Quantum Chemistry 1 (CHEM 565) Fall 2018personal.psu.edu/guk15/qm1/qm1-v-section-3-5-spin_degrees_of_freedom.pdf · Quantum Chemistry 1 (CHEM 565), Fall 2018 5 of 29. 3.5 Spin States

Quantum Chemistry 1 (CHEM 565)Fall 2018

Gerald KniziaDepartment of ChemistryThe Pennsylvania State University

Quantum Chemistry 1 (CHEM 565), Fall 2018

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Grand Topic #3

3. Simple Model Problems

Free particles & Wave packets (Plane wave solutions; Superposition principle; Gaussian

wave packets)

Numerical solution of 1-particle quantum problems (Dimensionless EOM; Real-space

grid discretization; Time evolution; Quantum phenomena (eigenstates, tunneling, Husimi-

repr.); Pseudo-spectral methods; Basis function decomposition/1)

Particle in a box & co (1D Case: Steps, barriers, wells; Transmission/reflection,

resonance, tunneling; 3D Case: Separation of variables)

Vibrations & Harmonic oscillator (H.-o. (1D,3D), Non-harmonic oscillator (1D) & QM

approximation methods (subspace projections, perturbation expansion, variational method));

Polyatomic case

Spherical potentials & Angular momenta (Rotational spectra; Spherical harmonics

; Systems with spherical symmetry )

Spin & Two-level systems (Single spin ½; Combining orbital & spin-degrees of

freedom; Coupled spins)

( /∥ ∥)Y lm r r V ( ) = V (r)r


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3.5 Spin States and Spin-Orbitals

Real space and spin-space

We previously discussed one-particle wave functions in 3D space. Theseare square-integrable functions mapping a coordinate in 3Dspace into the complex numbers: .

The rationale for making such wave functions was: We could inprinciple ask the question „is there a particle at a specific position inspace?“ (this is associated with an imaginable measurement).

The particle's state vector must therefore associate a probabilityamplitude to it.

Various classical physical observables map to operators on this space(e.g., the position operator corresponds to a multiplicative operator on this space.

The momentum operator corresponds to , the orbital angular momentum operator to

).

ψ( ) : R↦ Cr r R3 ψ ∈ ( ,C)L2 R3

r

|ψ⟩⟨ |ψ⟩ = ψ( )r r

≡ ⋅ (⋅)Q x

≡ −ıℏP ∇

:= ×L R P


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We also saw that particles might have an intrinsic angular momentum. Thisis described by a vector-operator

which's cartesian components must fulfill the angular momentumcommutation relations

(where ). So far we know little else about the .

As an intrinsic particle property, this operator cannot be described withother operators, which act as operators on .

= + + ,S e xSx e ySy e zSz

[ , ] = ıℏSα Sβ ∑γ

εαβγSγ

α,β, γ ∈ {x, y, z} { }Sα

Sψ ∈ ( ,C)L2 R3


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(...the intrinsic angular momentum operator cannot possibly be described by operators

acting on real-space functions)

So how can the complete state space of a particle, including spin, bedescribed? as state space is not sufficient, as it cannot decribethe spin degree of freedom.

Before we go into the mathematical structure of -electron wave functions(next chapter), we need to take a closer look at this.

(The topics of -particle indistinguishability and spin are tightly coupled)

Let us first consider a hypothetical particle which has only a spin degree offreedom. (e.g., a particle fixed in place)

S

ψ ∈ ( ,C)L2 R3

N

N


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We saw that the angular momentum commutation relations imply that there is a basis of eigenstates

(where is integer or an half-integer) such that

That resulted only from the commutation relations—which hold for spin, too.

This implies that the spin has the same kind of spectrum.

If we can somehow come up with any vector space which supports thisspectrum, and allows for defining operators compatible with thecommutation relations, we are already good. We can just make it upbecause by being intrinsic properties, the operators automatically areindependent of all real-space based observables.

