian oper-ator are orthogonal. We can choose them all to be normalised to unity. Theeigenkets then form an orthonormal set.

(3). The set of eigenkets can be chosen so that they satisfy the completenessrelation (32)

These properties of Hermitian operators map on to the analogous propertiesof finite Hermitian matrices, e.g., eq.(41)

Note that the orthogonality property in (2) above refers only to distincteigenvalues. It frequently happens in physical applications that eigenvalues ofHermitian operators are degenerate, i.e., several linearly independent eigenketscan be found all corresponding to the same eigenvalue. If there are, say, Niof these, they can, by taking appropriate linear combinations of them, takento be an orthonormal set. All Ni must be included in the summation in thecompleteness relation.

1.2.10 Eigenket notations.

Collecting our results, let A be an Hermitian operator corresponding to some ob-servable. Its eigenvalues are the real numbers, αi, in general infinite in number.The corresponding eigenkets are | αi, ni〉 and satisfy

A | αi, ni〉 = αi | αi, ni〉. (42)

We have labelled the eigenkets corresponding to the ith eigenvalue with twoquantities, the value of the eigenvalue itself and a second label ni, that distin-guishes different members of an orthonormal set of degenerate eigenkets. Theway this set is chosen is not unique. In practice, the labels ni are frequentlychosen to be the eigenvalues of one or more operators that can be constructedfrom the dynamical variables of the system and differ from A. The number ofoperators involved depends on the number of degrees of freedom of the system.These operators must satisfy very special conditions before they can be used inthis way. They must correspond to compatible observables. What this conditionmeans is discussed below.

So far we have used a notation that assumes the spectrum of eigenvaluesis a discrete set. In practice many operators of physical significance may havea spectrum of eigenvalues which is completely or partially continuous. Forexample, the spectrum of the Hermitian operator corresponding to the particleposition on the x axis, the operator x, has a spectrum that includes all realvalues of x from −∞ to +∞. We label the corresponding eigenkets, | x〉, −∞ <x < +∞. These satisfy

x | x〉 = x | x〉. (43)

The ket | x〉 describes a state in which a measurement of the particle’s positionwould yield the value x with certainty. ( More accurately, the particle positiondensity for this state is zero except at x.)

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In the formulae of Sections 1.2.5 and 1.2.6 all the Kroneker delta functionshave to be replaced by Dirac delta functions and the summations over thediscrete variable i must be replaced by integrals over x.

〈x | x′〉 = δ(x− x′), (44)

| S〉 =

∫ +∞

−∞dx | x〉ψS(x), (45)

ψS(x) = 〈x | S〉, (46)∫ +∞

−∞dx | x〉〈x | = 1, (47)

〈S′ | S〉 =

∫ +∞

−∞dxψ∗S′(x)ψS(x), (48)

〈S | S〉 =

∫ +∞

−∞dx | ψS(x) |2 . (49)

In eqs(45) and (46) we have introduced a special notation, ψS(x) for the coeffi-cients in the expansion of | S〉 in the orthonormal set | x〉, −∞ < x < +∞.

ψS(x) = 〈x | S〉. (50)

This is the standard ψ notation for what is called ”the wave function” of thestate S in introductory courses in quantum mechanics. We see that the wavefunction ψS(x) is revealed as the collection of complex numbers that describethe state S in the basis of eigenvectors | x〉 and is just one of the many ways ofdescribing the the ket | S〉. The x basis is a convenient one for some purposes,but it has no more fundamental significance than any other basis.

Note that the interpretation of ”the wave function” as a position probabilityamplitude is consistent with the general picture given in Section 1.2.1 thatinterprets the coefficients in the expansion of | S〉 in a basis as probabilityamplitudes.

The formulation of quantum mechanics in terms of kets in a complex vectorspace shows us that the function ψS(x) is just one of many other ways of rep-resenting a physical state as a set of complex numbers. We get a different setevery time we use a different basis, but all these sets are equally valid descrip-tions of S and each set separately contains the maximum information we canhave about a physical state according to quantum mechanics.

1.2.11 The momentum basis.

As a concrete example of another basis we consider the eigenkets of the Her-mitian operator px corresponding to a particle’s momentum in the x direction.We use the letter p, −∞ < p < +∞ to denote the real eigenvalues of px so thatan eigenket is | p〉 and satisfies

px | p〉 = p | p〉. (51)

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A general state S can be expanded

| S〉 =

∫ +∞

−∞dp | p〉φS(p), (52)

where

φS(p) = 〈p | S〉. (53)

φS(p) might be called ”the wavefunction of S in the momentum basis” or ”inmomentum space”. Both functions φS(p) and ψS(x) give a complete descriptionof S. We can transform from one function to the other using

φS(p) = 〈p | S〉,

= 〈p | (∫ +∞

−∞dx | x〉〈x |) | S〉

=

∫ +∞

−∞dx 〈p | x〉〈x | S〉

=

∫ +∞

−∞dx 〈p | x〉ψS(x). (54)

This formula gives the value of φS(p) for a particular p as an integral over allvalues of x of ψS(x) and the inner product 〈p | x〉. The latter is a function of xand p but is independent of S.

The complex numbers 〈p | x〉 have three possible interpretations:(1) The matrix 〈p | x〉 defines the transformation from the x to the p basis

as in eq.(54). The matrix 〈x | p〉 = 〈p | x〉∗ defines the transformation from thep to the x basis through the inverse formula to eq.(54),

ψS(x) =

∫ +∞

−∞dp 〈x | p〉φS(p). (55)

(2) As a function of p, 〈p | x〉 is the wavefunction in the p basis of a state inwhich the particle is definitely at the point x (an eigenstate of operator x witheigenvalue x). See eq.(53).

(3) As a function of x, 〈x | p〉 = 〈p | x〉∗ is the wavefunction in the x basis ofa state in which the particle has definite momentum p (an eigenstate of operatorp with eigenvalue p). See eq.(50).

To obtain 〈x | p〉, or 〈p | x〉, we must input more information about the phys-ical meaning of the momentum and position operators. For present purposeswe will simply appeal to our knowledge from introductory quantum mechanicscourses that acting on a wave function ψSx the momentum operator is h

ıddx .

According to interpretation (3) above, 〈x | p〉 is the wave function at pointx of a state corresponding to a definite momentum p. The wave function corre-sponding to the ket px | p〉 should therefore be h

ıddx 〈x | p〉. On the other hand

px | p〉 = p | p〉 and so

h

ı

∂

∂x〈x | p〉 = p〈x | p〉. (56)

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This is a simple differential equation for the function of x, 〈x | p〉, for a fixed p.The solution is

〈x | p〉 =1√2πh

exp(ııp x

h), (57)

where we have chosen an arbitrary constant in front of the exponential so thatthe is normalisation gives∫ +∞

−∞dp〈x′ | p〉〈p | x〉 = δ(x′ − x). (58)

When the explicit formula (57) for 〈x | p〉 is inserted into the relations (54)and (55) we see that wavefunctions in the x and p basis are Fourier transformsof each other. The Heisenberg Uncertainty relation between position and mo-mentum are then seen to be just the well-known relation between between theextent in the relevant coordinates (x and p in this case) of functions that areFourier transform of each other (ψS(x) and φS(p) in this case).

A much more elegant and powerful derivation of the results (120) and (57)can be based on the correspondence between the commutation relation [x, px] =ıh and the Poisson bracket of the same dynamical variables in classical me-chanics. This is the derivation given in Dirac’s book on Quantum Mechanics.We will ask see later that the momentum operator is the generator of spatialtranslations. Momentum plays a similar role in classical mechanics.

The formula (54) displays the general structure of the relation between ketcomponents in different bases. In the case that the two bases are both de-scribed in terms of purely discrete labels this relation is readily interpretableas a relationship between column matrices of the two sets of components and amatrix whose elements composed of the inner products of the two sets of basiskets. Thus if the initial basis set is | ui〉 with components Ui and correspondingquantities in the second set are | vi〉 and Vi, then the analogue of eq.(54) is

Vi =∑j

Ti,jUj (59)

where the matrix elements Ti,j are given by

Ti,j = 〈vi | uj〉. (60)

It is straightforward to show that the the matrix T is unitary, T T † = 1. Thusbasis changes are governed by a unitary transform. It is customary to use thesame language when the bases involve of both discrete and continuous labels.

1.3 Compatible observables.

It may be possible to create a complete set of kets that are all eigenkets of twoHermitian operators A and B. If we denote a pair of eigenvalues of A and B by

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the real numbers α and β respectively, then a simultaneous eigenket is written| α, β〉 and satisfies

A | α, β〉 = α | α, β〉,B | α, β〉 = β | α, β〉. (61)

It is shown in text books that a necessary and sufficient condition for acomplete set of simultaneous eigenstates to exist A and B must commute,i.e.,

AB = BA, (62)

In general the commutator of two operators C and D is an operator definedby

[C, D] = CD − DC. (63)

Hence the condition (62) is equivalent to the statement that the commutatoroperator of [A, B] must be the zero operator.

