QM Lecture 9: Quantum Spin

**By: Tao Steven Zheng (郑涛)**

## **Spin**

In Lecture 8, it was mentioned that there are two types of angular momentum in quantum mechanics: the orbital angular momentum

L
^

{\displaystyle \hat{L} }

 , and the spin angular momentum

S
^

{\displaystyle \hat{S}}

 (more commonly called "spin"). Together they can be summed to get the total angular momentum

J
^

=

L
^

S
^

{\displaystyle \hat{J} = \hat{L} + \hat{S} }

. In this lecture, we will only be investigating spin. The formal definition of the spin operator

S
^

{\displaystyle \hat{S}}

 is analogous to the orbital angular momentum operation

L
^

{\displaystyle \hat{L} }

. For example, we have three components of spin

S
^

=
(

S
^

x

,

S
^

y

,

S
^

z

)

{\displaystyle {\hat {S}}=({\hat {S}}\_{x},{\hat {S}}\_{y},{\hat {S}}\_{z})}

, whereby

S
^

2

=

S
^

x

2

S
^

y

2

S
^

z

2

{\displaystyle {\hat {S}}^{2}={{\hat {S}}\_{x}^{2}}+{{\hat {S}}\_{y}^{2}}+{{\hat {S}}\_{z}^{2}}}

with corresponding eigenvalues

S
^

2

s
,
m
⟩

=

ℏ

2

s
(
s
1
)

s
,
m
⟩

S
^

z

s
,
m
⟩

=
ℏ
m

s
,
m
⟩

and

S
^

s
,
m
⟩

=
ℏ

s
(
s
1
)
−
m
(
m
1
)

s
,
m
1
⟩

S
^

−

s
,
m
⟩

=
ℏ

s
(
s
1
)
−
m
(
m
−
1
)

s
,
m
−
1
⟩

where

S
^

=

S
^

x

i

S
^

y

{\displaystyle {\hat {S}}\_{+}={\hat {S}}\_{x}+i{\hat {S}}\_{y}}

 and

S
^

−

=

S
^

x

−
i

S
^

y

{\displaystyle {\hat {S}}\_{-}={\hat {S}}\_{x}-i{\hat {S}}\_{y}}

.

Like orbital angular momentum, the three components of spin obey the following commutation relations

[

S
^

x

,

S
^

y

# ]
i
ℏ

S
^

z

[

S
^

y

,

S
^

z

# ]
i
ℏ

S
^

x

[

S
^

z

,

S
^

x

# ]
i
ℏ

S
^

y

.

{\displaystyle [{\hat {S}}\_{x},{\hat {S}}\_{y}]=i\hbar {\hat {S}}\_{z}\qquad [{\hat {S}}\_{y},{\hat {S}}\_{z}]=i\hbar {\hat {S}}\_{x}\qquad [{\hat {S}}\_{z},{\hat {S}}\_{x}]=i\hbar {\hat {S}}\_{y}.}

While the value of orbital angular momentum can vary, the spin is fixed for all time for a given particle. Thus, spin is an intrinsic property of particles, much like mass or charge. Different types of particles have different values of spin. For example, force-carrying bosons have integer spin

# s
1

{\displaystyle s=1}

, whereas fermions (like the electron) possess half-integer spin

# s
±
1

/

2

{\displaystyle s=\pm 1/2}

.

## **Stern-Gerlach Experiment**

Spin is an intrinsic property of particles. This property was deduced from an experiment conceived by Otto Stern (1921) and conducted by Walther Gerlach (1922). Their collective work resulted in what is now called the Stern-Gerlach experiment. In the original experiment, neutral particles such as silver atoms were emitted through an inhomogeneous magnetic field. The field is created by two magnets of opposite poles, one with an irregular shape that produces the inhomogeneity in the desired direction. Particles would then travel in a perpendicular direction. The force on a particle with a magnetic moment

μ

{\displaystyle \mu}

 is

F
→

=
∇

(

μ

B
→

)

{\displaystyle {\bf {{\vec {F}}=\nabla \left(\mu {\bf {\vec {B}}}\right)}}}

where

B
→

{\displaystyle {\bf {\vec {B}}}}

 is the magnetic field. Suppose the z-component of the magnetic field leads to a force in the z-direction given by

F

z

=
μ

d

B

z

d
z

{\displaystyle F\_{z}=\mu {\frac {dB\_{z}}{dz}}}

.