[ , ] = ıℏJ α J β ∑γ εαβγJ γ |j,m⟩

j

|j,m⟩J z

|j,m⟩J2

=

=

ℏm|j,m⟩

j(j+ 1)|j,m⟩.ℏ2

S

{ }Sα

{ }Sα


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Creating a spin-space

We can therefore create a state space capable of describing spin degreesof freedom by simply making up an abstract vector space with basisvectors

We define these basis vectors to be orthonormal:

(we need them orthonormal as they will act as eigenstates of hermitian operators)

On this space, we then simply define linear operators and by

{|s,m⟩, s = 0… ,m = −s…+ s}.smax

⟨ , | , ⟩ := .s1 m1 s2 m2 δs1s2δm1m2

Sz S2

|s,m⟩Sz

|s,m⟩S2

:=

:=

ℏm|s,m⟩

s(s+ 1)|s,m⟩.ℏ2


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Since the form a vector space basis, this definition is sufficient to

compute the action of and on arbitrary vectors .

E.g., we can decompose any uniquely into

Thus, the action of on is given by (linearity!)

Once we make up a set of and operators compatible with the

commutation relations and , that will be a sufficient suitabledefinition of the spin space! (we already know this to be possible)

{|s,m⟩}

Sz S2

| ⟩ ∈ span{|s,m⟩}ψspin

| ⟩ψspin

| ⟩ = |s,m⟩.ψspin ∑sm

csm

S2| ⟩ψspin

| ⟩ = ( |s,m⟩) = ℏs(s+ 1)|s,m⟩.S2

ψspin ∑sm

csm S2 ∑

sm

csm

Sx Sy

= + +S2

S2x S

2y S

2z


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(...We have the definitions)

One can define the actions of and on the basis states as follows:

This implicity also defines the action of and , if combined with

(full details: Cohen-Tannoudji, Quantum Mechanics vol 1 (1996), chapter VI, p.658 )

⟨ , | , ⟩s1 m1 s2 m2

|s,m⟩Sz

|s,m⟩S2

:=

:=

:=

δs1s2δm1m2

ℏm|s,m⟩

s(s+ 1)|s,m⟩.ℏ2

(i.e., basis is orthonormal)

S+ S−

|s,m⟩S+

|s,m⟩S−

:=

:=

ℏ |s,m+1⟩s(s+ 1) −m(m+ 1)− −−−−−−−−−−−−−−−√

ℏ |s,m−1⟩.s(s+ 1) −m(m− 1)− −−−−−−−−−−−−−−−√

Sx Sy

= ( + ı ) and = ( − ı ).S+ Sx Sy S− Sx Sy


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Creating a spin-space—All definitions combined

(1) Propose an orthornomal basis of spin states. (2) Define raising and

lowering operators , . (3) Then define:

This (non-unique) combination is the „standard basis of angular momentum“. Itfulfills all normalizaiton and commutation relations (why? explicit check.).

It allows forming explicit vector representations of the spin states andmatrix representations of the operators (HW11)

{|s,m⟩; m ∈ {−s, −s+ 1…, s− 1, s}, s ∈ {0, 1, 2,…}}:= ( + ı )S+ Sx Sy := ( − ı )S− Sx Sy

⟨ , | , ⟩s1 m1 s2 m2

|s,m⟩Sz

|s,m⟩S2

|s,m⟩S+

|s,m⟩S−

:=

:=

:=

:=

:=

(i.e., basis is orthonormal)δs1s2δm1m2

ℏm|s,m⟩

s(s+ 1)|s,m⟩.ℏ2

ℏ |s,m+1⟩s(s+ 1) −m(m+ 1)− −−−−−−−−−−−−−−−√

ℏ |s,m−1⟩.s(s+ 1) −m(m− 1)− −−−−−−−−−−−−−−−√

CN


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Experimental evidence shows that all elementary particles (in particular,

electrons), have a fixed value for the total spin quantum number .

A single electron always has . There are no other total spin quantumnumbers to consider.

This implies that the spin of a single electron can be described with a two-dimensional vector space, spanned by the two abstract basis vectors

These are called spin-up (or alpha) and spin-down (or beta), respectively.

s

s = 1/2s

|↑ ⟩

|↓ ⟩

:=

:=

s= ,m= ⟩∣∣12

12

s= ,m=− ⟩ .∣∣12

12


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The vectors in this space have the general form

where and are the expansion coefficients.