[A, B] = 0, (64)

This is a very special condition that is certainly not satis fied by all operators.Some pairs of operators have a commutator that is a non-zero complex num-

ber rather than a more complicated operator, e.g., [x, px] = ıh. In elementaryquantum mechanics courses it is explained how the corresponding observablessatisfy the Heisenberg Uncertainty Principle.

More generally, if a pair of operators have a non-zero commutator, whether acomplex number or another operator, we cannot find a complete set of eigenketsof these operators. Bearing in mind the basic interpretation, discussed in Section1.2.3, of the situation that occurs when a state is described by an eigenket, wesee that in general we can’t prepare a set of states in which the observablescorresponding to the two non-commuting operators have a definite value. Thisis exactly the situation that occurs with the different components of the angularmomentum operator for a particle or a system of particles., Jx, Jy, Jz, no twoof which commute.

1.4 Quantum Dynamics. Conservation Laws.

In quantum mechanics the way the state of a system evolves in time is deter-mined by the Schrodinger equation

ıh∂

∂t| S, t〉 = H | S, t〉. (65)

The details nature of the Hamiltonian operator H depends on the physical sys-tem being considered. H is often determined by taking the classical Hamiltonianthat is believed to describe the analogous classical system and replacing the dy-namical variables by linear operators with commutation relations determined

14

by the connection between Classical Poisson brackets and quantum commuta-tors. It also frequently happens that the system has no classical analogue, forexample because experiment has determined that the system must have degreesof freedom that have no classical analogue, so that the structure of the Hamil-tonian has to constructed on the basis of intuition or general principles such assymmetry requirements.

We emphasised in Section 1.1 that the Schrodinger equation is a causalequation. This means that if the ket | S, t〉 is known at time t, the Schrodingerequation completely determines the ket describing the system at time t + δ t,and so on for all time. To see this, we have, for sufficiently small δ t

| S, t+ δ t〉 = | S, t〉+ δ t∂

∂t| S, t〉

= | S, t〉+ δ t× 1

ıhH | S, t〉. (66)

We see that the Hamiltonian operator plays a fundamental role in determiningthe way the state ket changes with time.

The Hamitonian operator has a second fundamental role. Its eigenvaluesdefine the possible energy values the system can have. Its eigenkets | Ei〉 arestates in which the energy of the system has a definite value and satisfy

H | Ei〉 = Ei | Ei〉. (67)

1.4.1 Stationary states with definite energy.

We go through some of the algebra in this sub-Section because the derivationsuse many of the basic properties of kets, linear operators, Hermitian operatorsand inner products discussed in sub-Section 1.2.

If the system is prepared in an eigenstate of H it evolves in time in a veryspecial way. Suppose for example that the system is in state | E0 > at timet = 0. The ket

| S0, t〉 = exp(− ıE0t

h) | E0 > (68)

reduces to | E0〉 > at t = 0 and satisfies the Schrodinger equation exactly forall time.

To see this we first evaluate the left-hand-side of eq.(65) and obtain

ıh∂

∂ texp(− ıE0t

h| E0 > = ıh× (− ıE0

h) exp(− ıE0t

h| E0 >

= E0 × exp(− ıE0t

h) | E0 > . (69)

The right-hand-side of eq.(65) can be evaluated using the fact that H is a linearoperator and hence can be taken around the complex number exp(− ıE0t

h ) in

15

eq.(68) to give

H exp(− ıE0t

h) | E0 > = exp(− ıE0t

h)× H | E0 >

= exp(− ıE0t

h)× E0 | E0 >

= E0 × exp(− ıE0t

h) | E0 >, (70)

which agrees with the evaluation of the left-hand-side of (65) we gave in eq.(69).The equality given in eq.(70) also shows that a state prepared in an eigenstate

of H at t = 0 remains an eigenstate of H with the same eigenvalue for all time.As time progresses the only effect is to multiply the initial state by a complexnumber of magnitude unit.

Another property of the state (68) is that the expectation value of any timeindependent linear operator O doesn’t change with time because

〈S0, t | O | S0, t〉 = exp(− ıE0t

h)〈S0, t | O | E0〉

= exp(− ıE0t

h)(〈E0 | O† | S0, t〉)∗

= exp(− ıE0t

h)(exp(− ıE0t

h)〈E0 | O† | E0〉)∗

= exp(− ıE0t

h)× exp(+

ıE0t

h)(〈E0 | O† | E0〉)∗

= 〈E0 | O | E0〉, (71)

which is independent of time.A similar calculation shows that the probability amplitude for observing

the state (68) to be in an arbitrary state | S′〉 at time t is 〈S′ | S0, t〉 =exp(− ıE0t

h )〈S′ | E0〉 and so the probability is

| 〈S′ | S0, t〉 |2=| 〈S′ | E0〉 |2, (72)

which is also independent of time.In general, solutions of the Schrodinger equation of the form exp(− ıEit

h ) | Ei〉where | Ei〉 is eigenstate of H with eigenvalue Ei are known as stationary states.

Eq.(71) we gave the time dependence of the expectation value of an arbi-trary linear operator in a stationary state. We now show that the Hamiltonianoperator has the special property that its expectation value is in dependent oftime for ANY solution of the Schrodinger equation, not just stationary states.

Consider the expectation value 〈S, t | H | S, t〉 where | S, t〉 is an arbitrarysolution of eq.(5). We have

∂

∂t〈S, t | H | S, t〉 = (

∂

∂t〈S, t |)H | S, t〉+ 〈S, t | H(

∂

∂t| S, t〉). (73)

16

From the Schrodinger equation we deduce that the second term is

〈S, t | H(∂

∂t| S, t〉) = 〈S, t | H(

H

ıh| S, t〉)

=1

ıh〈S, t | H2 | S, t〉). (74)

The first term on the right-hand-side of eq.(73) can be expressed

(∂

∂t〈S, t |)H | S, t〉 = [〈S, t | H(

∂

∂t| S, t〉)]∗

= [〈S, t | H 1

ıhH | S, t〉)]∗

= − 1

ıh[〈S, t | H2 | S, t〉)]∗

= − 1

ıh〈S, t | H2 | S, t〉), (75)

where we have used the definition of an Hermitian operator and its properties.Comparing eqs.(74) and (75) we conclude that

∂

∂t〈S, t | H | S, t〉 = 0, (76)

because the two terms on the right-hand-side of eq.(73) cancel exactly. This isthe form that the conservation of energy shows up in Quantum Mechanics.

We can derive the same result by a different route that is very instructive.The eigenkets of the Hamiltonian form an orthonormal basis. Using them wecan write down a general solution of the Schrodinger equation in the form

| S, t〉 =∑i

Si(t) | Ei〉. (77)

The expansion coefficients Si(t) are functions of time. According to thegeneral principles given in Section 1.2.1 they are the probability amplitudes forfinding the system in a state | Ei〉 with energy Ei at time t. Following stepsvery similar to those we used in the verification that the stationary states arespecial solutions of the Schrodinger equation we find that the coefficients Si(t)must depend on time according to

Si(t) = exp(− ıEith

)× Si, (78)

where the constants Si have the meaning of the values of the Sit at t = 0. Weimmediately deduce that the probabilities | Si(t) |2 are independent of time andequal to their values at t = 0, namely | Si |2.

Assuming the state at t = 0 is normalised to unity so that∑i | Si |2= 1 the

state remains normalised for all time and the expectation value of the energycalculated from the time independent probability distribution | Si(t) |2=| Si |2, i = 1, 2, . . . , will be

∑iEi | Si |2 which is independent of time.

17

We have calculated two expressions for the expectation value of the energythat must be equal and hence

〈S, t | H | S, t〉 =∑i

Ei | Si |2 . (79)

This equality can be checked directly by explicitly inserting the expansion(77) for the ket | S, t〉 on the left-hand-side and using the linearity properties ofthe inner product and the linearity of H.

1.4.2 Time dependence of the expectation value of a general opera-tor.

We calculate the time dependence of the expectation value of an arbitrary op-erator, O, not necessarily the Hamiltonian. Using similar steps to those used inobtains the results (74) and (75) we find

∂

∂t〈S, t | O | S, t〉 =

1

ıh〈S, t | [O, H] | S, t〉. (80)

Thus, in general the time derivative of the expectation value of an operatorO is proportional to the expectation value of the commutator of O with theHamiltonian of the system.

1.4.3 Commutators, conservation laws and symmetries.

We have already referred to the importance of the commutator of two operatorsin Section 1.3. There it was stated that if the commutator of two operators iszero (we say they ”commute”) we can construct a set of basis states in whichboth operators have a definite value, namely, one of their eigenvalues. Thisimplies that if the operator corresponding to an observable commutes with theHamiltonian of the system then states in of definite energy exist in which theobservable also has a definite value. The result (80) means that for any state theexpectation value of the observable will be constant in time. Such observablesare called constants of the motion. We will see later how constants of the motionare also linked with symmetries of the system through their commutator withthe Hamiltonian.

