**Non-relativistic quantum description of particles
with spin**

The operator associated with the spin of a particle is a vector observable.
Its
components satisfy the commutation relations that define an angular momentum operator.
The
quantum state of a particle with spin is not completely specified by specifying its
wave function ψ(**r**) in orbital space E_{r}.
We must also specify its spin variables in spin space E_{s}. A given particle is
characterizes by a unique value of s, which is a positive integer or half integer.
(2s + 1) basis vectors span the spin space E_{s} of this particle. We can
choose these basis vectors to be {|s,m_{s}>, m_{s}= -s, -s+1, ..., s},
the eigenvectors of S^{2} and S_{z}. The state space of the
particle is the tensor product space E = E_{r} ⊗ E_{s}.
The spin observables commute with all the orbital observables.

For the electron E_{s} is two-dimensional. The observables {X,Y,Z,S^{2},S_{z}} form a C.S.C.O.
in E = E_{r} ⊗ E_{s}.
So do the observables {P_{x},P_{y},P_{z},S^{2},S_{z}}.

Since {|**r**>} forms a basis for E_{r} and {|ε>,|ε> = |+>,|->} forms a basis for E_{s}, {|**r**>
⊗ |ε> = |**r**,ε>}
forms a basis for E for a spin ½ particle.

Each vector |**r**,ε>
is a common eigenvector of X, Y, Z, S^{2}, and S_{z}.
We
have

X|**r**,ε> = x|**r**,ε>, Y|**r**,ε> = y|**r**,ε>, Z|**r**,ε> = z|**r**,ε>,
S^{2}|**r**,ε> = (3/4)ħ^{2}|**r**,ε>, S_{z}|**r**,ε> =
εħ|**r**,ε>.

<**r**',ε'|**r**,ε> = δ_{εε'}δ(**r** - **r**'),
(normalization) Σ_{ε}∫d^{3}r|**r**,ε><**r**,ε| =
I, (completeness).

For any vector |ψ> we have |ψ> = Σ_{ε}∫d^{3}r|**r**,ε><**r**,ε|ψ>
= Σ_{ε}∫d^{3}r ψ(**r**)|**r**,ε>, with ψ(**r**) = <**r**,ε|ψ>.

**Notation:
**Let ψ

be a **two component spinor**, with its adjoint [ψ]^{†}
= (ψ_{+}*(**r**),** **ψ_{-}*(**r**)).

For a spin ½ particle |ψ> is
completely specified by the two component spinor, just as for a spinless particle |ψ> is specified by the
wave function ψ(**r**). The inner product is defined through

<ψ|Φ> = Σ_{ε}∫d^{3}r|<ψ|**r**,ε><**r**,ε|Φ> = ∫d^{3}r(ψ_{+}*(**r**)Φ_{+}(**r**)
+ ψ_{-}*(**r**)Φ_{-}(**r**)) = ∫d^{3}r[ψ]^{†}[Φ].

Note: Matrix multiplication of the spinors precedes the spatial integration.

The state is normalized in the following way:

<ψ|ψ> = ∫d^{3}r(ψ_{+}*(**r**)ψ_{+}(**r**) + ψ_{-}*(**r**)ψ_{-}(**r**))
= 1.

The spinor associated with a tensor product vector |ψ> = |Φ> x |Χ>

with |Φ> = ∫d^{3}r Φ(**r**)|**r**> in E_{r}
and |Χ> = c_{+}|+> + c_{-}|-> in E_{s}
is

Note: Not all vectors in E are tensor product vectors.

Let |ψ> be the state of a spin ½ particle and let
|ψ’> be obtained from |ψ>
through the action of a linear operator A, A|ψ> = |ψ’>.

We can associate with each linear operator A
a 2 x 2 matrix [A] such that [A][ψ](**r**) = [ψ’](**r**).

- Let A be a
**spin operator**, say S_{+}. We have,

.

.

The matrix of a spin operator is its matrix in the { |+>, |-> } basis in E

_{s}.Similarly, if A = S

_{z}then.

- Let A be an
**orbital operator**, say X.We have

,

.

The matrix of an orbital operator is diagonal.

- Let A be a
**mixed operator**, say L_{z}S_{z}..

Similarly, if A = XS

_{+}**,**then.

**Rotation operators for a spin ½ particle**

Consider a system consisting of a single spin ½ particle. Neglect the spatial degrees
of freedom. Rotate the system ccw about the z-axis through an angle φ. If the state of the system is |α>
before the rotation, it is |α>_{R} after the
rotation, where |α>_{R }= U(R)|α>, U(R) =
exp(-iS_{z}φ/ħ) = exp(-iσ_{z}φ/2).

Before the rotation the expectation value of the operator S_{x} is <S_{x}> = <α|S_{x}|α>, after the
rotation it is <S_{x}>_{R} = _{R}<α|S_{x}|α>_{R}. We may expand

since σ_{z}^{2} = I. The expression for U^{†}(R)S_{x}U(R)
therefore may be written

In matrix notation we have

.