We can define the spin operators , , via

Explicit calculation will show that they fulfill the angular momentumcommutation relations . (and are consistent with the earlier def.)

| ⟩ = |↑ ⟩ + |↓ ⟩ψspin c↑ c↓

∈ Cc↑ ∈ Cc↓

Sx Sy Sz

Sx

Sy

Sz

=

=

=

(|↑ ⟩⟨ ↓ | + |↓ ⟩⟨ ↑ |)ℏ2

(ı |↑ ⟩⟨ ↓ | − ı |↓ ⟩⟨ ↑ |)ℏ2

(|↑ ⟩⟨ ↑ | − |↓ ⟩⟨ ↓ |)ℏ2

[ , ] = ıℏSα Sβ ∑γ Sγ


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Commonly is simply written as a complex two-vector (a `spinor“), wherethe two basis vectors and are identified with

The spin operators are then , where are the Pauli matrices:

Explicit calculation will show that and .

The , , fulfill all required commutation relations.

ψspin

|↑ ⟩ |↓ ⟩

|↑ ⟩ =: ( ) |↓ ⟩ =: ( )10

01

{ = ½ℏ }Sα σα { }σα

= ( ) = ( ) = ( )σx0110

σy0ı

−ı

0σz

100−1

= = = 1σ2x σ2y σ2z [ , ] = 2ıσα σβ ∑γ εαβγσγ

Sx Sy Sz


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Notes on the spin space

The electron's spin space is 2-dimensional complex vector space, spannedby and .

It may seem as if the -direction is handled in a preferred way over and .This is not the case. Spin polarization into arbitrary directions can beexpressed in this space. If a normalized 3D vector isgiven, then the spin operator in the direction of is given by

This operator can be shown to (a) be Hermitian, (b) have eigenvalues and .

One can thus transform any spin states in terms of eigenstates into eigenstates

|↑ ⟩ |↓ ⟩

z x y

n = ( , , ∈nx ny nz)T R3n

S n =

=

=

(n ⋅ )S

( + + ) ⋅ ( + + )nxe x nye y nye z Sxe x Sye y Sze z

+ + .nxSx nySy nzSz

−ℏ/2+ℏ/2

Sz {|↑ ⟩, |↓ ⟩}Sn {|↑ , |↓ }⟩n ⟩n


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Combined spin- and function space

The state space of real electrons (which are not fixed in space), must be capableof describing both real-space related observables (in ) and the spindegrees of freedom (in ).

In any combination! One could ask the question: „is there an electronat the specific position in space and which has the specific spin-projection at the same time?. So the state vector must assign aprobability amplitude to this.

As the state space of a single electron must describe the probabilitydistributions of any conveivable combinations of measurements in realspace and spin space, it must be given as the tensor product of bothspaces:

There are several (ultimately equivalent) ways of thinking about this space.

( ,C)L2 R3C2

r m |φ⟩

⟨ ,m|φ⟩ ∈ Cr

H1

= ( ,C) ⊗H1 L2 R3 C2


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Combined spin- and function space of an electron

1.: We can re-define the one-particle wave function space from to. That is, instead of mapping every point in real space to a

complex number, the functions in this space map every point in real space to a spinor (i.e., a vector of two complex numbers).

2.: We can introduce two separate functions, . One electronis then described by (here the ket „ “ only refers to the state vector in the spin space).

( ,C)L2 R3( , )L2 R3 C2 R3

R3

, ∈ ( ,C)ψ↑ ψ↓ L2 R3|ψ( )⟩ = ( )|↑ ⟩ + ( )|↓ ⟩r ψ↑ r ψ↓ r

|⋅⟩


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3.: We can introduce a combined spin-space coordinate (where and ). We define the one-particle space as

. I.e., as the space of functions which map everycombination of a point in space and a spin projection value

to a complex number

4.: We can form the abstract basis

where are the position eigenstates (Dirac- distributions centered on )and simply declare .

:= ( ,m)x r ∈r R3 m ∈ {−½,½}:= ( ⊗ {−½,½},C)H1 L2 R3

∈r R3m ∈ {½, −½}

| ,m⟩ := | ⟩ ⊗ |m⟩ ( ∈ ,m ∈ {½, −½})r r r R3

| ⟩r δ r := span{| ,m⟩}H1 r


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As mentioned, all those approaches are equivalent. We will later adopt theview: is the space of functions mapping acombined spin-position into the complex numbers.

This simplifies dealing with -electron wave functions.

One-particle functions , which depend on real-space position only, are called spatial orbitals.

One-particle functions , which depend on botha real-space position and a spin projection , are called spin-orbitals.