2 Symmetry Transformations.

This course is primarily about geometrical symmetries of systems descrbed bynon-relativistic Quantum Mechanics.

A symmetry transformation of a system is defined to be a transformationthat, when applied to any dynamically possible state of the system, leads toanother dynamically possible state. For example, in the classical mechanics ofpoint particles, the interchange of the position of two identical particles.

18

In quantum Mechanics, dynamically possible states are ket vectors that sat-isfy the Schrodinger equation ( 5). We repeat it here for convenience.

ıh∂

∂t| S, t〉 = H(r, p) | S, t〉, (81)

We consider a linear operator U that commutes with the Hamiltonian H,i.e.,

[U , H] = 0, (82)

or equivalently

UH = HU . (83)

Operating on both sides of the equality (81) with U and using its assumedlinearity and the property (83) we find

ıh∂

∂t(U | S, t〉) = H(r, p) (U | S, t〉). (84)

This result implies that if | S, t〉 is a dynamically possible state then so is thetransformed state | S′, t〉 = U | S, t〉 because it also satisfies the Schrodingerequation2.

We will require that | S, t〉 and | S′, t〉 have the same normalisation so that

〈S, t | S, t〉 = 〈S′, t | S′, t〉. (85)

This is statisfied if U is a unitary operator and therefore has the property

U−1 = U†. (86)

We see that in Quantum Mechanics symmetry transformations are representedby a unitary transformation3.

2.1 The transformation operator for translations.

We first consider a system of a single spinless particle confined to the x-axis.In a certain state | S〉 (we ignore the time label temporarily) the particle has awavefunction in the x-basis, ψS(x). We translate the system by the distance aalong the x-axis. The new state is | Sa〉 and we ask:

”What is the new wavefunction ψSa(x) that describes | Sa〉 in the x-basis?”

2Note that if U was anti-linear the left-hand side of (84) would be replced by

−ıh ∂∂t

(U | S, t〉) = ıh ∂∂(−t)

(U | S, t〉), which hints at why the time reversal transformation

operator is anti-linear.3In the case of the time reversal transformation the corresponding U is anti-unitary, which

means it satisfies (86) but is anti-linear. For anti-linear operators the meaning of Hermitianhas to be modified slightly. Instead of (38), anti-linear Hermitian operators satisfy

〈a | (O | b〉) = 〈b | (O | a〉).

19

A moments thought shows that the new wave function must satisfy

ψSa(x) = ψS(x− a), (87)

i.e., the new wavefunction at x must have the same value as the old wavefunctionat x− a. For example, if the most probable value of x is x0 in the state S, thenthe most probable value of x in the state Sa will be (x0 + a).

The result 87 is what we need to derive the form of the unitary operatorU(a) that gives the relation beween Sa and S in the formula

| Sa〉 = U(a) | S〉. (88)

We identify U(a) by recognising that the right-hand-side of (87) can be expandedusing Taylor’s theorem

f(x+ a) = f(x) + adf(x)

dx+a2

2!

d2f(x)

dx2+a3

3!

d3f(x)

dx3. . .+

an

n!

dnf(x)

dxn+ . . . . (89)

Applying this general result to eq.(87) gives

ψSa(x) = ψS(x) + (−a)dψS(x)

dx. . .+

an

n!

dnψS(x))

dxn+ . . .

= ψS(x) + (−a)ı

hpxψS(x)) +

(−a)2

2!(ı

h)2(px)2ψS(x) . . .

+(−a)n

n!(ı

h)n(px)nψS(x) + . . . , (90)

where in the last line we have replace the derivative operators ddx ,

d2

dx2 by powersof the momentum operator px using the relation

px =ı

hpx. (91)

Applying Taylor’s Theorem to the function exp(z) gives

exp(z) = 1 + a+a2

2!+a3

3!. . .+

an

n!+ . . . . (92)

This result can be used to rewrite the last line in eq.(90) as

ψSa(x) = exp(− ı apxh

)ψS(x) (93)

Comparing this formula, which is valid for all values of x, with eq.(88) we deducethat the kets describing S and Sa are related by this formula with

U(a) = exp(− ı apxh

). (94)

The nature of physical space is that translations in the x, y and z directionscommute so it is a simple step to write down the generalisation to the operatorthat produces a translation by a vector displacement a:

U(a) = exp(− ıa.ph

). (95)

20

From this explicit form it is straight forward to show that for an Hermitianmomentum operator U(a) is unitary.

The exponential structure of this expression is typical for operators thatcorrespond to symmetry operations. The role played by the momentum operatorshould be noted. We say that the momentum operator ”generates translations”.It appears in the formula multiplied by the parameter a which determines theamount of translation.

We showed earlier that if a transformation is a symmetry of a system thetransformation operator must commute with the Hamiltonian. In the case oftranslations this means that U(a) of eq(95) must commute with H for all a.This can only be true if the generator p itself commutes with H, [p, H] = 0. Wesee that this in turn implies that momentum is a conserved quantity.

To see what the vanishing of the commutator of an observable with theHamiltonian implies in a non-trivial case, we consider the Hamiltonian for par-ticles labelled 1 and 2 interacting through a force generated by a potential energyfunction V (r1 − r2). The Hamiltonian operator is

H12 =p2

1

2m1+

p22

2m2+ V (r1 − r2). (96)

The momentum operators clearly commute with the kinetic energy terms butthey do not individually commute with the potential energy term V and hencethey do not commute with H12. For example

[p1, V ] =h

ı∇r1V. (97)

Hence as long as V is not constant everywhere the momentum of the individualparticles are not constants of the motion.

However, the sum p1 + p2 does commute with H if V is a function only ofthe diplacement r1 − r2 as indicated in (96).

Because of the latter assumption

∇r1V (r1 − r2) = −∇r2V (r1 − r2), (98)

and hence the the terms involving V in [(p1 + p2, H] cancel and the total mo-mentum of the two particles is conserved.

Of course, if V is a constant everywhere the individual particle momentumoperators do commute with H and the the indivudual momenta are conserved,.This reflects the complete absence of forces in the system in this case and cor-responds classically to Newton’s First Law

Note that our assumption about V means that it unchanged by the transfor-mation r1 → r1 +a, r2 → r2 +a which is precisely the meaning of translationalinvariance in this case.

In the classical case the condition (98) is just the condition for the validityof Newton’s Third Law, ”Action and reaction are equal and opposite”.

In summary, we have seen that there is a strong connection between transla-tional symmetry and the conservation of momentum. This is our first example

21

of the connection between a symmetry and a conservation law for the operatorthat generates the corresponding transformation.

3 Transformation operator for rotations.

We turn next to the operator corresponding to rotations. We will find that thegenerator in this case is an angular momentum operator and the role playedby a is played by an angle and an axis of rotation. But before going furtherwe have to anticipate that rotations in 3 dimensions are more complicated thantranslations. This is associated with the property of physical space that leadsto the non-interchangeability of rotations about different axes in general. Inquantum mechanics this shows up as the non-commutation of the componentsof the angular momentum operator that generate those rotations.

In Appendix B we set out the details of the notations and conventions weuse for describing rotations about an axis in 3 dimensions.

When we rotate a vector about an axis its components along a fixed setof 3 orthogonal basis vectors change. This is the ”active” way of looking atrotations. In the ”passive” point of view we ask how the components of a fixedvector change when we rotate the orthogonal basis vectors instead. Both waysof considering a rotation are used depending on the context. In both cases weare looking at the relation between 2 sets of 3 numbers. It is shown in AppendixB how in both cases these relationships are most simply expressed in terms of3× 3 real orthogonal matrices.

It is fundamental to the derivation of these results to appreciate why wemean by a rotation in the 3-dimensional space we use in the non-relativisticdescription of physical phenomena. When a pair of vectors are both subjectedto the same rotation by the same angle about the same axis their lengths areunchanged and the angle between the 2 vectors is also unchanged.

We consider a basis {a1,a2,a3}, of vectors of unit length forming an orthog-onal coordinate system. Each of these bias vectors is subjected to a rotationR(α,n), through an angle α about some axis defined by the unit vector n,resulting in 3 new vectors {A1,A2,A3}, i.e,

aiR(α,n)→ Ai, i = 1, 2, 3. (99)

It is a consequence of the concept of rotation described in the previousparagraph that the set {A1,A2,A3} also forms an orthogonal basis of unitvectors.

It is shown in Appendix B that the same matrix Ri j(α,n) relates the originaland transformed vector components in both the ”active” and ”passive” pointsof view. This matrix can be expressed entirely in terms of the 9 inner productsof the basis vectors {a1,a2,a3} and {A1,A2,A3} through the formula

Ri j(α,n) = ai.Aj . (100)

In the following we will frequently omit the argument (α,n) in Ri j(α,n) if thiscan be done without ambiguity.

