Quantum Mechanics — Lecture notes for PHYS223

XVI Spin

The electron also has an intrinsic angular momentum 𝐒^, independently of its orbital angular momentum 𝐋^. This intrinsic angular momentum is called spin (the name suggests that the electron performs a rotation around itself, which is, however, actually not the case).

Figure 5: Spin.

XVI.1 Stern-Gerlach experiment

The spin was discovered in the Stern-Gerlach experiment in 1922. In this experiment, silver atoms are heated in an oven, from which they escape through a tiny hole. They travel into x direction and enter an apparatus where they pass through an inhomogeneous magnetic field 𝐁=bz𝐤. This field points into z direction and has a finite gradient Bz/z=b. If the atoms carry a magnetic moment 𝐦 this gives rise to a force

Fz=-VBz=mzb (279)

which deflects the atoms into z direction.

Figure 6: Set-up of the Stern-Gerlach experiment.

In the experiment it is observed that the atoms arrive on the screen at two spots, only. This is not what is classically expected, but also incompatible with quantisation of the orbital angular momentum 𝐋, which always yields an odd number 2l+1 of possible values of Fz since l is an integer. However, the experiment can be fully explained by introducing an intrinsic magnetic moment

𝐦^=-eme𝐒^, (280)

where 𝐒^ has the properties of angular momentum, but with l=ls=1/2 not an integer. Since the spin magnetic quantum number is restricted to ms=-ls,-ls+1,,ls this allows only two values ms=±12, hence

Sz=2 or Sz=-2. (281)

Given the force from Eq. (279) it can be deduced that the observed locations of the arriving atoms indeed correspond to these two values.

XVI.2 Spinor

The spin state of the electron is described by a two-component vector (called spinor)

ψ=(ψψ) (282)

where

|ψ|2 = probability that Sz=/2 (‘spin up’), (283)
|ψ|2 = probability that Sz=-/2 (‘spin down’). (284)

Hence ψ=(10) describes an electron with spin up and ψ=(01) describes an electron with spin down.

The scalar product of spinors is given by (see the mathematical appendix)

ψ|φ=ψ*φ+ψ*φ. (285)

Normalisation of the probability requires |ψ|2+|ψ|2=1, hence ψ|ψ=1.

XVI.3 Spin operators

The spin operator 𝐒^=S^x𝐢+S^y𝐣+S^z𝐤 consists of the three spin matrices

S^x=2(0110), (286)
S^y=2(0-ii0), (287)
S^z=2(100-1). (288)

The three matrices

σx=(0110),σy=(0-ii0),σz=(100-1) (289)

are called Pauli matrices.

Matrix multiplication (see the mathematical appendix) gives

𝐒^2=S^x2+S^y2+S^z2=342(1001). (290)

The commutation relations also follow from matrix multiplication,

[S^x,S^y]=iS^z,[S^y,S^z]=iS^x,[S^z,S^x]=iS^y,
(291)
[S^x,𝐒^2]=0,[S^y,𝐒^2]=0,[S^z,𝐒^2]=0. (292)

This is identical to the commutation relations of angular momentum, see Eqs. (264) and (265).

XVI.4 Eigenvalues and eigenvectors of 𝐒^2 and S^z

Since

𝐒^2(ψψ)=342(1001)(ψψ)=342(ψψ), (293)

all vectors are eigenvectors 𝐒^2. The eigenvalue 342 corresponds to angular momentum with l=ls=1/2, see Eq. (266).

Since

S^z(ψψ)=2(100-1)(ψψ)=2(ψ-ψ), (294)

the normalised eigenvectors of S^z are ψ=(10) (with eigenvalue /2) and ψ=(01) (with eigenvalue -/2). This again corresponds to angular momentum with l=ls=1/2, see Eq. (267).

Figure 7: Quantisation of spin.

XVI.5 Eigenvalues and eigenvectors of S^x and S^y

S^x(ψψ)=2(0110)(ψψ)=2(ψψ). (295)

Hence the normalised eigenvectors of S^x are ψ=12(11) (with eigenvalue /2) and ψ=12(1-1) (with eigenvalue -/2).

S^y(ψψ)=2(0-ii0)(ψψ)=2(-iψiψ). (296)

Hence the normalised eigenvectors of S^x are ψ=12(1i) (with eigenvalue /2) and ψ=12(1-i) (with eigenvalue -/2).

XVI.6 Expectation values

The expectation value of Sz is given by

S^z = ψ|S^zψ (297)
= 2(ψ*,ψ*)(100-1)(ψψ)
= 2(ψ*,ψ*)(ψ-ψ)
= 2(|ψ|2-|ψ|2),

Note that

P(Sz=/2) = |ψ|2, (298)
P(Sz=-/2) = |ψ|2. (299)

are the probabilities for spin up and down, respectively.

By similar calculations,

S^x = Reψ*ψ, (300)
S^y = Imψ*ψ, (301)

where Re denotes the real part and Im denotes the imaginary part of the complex number ψ*ψ.