A spin orbital with describes all the degrees of freedom of anelectron (both intrinsic angular momentum and real-space observables)

= ( ⊗ {−½,½},C)H1 L2 R3= ( ,m)x r

N

ψ( ) ∈ ( ,C)r L2 R3r

ψ( ) ∈ ( ⊗ {−s,… ,+s},C)x L2 R3r m

ψ( )x = ( ,m)x r


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Notes on spin-orbitals

Since the non-relativistic Hamiltonian does not depend on spin, we canlargely decouple the spin- and spatial degrees of freedom.

We will normally make two spin orbitals and from one spatialorbital , which will individually have a trivial spin dependence:

This is what is meant with „an orbital is doubly occupied“. We haveone function of space , and make two spin-orbitals out of it. Bothof which are occupied.

( )ψ↑ x ( )ψ↓ x ψ( )r

( ,m)ψ↑ r

( ,m)ψ↓ r

=

=

{ ψ( )r

0if m = +½if m = −½

{ 0ψ( )r

if m = +½if m = −½

ψ( )r


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Notes on spin-orbitals

For many applications it is helpful to describe the spin orbitals with a basisset expansion.

Let be an orthonormal basis of (i.e., purely spatialfunctions). Then the set

is an orthonormal basis for the spin-orbitals. (i.e., the set of orbitals consisting

of: all orbitals in multiplied with an vector, and all orbitals in multiplied with

an vector)

We can therefore assume that we can express any spin orbital via, where are expansion coefficients and the are an

orthonormal basis of .

{| ⟩, r = 1...K}rspat ( ,C)L2 R3

{|r⟩} = {| ⟩ ⊗ |↑ ⟩} ∪ {| ⟩ ⊗ |↓ ⟩}rspat rspat

2K

K {|r⟩} |↑ ⟩ K {|r⟩}

|↓ ⟩

ψ( )x ψ( ) = r( )x ∑r ψr x ∈ Cψr |r⟩

= ( ⊗ {½, −½},C)H1 L2 R3


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Summary so far

We noted that the intrinsic angular momentum operator must be completelyindependent of real-space based operators like , , or . It musttherefore act on its own vector space, independent of the real-space operators...which we can therefore simply make up.

We considered possible definitions of this abstract spin space, by introducing a

basis , and defining how , , and act on it.

We noted that actual elementary particles only have a single value of . Forelectrons it is , meaning

We discussed that a complete description of a particle with spin requiresassigning a probability amplitude to every combination of point in real-space

and spin projection . We therefore defined spin-orbitalswhich are square-integrable functions of : .

We made a basis of the spin-orbital space by combining a real-space basis of with the trivial spin functions and .

SQ P = ×L Q P

{|s,m⟩} Sx Sy Sz S2

s

s = ½ m ∈ {½, −½}

∈ Rr m ∈ {½, −½} |φ⟩:= ( ,m)x r ψ( ) = ψ( ,m)x r

{|r⟩}(R,C)L2 |↑ ⟩ |↓ ⟩


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More comments on spin space

For the concrete case of the electrons, the spin space is the 2D complexvector space defined as the linear span of an abstract basis of two elements

The scalar product between arbitrary vectors by defining its value on thebasis vectors:

So we get a 2-dimensional complex inner product space

{|↑ ⟩, |↓ ⟩}

⟨ ↑|↑ ⟩⟨ ↓ |↓ ⟩⟨ ↑ |↓ ⟩⟨ ↓ |↑ ⟩

:=:=:=:=

1100


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An arbitrary element of this space can be written as

Such a vector thus assigns a complex scalar to the basis elements and :

| ⟩ = |↑ ⟩ + |↓ ⟩ψspin c↑ c↓

|ψ⟩ |↑ ⟩|↓ ⟩

⟨ ↑ |ψ⟩ = ⟨ ↓|s⟩ =c↑ c↓


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Electrons can be localized. Imagine a situation in which electrons are fixedto a 1D, 2D, or 3D lattice (e.g., in a solid or thin film):

1 2 3 4 5

Imagine that each such lattice site has one spin degree of freedom, whichcan be, individually, a vector in

This system can be described by the product space

i

:= span{|↑ , |↓ }Si ⟩i ⟩i

⊗ ⊗ ⊗…S1 S2 S3


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Vectors in the space assign one complex probabilityamplitude to every conceivable combination of vectors , for eachlattice site:

If there are lattice sites, a vector in this space is thus described by the values which denote the joint probabilityamplitude of finding the given combination of and for all latticepoints at the same time

|Ψ⟩ ⊗ ⊗ ⊗…S1 S2 S3|↑ ⟩ |↓ ⟩

|Ψ⟩ =(⟨ ↑ ⊗ ⟨ ↑ ⊗ ⟨ ↓ ⊗… )|1 |2 |3 :=⟨ ↑ ↑ ↓…|

c↑ ↑ ↓…

N 2N

{⟨ ↑↑↑ …|Ψ⟩, ⟨ ↓↑↑ …|Ψ⟩,…}|↑ ⟩i |↓ ⟩i

i = 1…N


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In a physical state of the spin lattice (e.g., a ground state), the probabilityampliudes of the different spin configurations may differ significantly.