22

In detail:(i) Active viewpoint:Vector W is obtained from vector V by a rotation R(α,n). The components

of the two vectors in the ai, i = 1, 2, 3, basis are, respectively, Wi, i = 1, 2, 3,and Vi, i = 1, 2, 3, . They are related by

Wi =∑j

Ri jVj . (101)

(ii) Passive viewpoint:Vector W is obtained from vector V by a rotation R(α,n). The components

of the vector V in the ai, i = 1, 2, 3, basis are Vi, i = 1, 2, 3,. The components

of the same vector V in the Ai, i = 1, 2, 3, basis, where aiR(α,n)→ Ai, are

V ′i , i = 1, 2, 3,. The 2 sets of numbers Vi, i = 1, 2, 3, and V ′i , i = 1, 2, 3, arerelated by

V ′i =∑j

Ri jVj , (102)

where R is the transpose of matrix R with elements

Ri j = Rj i. (103)

It is also shown in Appendix B that R = R−1, so that R describes therotation inverse to R(α,n). In other words the matrix involved in the pas-sive viewpoint is the matrix corresponding to the inverse rotation in the activeviewpoint.

3.1 Rotating a state.

A ket vector | S〉 describes a state S of a physical system (we omit the timevariable temporarily) that exists in 3-dimensional physical space. If we rotatethe state in this space by a rotation R(α,n) we obtain a new state S′ describedby a new ket | S′〉. The two states will be related by an operator R(α,n) suchthat

| S′〉 = R(α,n) | S〉. (104)

We approach the problem of determining the operator R(α,n) by first consid-ering the case of a spineless single particle and the description of S and S′ interms of the wave functions ψS(x, y, z) and ψS′(x, y, z) where (x, y, z) are thecoordinates of a general position vector r along the basis vectors ai, i = 1, 2, 3,i.e.,

r = xa1 + ya2 + za3. (105)

In terms of the kets | S〉 and | S′〉 are given by

ψS(x, y, z) = 〈x, y, z; a | S〉ψS′(x, y, z) = 〈x, y, z; a | S′〉. (106)

23

Note that in the 3-dimensional case we have to label the position eigenkets| x, y, z; a〉 with an extra label ”a” to indicate that the coordinates x, y, z, referto the axes ai, i = 1, 2, 3.

We can imagine the complex number that the function ψS(x, y, z) assigns tothe point (x, y, z) as written on a flag fixed to that point. What we mean byrotating the state S by R(α,n) is to move this flag, with the number written onit, to the point (Rx,Ry,Rz) reached from (x, y, z) by the rotation R(α,n) 4 .The number now written on the flag at the point (Rx,Ry,Rz) is ψS(x, y, z). Butthis is the number that is to be assigned to (Rx,Ry,Rz) by the wavefunctionψS′ that by definition describes the rotated state S′. Clearly, we must have

ψS′(Rx,Ry,Rz)) = ψS(x, y, z), (107)

for all points (x, y, z). Equivalently, we can write this as

ψS′(x, y, z) = ψS(R−1x,R−1y,R−1z)), (108)

where (R−1x,R−1y,R−1z)) is the point obtained from the point (x, y, z) by theinverse rotation R−1 = R(−α,a3).

Eq.(107) is the equivalent for a rotation to the relation given in eq.(87) for atranslation. We proceed in an analogous way to find the operator that generatesa rotation.

We start by considering a rotation about the z-axis (a3) by an angle α5.The matrix Ri j(α,a3) that appears in eq.(107) is given by cosα − sinα 0

sinα cosα 00 0 1

, (109)

The inverse rotation R−1 is obtained by replacing α by −α or, equivalently,taking the transpose of matrix Ri j(αn), and we find

R−1x = cosα x+ sinα y,

R−1y = cosα y − sinα x,

R−1z = z. (110)

Using these results the relation (108) can be written

ψS′(x, y, z) = ψS(cosα x+ sinα y, cosα y − sinα x, z)). (111)

Following the method we used for translations we want to expand the right-hand-side of eq.(111) in powers of α using Taylor’s theorem.

4We use an abbreviated notation in which , e.g., Rx means the x component of the vectorobtained by rotating r = xa1 + ya2 + za3

5The angle α is positive if, looking along the positive axis of rotation away from the planeof rotation, the sense of rotation is clockwise.

24

To obtain the term of first order in α we use

cosα = 1− α2

2+ . . . ,

sinα = α− α3

6+ . . . , (112)

and neglect all terms of order α2 or higher. We obtain

ψS′(x, y, z) = ψS(x+ αy, y − αx, z)

= ψS(x, y, z) + αy∂ψS(x, y, z)

∂x− αx∂ψS(x, y, z)

∂y+ . . .

= ψS(x, y, z)− ıαxpyh

ψS(x, y, z) +ıαypxh

ψS(x, y, z) + . . .

= ψS(x, y, z)− ıα

h(xpy − ypx)ψS(x, y, z) + . . . (113)

where the neglected terms are all of order α2 or higher. In the last two linesof eq.(113) we have used the standard expressions for the components of themomentum operator in terms of partial derivatives:

px =h

ı

∂

∂x, py =

h

ı

∂

∂y, pz =

h

ı

∂

∂z. (114)

We recognise that the combination (xpy − ypx) is the z-component of thevector r∧p, which is the classical expression for the orbital angular momentumof the particle, ` = r ∧ p. The expression (113) can therefore be written

ψS′(x, y, z) = (1− ıα

hˆz + . . .)ψS(x, y, z) + . . . , (115)

which can be compared with eq.(90) in the translation case.For an infinitesimal rotation angle δα, when we can ignore quadratic and

higher powers, we deduce from (115) that the rotation operator for an infinites-imal rotation about the z -axis is

R(δα,~a3) = 1− δαˆz. (116)

We can find operator for a finite rotation by first noting that roatations aboutan axis satisfy

R(α,~a3)R(β,~a3) = R(α+ β,~a3), (117)

for arbitrary finite angles α and β. In particular

R(α,~a3)R(δα,~a3) = R(α+ δα,~a3), (118)

and hence, using eq.(116),

R(α+ δα,~a3)− R(α,~a3) = −ıδαˆzR(δα,~a3). (119)

25

In the limit δα→ 0 this tells us

dR(α,~a3)

dα= −ıˆzR(δα,~a3). (120)

The unique solution to this differential equation satisfying R(α = 0,~a3) = 1 isthe operator

R(α,~a3) = exp(−ıαˆz). (121)

We can now immediately write down the rotation operator for a generalrotation by an angle α about an arbitrary axis n:

R(α,~n) = exp(−ıαˆ.n). (122)

The operator ˆ.n means (ˆxnx + ˆ

yny + ˆznz) where nx, ny, nz are the compo-

nents of n. However, this is not a practical way forward because the angularmomentum operators don’t commute and we have to follow a different path.

3.2 Angular momentum conservation.

We saw in Section 1.3 that if if the operator that generates a transformationcommutes with the Hamiltonian of the system, H, then the transformationcorresponds to a symmetry transformation. Hence if the Hamiltonian commuteswith the operator R(α,~n) the system is invariant under a rotation about theaxis n. If this is true for all α then eq.(122) tells that this can only be true if theHamiltonian commutes with the component of the angular momentum operatoralong n, and there must exist a basis in which the system has a definite energy(eigenvalue of H) and the angular momentum in the direction n ( an eigenvalueof ˆ.n). To proceed further we must learn about the eigenvalue spectrum of ˆ.

4 Eigenvalue spectrum of the angular momen-tum operators.

4.1 Eigenfunctions of ˆz.

Using the techniques of Appendix C we can express the orbital angular momen-tum operators for a single particle in terms of differential operators with respectto the polar and azimuthal angles θ and φ. The operator for ˆ

z is particularlysimple (ˆ

z = hı∂∂φ ) and leads to a straightforward differential equation for the

corresponding eigenvalue equation.The condition that m`h is an eigenvalue and that ψm`

(θ, φ) is an eigenfunc-

tion of ˆz is

h

ı

∂ψm`

∂φ= m`hψm`

. (123)

26

The solution to this equation is

ψm`(θ, φ) = exp(ım`φ)f(θ), (124)

where f(θ) is a function of θ only.The condition that ψm`

(θ, φ) does not jump in value at φ = 2π, which is thesame point in space as φ = 0, forces m` to be a positive or negative integer orzero.

The wave function (124) predicts that in a state with definite ˆz all values

of φ are equally probable. This is an extreme case of the Uncertainty Relation∆`z ∆φ ≥ h

2 .