The probabilities for spin measurements in x or y directions are given by

P(Sx=/2) = 12|ψ+ψ|2, (302)
P(Sx=-/2) = 12|ψ-ψ|2, (303)

and

P(Sy=/2) = 12|ψ-iψ|2, (304)
P(Sy=-/2) = 12|ψ-iψ|2, (305)

respectively.

XVI.7 Polarisation vector and Bloch sphere

A convenient alternative way to represent a spinor state is given by the 3-dimensional real polarisation vector

P = (σx,σy,σz) (306)
= (2Reψ*ψ,2Imψ*ψ,|ψ|2-|ψ|2). (307)

If ψ is normalised then the polarisation vector is of unit length, i.e., it is restricted to the surface of a sphere, the so-called Bloch sphere.

XVI.8 Stationary Schrödinger equation

The most general stationary Hamiltonian for a spin is a hermitian 2×2-dimensional matrix. Such a matrix can always be written as

H=a0I+axσx+ayσy+azσz (308)

where a0, ax, ay, and az are real constants, and

I=(1001) (309)

is the 2×2-dimensional identity matrix. The stationary Schrödinger equation Eψ=Hψ is solved by the eigenvectors of H. These can be found by writing the vector 𝐚=(ax,ay,az)=a𝐧, where 𝐧 is a unit vector and a=|𝐚|. The eigenvectors can then be written as

ψ+ = 12(1+nz)(1+nznx+iny), (310)
ψ- = 12(1+nz)(-nx+iny1+nz). (311)

On the Bloch sphere, these eigenvectors correspond to the polarisation vectors

P±=±𝐧. (312)

The corresponding eigenvalues

E±=a0±a (313)

determine the energy of these states.

XVI.9 Spin in Dirac notation

In order to describe spin in the Dirac notation, we introduce the normalised and mutually orthogonal states | for “spin up” and | for “spin down”, with

S^z|=2|,S^z|=-2|. (314)

Accordingly, we can write

S^z=2||-2||. (315)

The other spin components are then given by the operators

S^x=2||+2||,S^y=-i2||+i2||, (316)

and the total spin is given as

𝐒^2=S^x2+S^y2+S^z2=324(||+||). (317)

The eigenstates of S^x are |±x=1/2(|±|), while those of S^y are |±y=1/2(|±i|); the associated eigenvalues are ±/2. All the expressions for expectation values and probabilities given above can now be seen as special incarnations of our general rules for quantum mechanics, evaluated in the basis |, |.

In analogy to orbital angular momentum, we can also introduce the ladder operators

S^+=S^x+iSy=||,S^-=S^x-iSy=||, (318)

which are again non-hermitian and related by S^+=S^-. These now convert or annihilate spin eigenstates, according to

S^+|=|,S^+|=|𝟘,S^-|=|,S^-|=|𝟘. (319)

XVI.10 Mathematical appendix: two-dimensional linear algebra

XVI.10.1 Vectors and matrices

A vector 𝐯 in two dimensions is given by two numbers v1 and v2, called components,

𝐯=(v1v2). (320)

Matrices A, B etc are given by four numbers, such as

A=(a11a12a21a22), (321)
B=(b11b12b21b22). (322)

We consider the case that the matrix and vector components can be complex numbers.

XVI.10.2 Addition of vectors and matrices

Vectors and matrices are added element by element,

𝐯+𝐰=(v1+w1v2+w2), (323)
A+B=(a11+b11a12+b12a21+b21a22+b22). (324)

XVI.10.3 Scalar product

The scalar product

𝐯|𝐰=v1*w1+v2*w2 (325)

can also be written as the product

𝐯|𝐰=𝐯𝐰, (326)

where 𝐯=(v1*,v2*) denotes the hermitian conjugate.

A vector is said to be normalised when 𝐯|𝐯=1.

XVI.10.4 Multiplication of a vector and a matrix

When a vector 𝐯 is multiplied by a matrix A, the result is a new vector A𝐯 with components

A𝐯=(a11v1+a12v2a21v1+a22v2). (327)

One can also multiply two matrices A and B, which results in another matrix

AB=(a11b11+a12b21a11b12+a12b22a21b11+a22b21a21b12+a22b22). (328)

Note that this is different from element-by-element multiplication.

XVI.10.5 Eigenvalues and eigenvectors

Given is a matrix A. The eigenvalue equation

A𝐯=λ𝐯 (329)

requires to find vectors 𝐯 such that the matrix multiplication only changes the length of the vector (multiplication by the constant λ), but not its direction. The number λ is called eigenvalue. In general, the eigenvalues are found from the equation det(A-λ)=0, where det is the determinant, det=b11b22-b12b21. This gives the two solutions

λ±=a11+a222±a12a21+14(a11-a22)2. (330)

The corresponding eigenvectors can be written as

𝐯±=(a12λ±-a11). (331)

XVI.10.6 Revision Quiz

For

𝐯=(32i),𝐰=(2-i),A=(1i-i1),B=(1i22),

calculate

(i) 𝐯+𝐰,

(ii) 2𝐯+3i𝐰,

(iii) 𝐯|𝐯,

(iv) 𝐯|𝐰,

(v) A𝐯,

(vi) B𝐰,

(vii) the eigenvalues and eigenvectors of A.