For example, if a system is ferromagnetically coupled (i.e., adjacent spins„like“ to point to the same direction), then the probability amplitudes

might have large absolute values, while

absolute value may be near zero.

But in principle every combination of spin directions is conceivable, and thusgets assigned a probability amplitude. For this reason, such probabilityamplitudes of interacting systems are represented in the product space.

⟨ ↑↑↑↑↑ … |Ψ⟩ and ⟨ ↓↓↓↓↓ … |Ψ⟩

⟨ ↓↑↓↑↓ … |Ψ s⟩′


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Comments on operators in multi-particle spin-spaces

The operators we defined before (for a single particle) can be retained, butthere is now one of them for each lattice site : ( )

An operator acts only on the state space of site --- that means itacts as identity on all other sites (not as zero!).E.g., if we apply to a state , we get:

The other degrees of freedom remain unchanged, but they are still there!

This is very similar to the separation example from HW10.

{ }Sα

i { }S i,α i = 1,… ,N

{ }S i,α i

S3,z | ↓↑↓↑↓ …⟩

| ↓↑↓↑↓ …⟩S3,z =

=

=

(|↓ ⊗ |↑ ⊗ |↓ ⊗ |↑ ⊗ |↓ ⟩… )S3,z ⟩1 ⟩2 ⟩3 ⟩4

|↓ ⊗ |↑ ⊗ ( |↓ )⊗ |↑ ⊗ |↓ ⟩…⟩1 ⟩2 S3,z ⟩3 ⟩4

↓↑ ( |↓ ⟩) ↑↓ …⟩∣∣ Sz


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Comments on operators in multi-particle spin-spaces

However, one can also define the operators of total spin projection and totalspin by summing over the lattice sites:

In many cases these are the more relevant operators.

In this case, operator application involves more terms. E.g., if we apply to a state , we get:

Operators like (total spin squared) then become quite complicated!Identifying their eigenfunctions will be non-trivial ( HW11 for N=2 case).

Stotz := ∑

i=1

N

Si,z Stot := ∑i=1

N

Si

Stotz

| ↓↑↓↑↓ …⟩

| ↓↑↓↑↓ …⟩Stotz = ( |↓ ⟩) ↑↓↑↓ …⟩+ ↓ ( |↑ ⟩) ↓↑↓ …⟩+ ↓↑ ( |↓ ⟩) ↑↓ …⟩+∣

∣ Sz∣∣ Sz

∣∣ Sz

↓↑↓ ( |↑ ⟩) ↓ …⟩+ ↓↑↓↑ ( |↓ ⟩)…⟩+…∣∣ Sz

∣∣ Sz

S2tot

→


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Tentative Schedule

Mathematical prerequisites [4 weeks]: Linear Algebra, Probability Theory

Principles of quantum mechanics [3 weeks]: State space, Operators of observables, Born

rule, Time-dependent and time-independent Schrödinger equation, Constants of motion and

conserved quantities, Probability flux, Boundary conditions, vs. : Pure and mixed states,

Entropy & statistical ensembles

Simple model problems [4 weeks]: (Semi-)Free particle, Particle in a box,Harmonic oscillator, Spherical potential, Spin ½

Many-particle wave functions [2 weeks]: Fermions and Bosons,Occupation number vectors, Fock space, 2nd Quantization

The Hartree-Fock approximation [2 weeks]: Meaning,Interpretation/bonding, Uses, Breakdown

|ψ⟩ ρ


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Documents

Quantum Chemistry 1 (CHEM 565) Fall 2018personal.psu.edu/guk15/qm1/qm1-v-section-3-5-spin_degrees_of_freedom.pdf · Quantum Chemistry 1 (CHEM 565), Fall 2018 5 of 29. 3.5 Spin States