4.2 Angular momentum commutators.

It is shown in Appendix C.2 that ˆx, ˆ

y, ˆz do not commute with each other so

that we can not create a basis in which any 2 of them have a definite value. Ofcourse, we can separately find eigenstates of ˆ

x and ˆy. The basic equivalence

of the x y and z directions leads us to expect that the eigen values of ˆx and ˆ

y

will also have the form mh with m a positive or negative integer or zero.There does exist another operator that commutes with all three components

of ˆand that is ˆ2= ˆ2

x+ ˆ2y + ˆ2

z,, i.e., the operator corresponding to the squareof the magnitude of the orbital angular momentum. For example

[ˆ2x,ˆz] = ˆ2

xˆz − ˆ

zˆ2x

= ˆx([ˆx, ˆ

z] + ˆzˆx)− ˆ

zˆ2x

= ˆx[ˆx, ˆ

z] + ˆx

ˆzˆx − ˆ

zˆ2x

= ˆx[ˆx, ˆ

z] + [ˆx, ˆz]ˆx

= ˆx(−ıˆy) + (−ıˆy)ˆ

x

= −ı(ˆx

ˆy + ˆ

yˆx). (125)

Similarly we find

[ˆ2y,ˆz] = +ı(ˆ

xˆy + ˆ

yˆx). (126)

Adding eqs.(125) and (126) and using [ˆ2z,ˆz] = 0 we obtain

[ˆ2, ˆz] = 0. (127)

Similarly

[ˆ2, ˆx] = 0, (128)

and

[ˆ2, ˆy] = 0. (129)

27

In summary, the commutation relations between ˆ2, ˆ

x, ˆy and ˆ

z imply

that we can find a basis of states | α, β〉 that are labelled by eigenvalues of ˆ2

(denoted by α) and one of the components ˆx, ˆ

y or ˆz. We can select any one of

the components. We shall choose ˆz, (eigenvalues denoted β),but this choice is

entirely conventional. It has the convenience that with the usual definitions ofthe angles θ and φ eigenfunctions of the orbital angular momentum of a singleparticle are simple functions of φ (see eq.(124)). In general these eigenkets willsatisfy

ˆ2| α, β〉 = α | α, β〉

ˆz | α, β〉 = β | α, β〉. (130)

Eigenkets corresponding to distinct eigenvalues will be orthogonal and we willassume all states are normalised to unity. Assuming a descrete eigenvalue spec-trum for definiteness we have

〈α′, β′ | α, β〉 = δα′,αδβ′,β . (131)

In Appendix C.1 it is shown how the operator corresponding to ˆ2for a

single particle can be expressed in terms of partial derivatives with respect to theangles θ, φ. The first of the eigenket eqs. (130), when expressed in the positionbasis with spherical polar coordinates becomes a complicated 2nd order partialdifferential equation. This approach to finding the eigenvalue spectrum of theangular momentum operators is described in many text books. We will follow adifferent trajectory here and use instead a very beautiful and powerful methoddue to Dirac. The power of this method can be appreciated by imagining thestandard approach to the problem of finding the eigenvalue spectrum for theorbital angular momentum of a many particle body system whose componentswould be expressed as linear combination of operators, one for each particle,involving the individual particle coordinates separately, e.g.,

ˆ= r1 ∧ p1 + r2 ∧ p2 + . . . . (132)

In Dirac’s method all we need to assume about the operators ˆ2, ˆx, ˆ

y andˆz is that they are Hermitian and obey the commutation relations ˆ∧ ˆ= ıhˆ.

We will not need expressions for these operators as differential operators or solveany complicated differential equations. It is straightforward to show that as theindividual terms in the sum in eq.(132) are Hermitian and satisfy commutationrelations so does the sum ˆ.

4.3 Dirac’s method

The key step in Dirac’s development is the use of the raising and loweringoperators ˆ± = ˆ

x ± ıˆy and the commutation relations eqs.(212) and (213) ofAppendix C.2.

28

In the following we will set h = 1 in all formulae. This amounts to measuringall angular momentum eigenvalues in units of h.

At this point all we know about the eigenvalues α and β is that they mustbe real numbers. In addition, we can assume that α is positive because it is an

eigenvalue of the square of an Hermitian operator, ˆ2.

Consider the state defined by

| α, β〉+ = ˆ+ | α, β〉, (133)

where | α, β〉 is the simultaneous eigenket of ˆ2and ˆ

z introduced in eq.(130).The existence of the orthogonal set | α, β〉 and the reality of the eigenvalues αand β is guaranteed by the assumed Hermiticity of ˆ.

First of all we show that | α, β〉+ is an eigenstate of ˆ2with eigenvalue α

because ˆ2commutes with ˆ

+ and so

ˆ2| α, β〉+ = ˆ2 ˆ

+ | α, β〉

= ˆ+

ˆ2| α, β〉

= ˆ+α | α, β〉

= α | α, β〉+. (134)

A second property of the state | α, β〉+ follows from the commutation rela-

tions (212) and (213) between ˆ)z and ˆ± that follow from the basic relationˆ∧ ˆ= ıˆ.

ˆz | α, β〉+ = ˆ

zˆ+ | α, β〉

= ([ˆz, ˆ+] + ˆ

+ˆz) | α, β〉

= (+ˆ+ + ˆ

+β) | α, β〉= (1 + β)ˆ

+ | α, β〉= (1 + β) | α, β〉+. (135)

We see that starting with an eigenket of ˆz with eigenvalue β we have created

a new eigenket, | α, β〉+, with eigenvalues β + 1 of ˆz and α of ˆ2

.

Acting with ˆ+ again on | α, β〉+ we can create a new eigenket with eigenvalue

β + 2, and so on. Proceeding in this way we can create a ladder of eigenketswith ˆ

z eigenvalues differing by unity as we go up the ladder, all states in the

ladder having the same eigenvalue, α, of ˆ2.

In a similar way we can show that by operating with ˆ− we can create astate | α, β〉− = ˆ− | α, β〉 with eigenvalue β − 1 and go on to create a laddergoing down by unity each time.

The natural question is whether this ladder goes on indefinitely upwards anddownwards, or does the ladder have a top and a bottom. In fact the ladder hasa top. We show this by evaluating the norm of the ket | α, β〉+.

29

Using the result (ˆ+)† = ˆ− we find

+〈α, β | α, β〉+ = 〈α, β | ˆ− ˆ+ | α, β〉

= 〈α, β | ˆ2x + ˆ2

y + ı[ˆx, ˆy] | α, β〉

= 〈α, β | ˆ2− ˆ2

z − ˆz | α, β〉. (136)

The operator appearing in the last line can be replaced by a number by using

the fact that | α, β〉 is an eigenket of ˆ2and ˆ

z. We obtain

+〈α, β | α, β〉+ = α− β2 − β. (137)

The innerproduct on the left-hand-side must be positive or zero, but, for afixed α, if β becomes large enough the number on the right can clearly becomenegative. Hence the ladder must have a top and there must be a β that makesthe right-hand-side vanish. This happens when

α− β2 − β = 0. (138)

Solving this quadratic equation for βmax gives the positive number

βmax =

√4α+ 1− 1

2. (139)

The other root of the quadratic equation (138) is found to be necessarily lessthan βmax given by eq.(139).

In an analogous fashion we find that the ladder must have a bottom andthat the minimum value of β is just −βmax.

We now imagine starting at the bottom of the ladder and going upwards byunit steps in β using the operator ˆ

+ . Clearly we must eventually get to βmax.But if we get to βmax by unit steps from −βmax this is only possible if 2βmax

is an integer.We use the notation βmax = j, where j is always either a positive integer

or half a positive odd integer, i.e., j = 0, 1/2, 1, 3/2, 2, . . .. From eq.(139) wecalculate

α = j(j + 1). (140)

We have proved that if a set of three Hermitian operators satisfy the commu-tation relations j ∧ j = ıj then there exists a basis of simultaneous eigenstates

of j2

and jz. We will label them | j,m〉 were j label tells us that the eigen-

value of j2

is j(j + 1) and m is the eigenvalue of jz. The value of m musthave a value between −j and +j and connected to j by integer steps, i.e.,m = −j,−j + 1,−j + 2, . . . j − 1,+j.

j2| j,m〉 = j(j + 1) | j,m〉,

jz | j,m〉 = m | j,m〉, m = −j,−j + 1,−j + 2, . . . j − 1,+j. (141)

30

In the following we will use this notation when we are discussing the generalproperties of a set of three Hermitian operators jx, jy, jz, satisfying the com-

mutation relations j ∧ j = ıj. The notation ˆ wil be used exclusively for thecase of single particle orbital angular momentum.

In quantum mechanics, when we say that a system has angular momentumjh, we mean that a measurement of the magnitude of its angular momentumwill give h

√j(j + 1. A measurement of a component of the angular momentum

along any direction will yield a maximum of +jh and a minimum of −jh, orany value in between and connected by an integer number of h steps.