This force will cause particles in the beam to be deflected. Stern and Gerlach originally hypothesized that the deflections would be random because, classically, one would expect the particles subjected to this force to experience a deflection that varied continuously between

μ

{\displaystyle +\mu }

 and

−
μ

{\displaystyle -\mu }

. This was not known initially at the time, but it turns out that the magnetic dipole moment of the particle is proportional to its spin. However, when the experiment is actually carried out, the emitted beam splits exactly in two as if it were emitted directly from the apparatus. One beam heads in the positive z-direction while the other heads in the negative z-direction. These are the two spin states of

# m
1

/

2

{\displaystyle m=+1/2}

 and

# m
−
1

/

2

{\displaystyle m=-1/2}

. The results are quantized! What is most astonishing, however, is that the Stern-Gerlach experiment was performed several years before George Uhlenbeck (1925) and Samuel Goudsmit (1927) formulated their hypothesis of the existence of the electron spin!

## **Pauli Matrices**

The **Pauli matrices**

σ

=

(

σ

x

,

σ

y

,

σ

z

)

{\displaystyle \mathbf {\sigma } =\left(\sigma \_{x},\sigma \_{y},\sigma \_{z}\right)}

 , named after Wolfgang Pauli, are a set of three

2
×
2

{\displaystyle 2 \times 2}

 Hermitian matrices:

σ

x

=

(

0

1

1

0

)

σ

y

=

(

0

−
i

i

0

)

σ

z

=

(

1

0

0

−
1

)

.

{\displaystyle \sigma \_{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\qquad \sigma \_{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\qquad \sigma \_{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}

The above matrices were discovered by solving the Pauli equation

[

1

2
m

(

σ

⋅
(

p
^

−
q

A
→

)

)

2

q
ϕ

]

ψ
# ⟩
i
ℏ

∂

∂
t

ψ
⟩

{\displaystyle \left[{\frac {1}{2m}}({\boldsymbol {\sigma }}\cdot (\mathbf {\hat {p}} -q\mathbf {\vec {A}} ))^{2}+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }

where

m

{\displaystyle m}

 is the mass of the particle,

q

{\displaystyle q}

 its electric charge, in an electromagnetic field described by the magnetic vector potential

A
→

{\displaystyle \mathbf {\vec {A}} }

 and the electric scalar potential

ϕ

{\displaystyle \phi}

. The Pauli equation is a reformulation of the Schrödinger equation for spin-1/2 particles, which takes into account the interaction of the spin of a quantum particle with an external electromagnetic field.

The Pauli matrices are closely related to spin operators of spin-1/2 particles by a factor:

S
^

x

=

ℏ
2

(

0

1

1

0

)

S
^

y

=

ℏ
2

(

0

−
i

i

0

)

S
^

z

=

ℏ
2

(

1

0

0

−
1

)

.

{\displaystyle {\hat {S}}\_{x}={\frac {\hbar }{2}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}\qquad {\hat {S}}\_{y}={\frac {\hbar }{2}}{\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\qquad {\hat {S}}\_{z}={\frac {\hbar }{2}}{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}

## **Problems**

### **Problem 1: Commutation Relations of the Pauli Matrices**

Calculate the commutation relations of the Pauli matrices

(
1
)

[

σ

x

,

σ

y

# ]
2
i

σ

z

(
2
)

[

σ

y

,

σ

z

# ]
2
i

σ

x

(
3
)

[

σ

z

,

σ

x

# ]
2
i

σ

y

{\displaystyle {\begin{aligned}&(1)\;[\sigma \_{x},\sigma \_{y}]=2i\sigma \_{z}\\[5pt]&(2)\;[\sigma \_{y},\sigma \_{z}]=2i\sigma \_{x}\\[5pt]&(3)\;[\sigma \_{z},\sigma \_{x}]=2i\sigma \_{y}\end{aligned}}}

### **Problem 2: Constructing Spin Matrices using Ladder Operators**

Construct the spin operators

S
^

x

{\displaystyle {\hat {S}}\_{x}}

 and

S
^

y

{\displaystyle {\hat {S}}\_{y}}

 using the eigenvalues of the raising and lowering ladder operators for spin:

S
^

s
,
m
⟩

=
ℏ

s
(
s
1
)
−
m
(
m
1
)

s
,
m
1
⟩

S
^

−

s
,
m
⟩

=
ℏ

s
(
s
1
)
−
m
(
m
−
1
)

s
,
m
−
1
⟩

### **Problem 3: Pauli Matrix Relations**

Prove the following relation for the Pauli matrices

σ

x

2

=

σ

y

2

=

σ

z

2

=
I

{\displaystyle \sigma \_{x}^{2}=\sigma \_{y}^{2}=\sigma \_{z}^{2}=I}

where

# I

(

1

0

0

1

)

{\displaystyle I={\begin{pmatrix}1&0\\0&1\end{pmatrix}}}

 is the

2
×
2

{\displaystyle 2 \times 2}

 identity matrix.