Therefore _{R}<α|S_{x}|α>_{R} = <α|U^{†}(R)S_{x}U(R)|α>
= <α|S_{x}cosφ - S_{y}sinφ|α>

= <S_{x}>cosφ - <S_{y}>sinφ.

Similarly _{R}<α|S_{y}|α>_{R} = <S_{y}>cosφ
+ <S_{x}>sinφ.

The average value of S_{z} does not change, since it commutes with U(R).
The expectation value of the spin operator **S** behaves as though it were a
classical vector under rotation.

We have <S_{i}>_{R} = ∑_{i}R_{ij}<S_{j}>,

where R is the 3 x 3 rotation matrix for the rotation
being considered. For an arbitrary system we can generalize to

<J_{i}>_{R} = ∑_{i}R_{ij}<J_{j}>.

[Remember **S** is a vector operator.
While the
expectation values of the components of **S** behave as the
components of a classical vector under rotation, the components of **S**
itself behave like a classical vector rotated backward, **S**' = R^{-1}**S**.]

**How does the state vector of the spin ½ particle behave under rotation?
**Let |α> = c

|α>

For φ = 2π we have |α>

A 360

**Can it be observed?
**The minus sign is a change in the phase of the state vector. If all state vectors are
multiplied by a minus sign there will be no observable effect. We can only observe changes
in relative phase. We must compare a rotated ket to an unrotated ket and look for
interference effects. Assume a spin ½ particle can move from point A to point B along two
different path. Assume that when we observe the particle at point B we do not know which
path it actually took to get there. Assume that path 1 does not rotate the state vector
and path 2 does rotate the state vector.

Let {|ψ(1)>,|ψ(2)>}
be the eigenstates of the observable that determines the path of the particle. Assume that at point A the state vector is

|α(0)> = 2^{(-1/2)}(|ψ(1)> + |ψ(2)>).

At point B the state vector is

|α(t)> = 2^{(-1/2)}U(t,t_{0})(|ψ(1)> + |ψ(2)>).

Assume the evolution operator rotates |ψ(2)> but
not |ψ(1)>. If |ψ(2)>
is rotated through 360^{o} then the state vector at point B is

|α(t)> = 2^{(-1/2)}(|ψ(1)> - |ψ(2)>).

The probability of finding the particle at point B is

|<**r**_{B}|α(t)>|^{2 }= ½|<**r**_{B}|ψ(1)> - <**r**_{B}|ψ(2)>|^{2}.

If without rotation the probability of finding the particle at point B is
1, then with rotation we have destructive interference.

**How can we rotate the state vector?
**The Hamiltonian of a spin ½ particle in a magnetic field

The evolution operator therefore is U(t,0) = exp(-iωS

The evolution operator equals the rotation operator with φ = ωt. This explains spin precession. In a magnetic field

<S

<S

Assume that in a neutron interferometry experiment path 1 goes through a field free
region and path 2 goes through a region with a static magnetic field
**B** = B**k**.

Then |α(t)> = 2^{(-1/2)}(|ψ(1)> + exp(-i(ωT/2)|ψ(2)>).
The intensity at
point B is therefore is proportional to C + Dcos(ωT/2).
We can vary the angle φ = ωT
by varying the magnetic field strength. The intensity at point B is predicted to vary
sinusoidally with a period 4π/ω. These
predictions have been experimentally verified.

.

After the rotation the state vector is |ψ>_{R }= U(R)|ψ>, where U(R) =^{ r}U(R)
⊗ ^{s}U(R). U(R) is a **mixed operator**.

The matrix for ^{r}U(R) is
and the matrix for ^{s}U(R) is
.

We therefore have

.

How can we characterize a rotation? We need three real numbers to characterize a
general rotation. We can, for example, specify the direction of the rotation axis (two
angles) and the rotation angle. We can also specify the 3 x 3
rotation matrix R. R has 9 elements. R is an orthogonal matrix. RR^{t }= R^{t}R = I. The orthogonality condition results in a set of 6 independent equations for 9 unknowns.
Therefore R contains 3 independent numbers.

The set of all multiplication operation with orthogonal matrices forms a group. This group has the name SO(3). (S stands for special (determinant = 1), O stands for orthogonal, 3 stands for three dimensions.)

We can also characterize a rotation by specifying the 2 x 2
unitary matrix U(R) which is the matrix representations of the rotation operator in
E_{s}. The most general form of such a rotation matrix is

with |a|^{2} + |b|^{2} = 1 and a and b complex
numbers. Again we have 3 independent elements. The matrices U(R) also form a group.
This group has the name SU(2). (S stands for special, U stands for unitary, 2
stands for dimensionality 2.) The groups SO(3) and SU(2) are not isomorphic.
A 2π rotation and a 4π rotation are
represented by the same R matrix but by different U(R) matrices. U(a,b)
and U(-a,-b) correspond to the same 3 x 3 rotation matrix
in the SO(3) language. For a given R the corresponding U(R) is double
valued.