Some examples (in units of h):

j = 0 : Magnitude :0. m = 0.

j =1

2: Magnitude :

√3

4. m = −1

2,+

1

2.

j = 1 : Magnitude :√

2. m = −1, 0,+1.

j =3

2: Magnitude :

√15

4. m = −3

2,−1

2,+

1

2,+

3

2.

j = 2 : Magnitude :√

6. m = −2,−1, 0,+1,+2.

j = 10 : Magnitude :10.48809 . . . m = −10,−9, . . .+ 9,+10. (142)

Note that when j is large compared to unity the quantum mechanical resultsapproach the classical expectation that the magnitude of the angular momentumcoincides with its maximum projection on an axis.

4.4 Properties of the angular momentum eigenkets.

In our general notation, the property derived in eq.(135) can be written

jz | j,m〉+ = (1 +m) | j,m〉+, (143)

where the state | j,m〉+ is defined by (see eq.(133))

| j,m〉+ = j+ | j,m〉. (144)

Eq.(143) implies that | j,m〉+ is an eigenket of jz with eigenvalue m+1. It musttherefore be proportional to | j,m+ 1〉.

| j,m〉+ = C+jm | j,m+ 1〉, (145)

We can calculate the constant C+jm from the norm of | j,m〉+ we worked out in

eq.(137). In the present notation this reads

+〈j,m | j,m〉+ = j(j + 1)−m2 −m. (146)

Comparing with eq,(145) we see that

| C+jm |

2= j(j + 1)−m2 −m, (147)

31

where we have used the normalisation condition 〈j,m+ 1 | j,m+ 1〉 = 1.Using similar steps we find that

| j,m〉− = j− | j,m〉 = C−jm | j,m− 1〉, (148)

where

| C−jm |2= j(j + 1)−m2 +m. (149)

We recall that these results for C±jm |2 were derived using only the commu-

tation relations j ∧ j = ıj and the assumed normalisation of the states | j,m〉.Therefore they will reflect properties of the kets describing any system, nucleus,atom molecule or elementary particle in states that can be characterised by adefinite values of angular momentum and projection on an axis. In practicemany-body states of definite j will have other labels that may be the eigenval-

ues of other operators that commute with j2

and jz. We have omitted theselabels in te development so far.

The formulae (147) and (149) for the squared moduli fixes the constantsC±jm to within an overall constant of magnitude unity that may depend on jand m. This is consistent with the fact that the eigenvalue equations (141)say nothing about the normalisation of the states | j,m〉. We have assumed anormalisation condition 〈j,m + 1 | j,m + 1〉 = 1 but this also only determinesthe states within overall phase. The constants C±jm connect states with differentvalues of m we can therefore insist that the kets | j,m〉 be chosen in such a waythat the C±jm are the positive square roots of the right-hand-sides of eqs.(149).This is known as the Condon and Shortley phase convention and is the choicemade universally. With this convention we have from eqs.(145), (147), (148)and (149)

jx ± ıjy | j,m〉 =√j(j + 1)−m(m± 1) | j,m+ 1〉,

j2| j,m〉 = j(j + 1) | j,m〉,

jz | j,m〉 = m | j,m〉, m = −j,−j + 1,−j + 2, . . . j − 1,+j,

〈j′,m′ | j,m〉 = δj′,jδm′,m. (150)

In this equation we have brought together the properties of what we shall referto as a standard set of angular momentum eigenstates. Subsequent discussionsof the properties of angular momentum states will assume that they conform toeq.(150) unless otherwise stated.

5 Orbital angular momentum eigenfunctions inposition space. Spherical Harmonics.

In this Section we show how to construct explicit formulae for the eigenstatesin the position basis of the one-particle orbital angular momentum eigenstates

32

we discussed in Appendix C.1. We obtain a simple differential equation forthe state | `,m` = `〉 and show others cam be worked out using properties of

lowering operator ˆ−.According to eq.(150) the state | `,m` = `〉 satisfies

ˆ+ | `,m` = `〉 = 0. (151)

Using spherical polar coordinates in the position basis and using the explicitform for ˆ

+ given in Appendix C.1, eq.(198), this equation reads

ıh exp(ıφ)(−ı ∂∂θ

+ cot θ∂

∂φ)〈θ, φ | `,m` = `〉 = 0, (152)

where we have suppressed the radial coordinate, r in the basis ket | r, θ, φ〉.We showed in Section 4.1, eq.(124) that the function 〈θ, φ | `,m` = `〉 must

have the form

〈θ, φ | `,m` = `〉 = exp(ı`φ)f(θ), (153)

so that the derivative with respect to φ in eq.(152)can be replaced by ı`.Eq.(152) therefore becomes

h exp(ı(`+ 1)φ)(∂

∂θ− ` cot θ)〈θ, φ | `,m` = `〉 = 0, (154)

or, cancelling non-zero factors we obtain

(d

dθ− ` cot θ)f(θ) = 0. (155)

This equation has the general solution

f(θ) = C(sin θ)`, (156)

where C is independent of θ and φ.Our result for the state | `,m` = `〉 is therefore

〈θ, φ | `,m` = `〉 = C`(sin θ)` exp(ı`φ). (157)

We choose the constant C` so that this function is normalised according to∫ 2π

0

dφ

∫ π

0

sin θ dθ | 〈θ, φ | `,m` = `〉 |2= 1. (158)

With this choice a ket of unit norm is created when combined with a radialfunction R(r) normalised so that∫ ∞

0

r2dr | R(r) |2= 1. (159)

We have used the standard form for the infinitesimal volume in position spacein terms of spherical polar ordinates, r2dr dφ sin θ dθ.

The condition (158) gives

| C` |2=2`+ 1

4π

(2`)!

22`(`!)2. (160)

33

5.1 Physical interpretation of the state | `,m` = `〉.The squared modulus of the function (157) gives the distribution in angle ofthe position probability density of finding a particle in a state of orbital angu-lar momentum of magnitude

√`(`+ 1)h and maximum component `h in the

positive z direction. A very similar calculation for the state | `,m` = −`〉 us-

ing ˆ− | `,m` = −`〉 = 0 gives the same θ dependence as (157) but with φdependence exp(−ı`φ).

In both cases, and in common with all eigenstates of ˆz, the position prob-

ability density is independent of φ. The θ dependence is also the same in bothcases and has a maximum at θ = π

2 , i.e. for all points on the x− y plane. As `increases this maximum peaks more and more sharply because for angles otherthat θ = π

2 , | (sin θ) |2` becomes vanishingly small as ` → ∞, whereas exactlyat θ = π

2 , | (sin θ) |2`= 1 for all `.This limiting case of motion confined to the x−y plane is just what we would

expect for a classical particle with its angular momentum r ∧ p pointing in thepositive or negative z-directions. This is a good example of how underlying thesometimes strange formulations of the quantum world there is a classical worldstruggling to get out.

5.2 States with | m` |< `.

Other states in the ladder can be calculated by repeated action with ˆ− on| `,m` = `〉 and using the first of equations (150) to get the normalisationcorrectly. Alternatively one can go up the ladder starting from | `,m` = −`〉and using ˆ

+.We note that, in a similar way to the derivation of eq.(154), when acting on

the function 〈θ, φ | `,m`〉 the operator ˆ− has the form

ˆ− → −h exp(ı(m` − 1)φ)(∂

∂θ+ ` cot θ). (161)

The factor exp(ı(m`−1)φ) becomes the correct φ dependence for the next statedown the ladder. Therefore, starting at the top of the ladder with eq.(157) for| `,m` = `〉 and choosing the constant C` to be a real number satisfying (160),all the functions 〈θ, φ | `,m`〉 will have the form of a real function of θ multipliedby exp(ım`φ).

We have to make an arbitrary choice of phase, which could depend on `, indefining C` from eq.(160). The choice

C` = (−1)`√

2`+ 1)

4π

√(2`)!22`(`!)2, (162)

leads to what are known as the standard spherical harmonics, Y`,m`(θ, φ). A

general expression for the dependence on θ can be given in terms of the realfunctions of cos θ known as Associated Legendre Functions, Pm`

` (cos θ). We usethe definitions given in Messiah [1], Appendix B IV.

34

For m` ≥ 0 the spherical harmonics are given by

Y`,m`(θ, φ) = (−1)m`

√(2`+ 1)

4π

(`−m`)!

(`+m`)!Pm`

` (cos θ) exp(ım`φ). (163)

For m` < 0 we can use a relation that arises from the connection between,e.g., the effect of ˆ

+ acting on | `,m` = −` and the effect of ˆ− acting on| `,m` = ` discussed above. The θ parts of these operators are proportional.We saw earlier that | `,m` = −` and | `,m` = +` have the same dependence onθ, namely, (sin θ)`, so the θ dependence of | `,m` = −`+ 1〉 and | `,m` = `− 1〉will be real and also have proportional θ dependences. This relationship will beretained all the way up and down the ladder. The φ dependent parts of stateswith m` values of opposite sign are always complex conjugates of each other,i.e.,exp(−ım`φ) = (exp(+ım`φ))∗.

