Quantum theory of spin: algebraic approach• in analogy with classical physics, we believe that a spinning mass carries intrinsic angular momentum, along with the angular momentum of its center-of-mass, which is L• we call this electron spin angular momentum S• Stern and Gerlach experimentally demonstrated, by putting a beam of electrons in a non-uniform magnetic field, that electrons seemed to carry two possible values of one spin component, so the suggestion is that electrons have spin quantum number s = ½ and a z-component of spin quantum number m = ± ½ ‘spin-up’ or ‘spin-down’• we will work almost exclusively with operator algebra, and push the analogy with what we have seen already to the hilt• the explicit representation for spin will be 2 x 2 matrix algebra• we have a new notation now for our humble electron’s spin• | s m > is the ‘spin state’ for an electron with spin angular momentum (√3ħ /2) and spin z-component ±ħ /2

What do our spin operators look like?

• what does this tell us? Same operator behavior:

• start with commutation relations: assume verysame structure

yxzxzyzyx SiSSSiSSSiSScyclicity ˆˆ,ˆˆˆ,ˆˆˆ,ˆby

• eigenstates (simultaneously measurable) of those two operators• while we’re on the subject, we have raising and lowering operators

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1

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• otherwise results are zero: can’t raise m = ½ or lower m = ½• that √ prefactor is exactly zero as a guarantee • in fact, s may be an integer (like l was) or a ‘half-integer’ (as here)

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124

322 :ˆ)1(:ˆ

What do our spin states look like, and are they eigenstates?

• the basis spinors are eigenfunctions of S2 and Sz operators

• the eigenvalues are s(s+1) ħ2 and mħ, respectively• in this notation, the operators are 2 x 2 matrices• inner (scalar) products are row vectors times column vectors• to get of S2 as a matrix, look at its effect on ‘up’ and ‘down’ spin

• what are our spin states? The general state is a spinor •we have an alternate, more explicit, column vector notation

spinors thedefines which1

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What do more of our spin operators look like?• to get Sz as a matrix, look at its effect on ‘up’ and ‘down’ spin• we do it in class! Result:

• now get the raising and lowering operators! Results:

10

01

and 01

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01

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0

0

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0 and 00

0

1

00:ˆ

sec

fe

dcS

fdfe

dcS

• to get Sx and Sy as a matrix recall that the raising and lowering

operators were expressed in terms of them• we also define the 3 Pauli spin matrices {} • with the identity matrix I the four are said to span spin space, since any complex 2 x 2 matrix can be expressed as a linear combo

Now to get the last couple of them by working backwards, and define the Pauli spin matrices

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Effect of Sx and Sy on eigenvectors of Sz • what is the effect of (say) Sx on an eigenspinor of Sz ?

• so that’s interesting… it ‘flips’ the basis spinor• what is the effect of (say) Sy on an eigenspinor of Sz ?

• so that’s even more interesting… it ‘flips’ the basis spinor and it does a strange sign/imaginary operation on it, depending• note that we can normalize any spinor by taking its inner product with its adjoint (in this representation, columnrow and c.c.)• example of how this works:

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Eigenvectors for Sx

• what are the eigenspinors and eigenvalues of Sx ?

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• but any multiple of an eigenvector is also an eigenvector of the same eigenvalue, so let’s rescale by dividing by (bad news if = 0)

normalized when 1

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Eigenvectors for Sy

• all operators have real eigenvalues and are observable• the normalized eigenspinors for Sz are obvious building blocks for

an arbitrary spinor• however, the normalized eigenspinors for Sx and for Sy are

equally good: we need a ‘Fourier’s trick’ in spin space!!

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Decomposing an arbitrary spinor into a basis set

• [notation modified for consistency with x and y]

• assume is normalized:

down- sit'that and

up,-spin sit'y probabilit is

1

2

2

22

• how to express as a linear combination of (say) eigenspinors of Sx

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• we can also express as a linear combination of eigenspinors of Sy

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Do these matrices obey the requisite commutation relations?

• for example, does Sx commute with Sz ?

YES! 0

0

02

20

0010

0100

0001

1000

01

10

10

01

10

01

01

10,

24

2

4

2

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2

4

2

4

2

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• for example, does S2 commute with Sz ?

YES! 00

00

00

00

1000

0001

1000

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01

10

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10

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10

01,

8

33

8

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How does one combine angular momentum?• this might be a situation where two electrons need to have the total spin added• or it might be a situation where you have one electron and you want to add its spin to its orbital angular momentum• or you might want to combine an electron’s angular momentum with the angular momentum of the nucleus…• classically, it is merely the vector sum of each part – changes with time, if torques act, etc.• quantally, it is a more probabilistic cloud of somewhat quantized possibilities• the classical picture of adding two vectors is helpful, but one must realize that in the end the sum is quantized both in length and in z-component, and that there will usually be more than one way to achieve a given length/z-component combo• we have two angular momenta: S(1) and S(2) represented by operators of the usual form (operators, matrices..)• to start, assume two s = ½ ‘spins’

What will our notation be? What can we cook up?

• if this is really a s = 1 situation, then where is s =1, m = 0 ? • problem lies in the two middle states, which appear twice• NOT eigenstates of the operator S2 = (S(1) + S (2) )·(S(1) + S(2))• realize that the ‘1’ operators only operate on the first argument of the ket, and the ‘2’ operators only on the second argument

• a naïve attempt to create a couple of angular momenta together

1 out maxed 1; ; downwards' up lined' spinsboth :

0 out cancelled 0; ; aligned' oppositely' spins :

0 out cancelled 0; ; aligned' oppositely' spins :

1 out maxed 1; ; upwards' up lined' spinsboth :

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21

21

21

21

21

21

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sm

sm

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Check out whether one of those middle states works

• obviously, the other ‘naïve’ state will fail too• Are either of them eigenstates of Sz= Sz

(1) + Sz(2) ?

• NO• are the first and last states eigenstates of either operator?

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2:

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2

2

2

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2

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What to do with the middle states, and recover the missing combination to boot

• what can we cook up? Linear combinations of those two states!!• strategy: try lowering that ‘top’ state

1 factor) 21/ (needs normalizednot but lowered

)()(

)1(1)1(1

:

21

21

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!! 1 with )1(

2

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2

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23

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)(

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2

24

23

24

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2

12

sss

ii

S

Another, singlet state: m = s = 0

• all four are orthornormal to one another• all are eigenstates of Sz= Sz

(1) + Sz(2) as well as of S2

• the Clebsch-Gordan coefficients tell how to assemble more general linear combinations of two angular momenta

• the new state had a + sign in it.. Try a ..

• the triplet states have s = 1, m = 1, 0, 1

11;01;112

1 msmsms

• the singlet state has s = 0, m = 0 2

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2

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1

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