When this idea is followed through in detail we find

Y ∗`,m`= (−1)m`Y`,−m`

, (164)

and using this for m` < 0, or equivalently when m` = − | m` |, we find

Y`,−|m`|(θ, φ) =

√(2`+ 1)

4π

(`− | m` |)!(`+ | m` |)!

P|m`|` (cos θ) exp(−ı | m` | φ). (165)

The absence of the phase factor (−1)m` on the right-hand-side of eq.(165) shouldbe noted.

The physical meaning of the relation (164) and the involvement of the com-plex conjugate can be traced to the fact that state with opposite signs for m`

are related by the anti-linear time reversal transformation.The expressions (163) and (165) are widely used in practical applications as

the angular part of the position representative, 〈θ, φ | `,m`〉, of a one particlestate with orbital angular momentum quantum numbers `,m`. If (ˆ)2 andˆz commute with the Hamiltonian of the system, energy eigenfunctions have

the form, R`, E(r)Y`,m`(θ, φ), where the radial function is determined by the

values of `, the eigenenergy, E, and the detailed of the potential energy functionassociated with the environment of the particle.

5.3 Transformation of a general quantum state under spa-tial rotations.

We have discussed how the state of a spineless particle is transformed under arotation by an angle α about an arbitrary axis n and shown how the relevantrotation operator is connected with the orbital angular momentum operatorthrough the formula (122), which we reproduce here for convenience.

R(α,~n) = exp(−ıαn.ˆ). (166)

35

We generalise this formula to an arbitrary many particle state by definingwhat is meant by the angular momentum operator of such a state as the operatorj, satisfying the standard conditions (150), that generates a transformationunder a rotation from a state | S〉 to a state | S′〉 through the formula

| S′〉 = exp(−ıαn.j) | S〉. (167)

For example, a general state of two spineless particles labelled 1 and 2 will bedescribed by wave functions in the position basis that are functions of the posi-tions r1 and r2 The operators corresponding to the individual orbital angularmomentum operators of the particles about some origin will be ˆ

1 = r1 ∧ p1

and ˆ2 = r2 ∧ p2.

For this system the operator j will be

j = ˆ1 + ˆ

2, (168)

and the operator R(α,~n) will act in the space of the variables r1 and r2. Therotation generated by R(α,~n) induces the same rotation to both these variables.Conservation of j would be associated with the vanishing of the commutatorsof the Hamiltonian of the system with the componentss of j and imply that theHamiltonian was invariant under the simultaneous rotation of all the dynamicalvariables of particles 1 and 2.

The importance of the way of thinking presented here and in the followingSections is that many properties of states of definite angular momentum j ofany system can be understood without detailed knowledge of explict expressionsfor the operator j or the corresponding eigenkets in terms of the dynamicalvariables of the system. This is obviously a great advantage when dealing withmany-body systems.

5.4 Intrinsic spin-12particles.

We found that the commutation relations j ∧ j = ıhj are consistent with aneigenvalue spectrum in which j was an integer or a half-integer. We found thatorbital angular momementum eigenvalues had to be integer to get an physicallyacceptable wave functions. How is the half-integer possibility realised in physics?

In the 1920’s it was discovered as a result of the study of the behaviour ofatoms in a magnetic field that the correct physical picture is that position andmomentum do not exhaust the list of dynamical variables needed to describean electron. It was found necessary to introduce an extra degree of freedomdescribed by an operator sz that had only two eigenvalues + 1

2 and − 12 . The

difference between these two eigenstates can be observed by placing the electronin a magnetic field in the z direction when it would be found that the two stateshave equal and opposite values of an intrinsic magnetic moment and hence equaland opposite energies proportional to the strength of the field.

This extra degree of freedom is referred to as intrinsic spin. The experimentalobservations mean that a basis in the ket space of an electron must be taken to

36

be | r, sz〉 and the wave function describing a ket | S, t〉 in this basis is

ψS(r, sz, t) =)〈r, sz | S, t〉. (169)

The physical interpretation of the number ψS(r, sz, t) for some value of r, and tand, e.g., sz = 1

2 , is that it is the probability amplitude for finding the electronat position r at time t with a magnetic moment −µe, where µe is intrinsicmagnetic moment of the electron (very close to a Bohr magneton, eh

2mec). Such

a staet is often referred to as ”spin-up”, with sz = − 12 being ”spin-down”.

References

[1] A. Messiah, Mecanique Quantique, Dunod, Paris 1960.

A Complex numbers.

The expression | z |2 denotes the squared modulus of the complex number z.If we write z in terms of its real and imaginary parts x and y we have

z = x+ ıy where ı =√−1.

The square modulus of z, | z |2, is defined as | z |2= x2 + y2, which isobviously positive or zero.

An alternative definition, which may not need explicit evaluation of x andy is to calculate z × z∗, where the complex conjugate z∗ is obtained from z byreplacing ı by −ı everywhere. Using the rules for multiplying complex numberswe have z × z∗ = (x + ıy)(x − ıy) = x2 + y2 in agreement with the earlierdefinition.

B The description of rotations in 3 dimensions.

B.1 Coordinate systems.

We define 2 sets of right-handed orthogonal unit vectors (basis vectors) or {ai}for short,and {A1,A2,A3}, or {Ai} for short. They are connected by a rotation,R(α,n), through an angle α about some axis defined by the unit vector n, i.e,

aiR(α,n)→ Ai, i = 1, 2, 3. (170)

Orthogonality means

ai.aj = δi,j , i, j = 1, 2, 3,

Ai.Aj = δi,j , i, j = 1, 2, 3,

δi,j = 0, i 6= j,

= 1, i = j, (171)

and right-handed means a1 ∧ a2 = a3, A1 ∧A2 = A3.

37

B.2 Rotating vectors.

B.2.1 Relation between components of a vector V in 2 coordinatesystems.

We express the vector V in terms of its components Vi in the {ai} coordinatesystem by

V =∑i

Viai,

(172)

where

Vi = V .ai, i = 1, 2, 3.

(173)

The same vector V can also be expressed in terms of its components V ′i inthe {Ai} coordinate system by

V =∑i

V ′iAi, (174)

where

V ′i = V .Ai, i = 1, 2, 3. (175)

To find the relation between the numbers Vi and V ′i we use

V ′i = V .Ai,

= (∑j

Vjaj).Ai

=∑j

VjRji, (176)

where the matrix of 9 numbers Rji is defined by

Rji = aj .Ai. (177)

The matrix Rji is determined entirely by the relation beteen the 2 coordinatesystems and does not depend on the particular vector V we are considering. Itis easily calculated by working out the set of scalar products of the 2 sets ofbasis vectors.

The same matrix turns up in the relation between the 2 sets of orthogonalvectors. The equation

Ai =∑j

(Ai.aj)aj (178)

38

expresses Ai in terms of its components in the {ai} coordinate system. Thisrelation can also be written in terms of Rji:

Ai =∑j

Rjiaj . (179)

Note that if we want to think of these relations in terms of columns ofnumbers, in the case of (176), or columns of vectors, in the case of (179), wehave to write them in terms of the transposed matrix R with elements definedby Rij = Rji. Equations (176) and (179) can then be written V ′1

V ′2V ′3

=

R11 R12 R13

R21 R22 R23

R31 R32 R33

V1

V2

V3

, (180)

and A1

A2

A3

=

R11 R12 R13

R21 R22 R23

R31 R32 R33

a1

a2

a3

. (181)

B.2.2 Relation between components of two vectors V and its rotatedversion W in a fixed coordinate system.

In this case we consider 2 different vectors V and W , where the latter is obtainedfrom the former by rotating it by α about n. Expressed in terms of theircomponents in the {ai} system we have

V =∑i

Viai,

W =∑i

Wiai, (182)

We want to find the relation between the sets of numbers V1, V2, V3, andW1,W2,W3.To do this we use the fact that because W is obtained from V by the samerotation that sends the {ai} into the {Ai}, the components of W in the {Ai}system must be the same numbers as the components of V in the {ai} system,i.e.,

W .Ai = V .ai, (183)

for all i. Expressing the vector W on the left-hand side in terms of its compo-nents in the {ai} system this relation reads∑

j

Wjaj .Ai = Vi, (184)

or

Vi =∑j

WjRji, (185)

39

where we have introduced exactly the same matrix with elements Rij that wedefined in the previous sub-section.

In matrix notation we can write this in terms of one-column matrices andthe transpose matrix Rij as V1

V2

V3

=

R11 R12 R13

R21 R22 R23

R31 R32 R33

W1

W2

W3

. (186)

We can convert this to a formula the gives the Wi in terms of the Vi (i.e., byinverting the result (186)) by using the fact that the inverse of the matrix R isR:

R R = 1, (187)

where the 1 on the right-hand side means the 3x3 unit matrix. This is provedas follows.

(R R)ij =∑k

RikRkj

=∑k

RikRjk

=∑k

(ai.Aj)(aj .Ak)

= ai.aj , (188)

where we have recognised the expression for the scalar product of ai and ai interms of their components in the {Ai} system. Hence from the orthogonalityof the ai we obtain

(R R)ij = ai.aj = δi,j , (189)

which is just eq.(187) expressed interms of matrix components.We use eq.(187) by multiplying both sides of eq.(186) with matrix R on the

left and obtain W1

W2

W3

=

R11 R12 R13

R21 R22 R23

R31 R32 R33

V1

V2

V3

, (190)

orW = RV, (191)

for short. Thus with our definition of R we have a nice way of thinking of therotation (”‘active”’ rotation): R acts on V to give the rotated vector W . In the”‘passive ”‘ formulation associated with eq.(180) the inverse matrix R−1 = Rappears.

40

B.3 Orthogonal matrices.

A matrix satisfying the condition (189), or equivalently R−1 = R, is called anorthogonal matrix. It is easily proved that such a matrix does not alter the norm√V 2

1 + V 22 + V 2

3 of a column vector. Since rotating a vector does not changeits length we see the this is why orthogonal matrices turn up in discussions ofrotations.

C Spherical polar coordinates.

The orbital angular momentum state of a particle is dependent only on theangular coordinates of the particles position or momentum and not on its radialseparation from the origin. In this situation spherical polar coordinates areconvenient. These coordinates involve two angles (θ, φ) and the radial distancer instead of the 3 Cartesian coordinates x, y, z, with respect to an origin andthe 3 orthogonal unit vectors of a right-handed system.

The angle θ of a point P is called the polar angle. It is the angle betweenz-axis (the zenith) and the line (the radius) from the origin to P . The angle φis the angle between the x-axis (the azimuth) and the projection of the radiuson the x− y plain. A convenient way of making the angles unique is to measureθ from the positive z− axis and φ from the positive x-axis. Value ranges arerestricted as follows.

0 ≤ θ ≤ π0 ≤ φ < 2π. (192)

Angle φ is positive for points with positive y coordinates. Points on the z axishave θ = 0 or θ = π and by convention φ = 0. The origin is assigned the valuesr = 0, θ = 0, φ = 0.

With these choices the Cartesian coordinates are related to r, θ, φ, by

x = r sin θ cosφ,

y = r sin θ sinφ,

z = r cos θ. (193)

C.1 Orbital angular momentum operators in polar coor-dinates.

Using the chain rule for partial differentiation of a function of several variableswe have

∂

∂φ=

∂x

∂φ

∂

∂x+∂y

∂φ

∂

∂y+∂z

∂φ

∂

∂z

= −r sin θ sinφ∂

∂x+ r sin θ cosφ

∂

∂y

= −y ∂∂x

+ x∂

∂y, (194)

41

and hence

h

ı

∂

∂φ= −ypx + xpy

= ˆz. (195)

It is shown in the main text that ˆz generates rotations about the z-axis. It is

clear from the definition of φ that a rotation about the z-axis by an angle αcorresponds to a change in φ by α so the result (195) is a very natural one.

Changes in θ do not correspond to a rotation about one of the Cartesiancoordinate axes so the formula analogous to (195) for ∂

∂θ is more complicated.We have

∂

∂θ=

∂x

∂θ

∂

∂x+∂y

∂θ

∂

∂y+∂z

∂θ

∂

∂z

= r cos θ cosφ∂

∂x+ r cos θ sinφ

∂

∂y− r sin θ

∂

∂z

= z cosφ∂

∂x+ z sinφ

∂

∂y− exp(−ıφ)(x+ ıy)

∂

∂z, (196)

where we have used exp(ıφ) = cosφ+ sinφ in the last term on the right-hand-side.

From the second equality in eq.(194) we have

cot θ∂

∂φ= − sinφ× z ∂

∂x+ cosφ× z ∂

∂y. (197)

Putting together eqs.(196) and (197) we obtain

ıh exp(ıφ)(−ı ∂∂θ

+ cot θ∂

∂φ) = ˆ

x + ıˆy, (198)

where we have used

ˆx =

h

ı(y∂

∂z− z ∂

∂y),

ˆy =

h

ı(z∂

∂x− x ∂

∂z),

ˆz =

h

ı(x∂

∂y− y ∂

∂x). (199)

Similarily we find

ıh exp(−ıφ)(+ı∂

∂θ+ cot θ

∂

∂φ) = ˆ

x − ıˆy, (200)

By taking linear combinations of the two formulae (198) and (198) we can

readily obtain separate formulae for ˆx and ˆ

y in terms of partial derivatives

with respect to θ and φ, but as we will see in the next sub-section ˆx ± ıˆy

are just as useful. We also show in the main text how these operators play animportant role as ”raising” (ˆ

+ = ˆx + ıˆy)) and ”lowering” (ˆ− = ˆ

x − ıˆy))operators.

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C.2 Commutation relation of angular momentum opera-tors.

We first derive the commutation relations for the ”raising” and ”lowering” op-erators from the explicit formulae (198) and (200). This is a useful exercise inevaluating complicated commutators.

For an arbitrary function ψ(r, θ, φ) we have

ˆ+

ˆ−ψ = −h2 exp(+ıφ)[−ı ∂∂θ

+ cot θ∂

∂φ] exp(−ıφ)(ı

∂ψ

∂θ+ cot θ

∂ψ

∂φ)

= −h2 exp(+ıφ)(exp(−ıφ))(−ı2 ∂2ψ

∂θ2+ ı(sin θ)−2 ∂ψ

∂φ− ı cot θ

∂2ψ

∂θ∂φ)

−h2 exp(+ıφ)(cot θ)(−ı exp(−ıφ))(ı∂ψ

∂θ+ cot θ

∂ψ

∂φ)

−h2 exp(+ıφ) cot θ exp(−ıφ)(ı∂2ψ

∂φ∂θ+ cot θ

∂2ψ

∂φ2) (201)

The various terms here arise as follows. The two terms in the square bracket inthe first line operate on everything to the right. The −ı ∂∂θ in the square bracket

give rise to all the terms on the second line. The cot θ ∂∂φ in the square bracketgive rise to all the terms on the third and fourth lines.

The expression (201) simplifies considerably. The terms ∂2ψ∂θ∂φ cancel and the

coefficient of ∂ψ∂φ combine using ı(sin θ)−2 − ı(cot θ)2 = ı. We obtain

ˆ+

ˆ−ψ = −h2(∂2ψ

∂θ2+∂ψ

∂θ+ (cot θ)2 ∂

2ψ

∂φ2+ ı

∂ψ

∂φ). (202)

Inspection of the first line of eq.(201) reveals that ˆ− ˆ+ψ can be evaluate simply

by changing the sign of φ everywhere in eq.(202). We obtain

ˆ− ˆ+ψ = −h2(

∂2ψ

∂θ2+∂ψ

∂θ+ (cot θ)2 ∂

2ψ

∂φ2− ı∂ψ

∂φ). (203)

Hence

ˆ+

ˆ−ψ − ˆ− ˆ+ψ = −2h2ı

∂ψ

∂φ

= 2hˆzψ, (204)

where the last equality follows from the result (195).This equality is valid for arbitrary wavefunctions ψ(r, θ, φ) so we can deduce

the operator commutaion relation

[ˆ+, ˆ−] = 2hˆz. (205)

We can make another important deduction from this result by noting that

ˆ+

ˆ− = (ˆx + ıˆy)(ˆ

x − ıˆy)

= (ˆx)2 + (ˆ

y)2 − ı(ˆx

ˆy − ˆ

yˆx), (206)

43

and

ˆ− ˆ+ = (ˆ

x − ıˆy)(ˆx + ıˆy)

= (ˆx)2 + (ˆ

y)2 + ı(ˆx

ˆy − ˆ

yˆx). (207)

Subtracting the two equations (206) and (207) using (205) we deduce

[ˆx, ˆy] = ıhˆ

z. (208)

The theoretical framework makes no distinction between the labels used forthe coordinate axis so we can immediately deduce the other two commutationrelations

[ˆz, ˆx] = ıhˆ

y, (209)

and

[ˆy, ˆz] = ıhˆ

x. (210)

The three relations (208), (209) and (210) can be conveniently summarised inthe vector relation

~` ∧ ~` = ı

~`. (211)

Using these commutation relations we can also deduce the crucial raisingand lowering properties of ˆ

+ and ˆ−. We have

[ˆ+, `z] = [`x, `z] + ı[`y, `z] = −ı`y + ı(ı`x) = −`+, (212)

and similarily

[ˆ−, `z] = +`−. (213)

These relations are used in Section 4.3 to obtain the eigenvalue spectra of theangular momentum operators.

44