Lecture 33:
Quantum Mechanical Spin
Phy851 Fall 2009Intrinsic Spin
• Empirically, we have found that most particles have an additional internal degree of freedom, called ‘spin’
• The Stern-Gerlach experiment (1922):
• Each type of particle has a discrete number of internal states:
– 2 states --> spin _ – 3 states --> spin 1 – Etc….
Interpretation
• It is best to think of spin as just an additional quantum number needed to specify the state of a particle.
– Within the Dirac formalism, this is relatively simple and requires no new physical concepts
• The physical meaning of spin is not well-understood
• Fro Dirac eq. we find that for QM to be Lorentz invariant requires particles to have both anti-particles and spin.
• The ‘spin’ of a particle is a form of angular momentum
Spin Operators
• Spin is described by a vector operator:
• The components satisfy angular
momentum commutation relations:
• This means simultaneous eigenstates of S2 and S z exist: z z y y x x
e
S
e
S
e
S
S
r
=
r
+
r
+
r
y x z x z y z y xS
i
S
S
S
i
S
S
S
i
S
S
h
h
h
=
=
=
]
,
[
]
,
[
]
,
[
2 2 2 2 z y xS
S
S
S
=
+
+
s ss
s
s
m
m
s
S
2,
=
h
2(
+
1
)
,
s s zs
m
m
s
m
S
,
=
h
,
Allowed quantum numbers
• For any set of 3 operators satisfying theangular momentum algebra, the allowed values of the quantum numbers are:
• For orbital angular momentum, the
allowed values were further restricted to only integer values by the requirement that the wavefunction be single-valued • For spin, the quantum number, s, can
only take on one value
– The value depends on the type of particle – S=0: Higgs
– s=1/2: Electrons, positrons, protons, neutrons, muons,neutrinos, quarks,… – s=1: Photons, W, Z, Gluon – s=2: graviton
{
0
,
,
1
,
2,
K
}
3 2 1∈
j
{
j
j
j
}
m
j∈
−
,
−
+
1
,
K
,
{
s
s
s
}
m
s∈
−
,
−
+
1
,
K
,
Complete single particle basis
• A set of 5 commuting operators whichdescribe the independent observables of a single particle are:
– Or equivalently:
• Some possible basis choices:
• When dealing with a single-particle, it is permissible to drop the s quantum
number z
S
S
R
r
,
2,
z zS
S
L
L
R
,
2,
, 2,
{
r
r
,
s
,
m
s}
{
p
r
,
s
,
m
s}
{
r
,
l
,
m
l,
s
,
m
s}
{
n
,
l
,
m
l,
s
,
m
s}
Intrinsic Magnetic Dipole
Moment
• Due to spin, an electron has an intrinsic magnetic dipole moment:
– ge is the electron g-factor – For an electron, we have:
– The is the most precisely measured physical quantity
• For most purposes, we can take ge≈ 2, so that
• For any charged particle we have:
S
m
e
g
e e er
r
2
−
=
µ
015
0000000000
.
0
622
0023193043
.
2
±
=
eg
S
m
e
e er
r
−
=
µ
S
M
q
g
r
r
2
=
µ
has a differentEach particleHamiltonian for an electron in
a magnetic field
• Because the electron is a point-particle, the dipole-approximation is always valid for the spin degree of freedom
• Any `kinetic’ energy associated with S2 is absorbed into the rest mass
• To obtain the full Hamiltonian of an electron, we must add a single term:
)
(
R
B
S
m
e
H
H
er
r
r
⋅
+
→
[
(
)
]
(
)
(
)
2
1
2R
B
S
m
e
R
Ö
e
R
A
e
P
m
H
e er
r
r
r
r
r
r
⋅
+
−
+
=
Uniform Weak Magnetic Field
as a perturbation
• For a weak uniform field, we find
• With the addition of a spherically symmetric potential, this gives:
– If the zero-field eigenstates are known
– The weak-uniform-field eigenstates are:
(
z z)
e eS
L
m
B
e
m
P
H
2
2
2
0 2+
+
=
(
z z)
e eS
L
m
B
e
R
V
m
P
H
2
2
)
(
2
0 2+
+
+
=
l ll
l
m
E
n
m
n
H
0,
,
=
0,n,
,
{
n
,
l
,
m
l,
m
s}
(
s)
B n m m nE
B
m
m
E
s 0, 02
, , l=
+
µ
l+
s m m n sE
n
m
m
m
m
n
H
s,
,
,
,
,
,
l
l , ,l
l l=
e Bm
e
2
h
=
µ
‘Bohr Magneton’Wavefunctions
• In Dirac notation, all spin does is add two extra quantum numbers
• The separate concept of a ‘spinor’ is unnecessary • Coordinate basis: – Eigenstate of • Projector: • Wavefunction:
{
r
r
,
s
,
m
s}
zS
S
R
r
,
2,
s s s s m Vm
s
r
m
s
r
r
d
I
s,
,
,
,
3r
r
∑ ∫
− ==
( )
ψ
ψ
msr
r
:
=
r
r
,
s
,
m
sSpinor Notation:
• We think of them as components of a length 2s+1 vector, where each
component is a wavefunction • Example: s=1/2
• Spinor wavefunction definition:
• If external and internal motions are not entangled, we can factorize the spinor wavefunction:
( )
ψ
ψ
msr
r
:
=
r
r
,
s
,
m
s( )
ψ
ψ
↑r
r
:
=
r
r
,
s
,
12( )
ψ
ψ
↓r
r
:
=
r
r
,
s
,
−
12[ ]
( )
( )
( )
=
↓ ↑r
r
r
r
r
r
ψ
ψ
ψ
:
( )
( )
=
−r
r
r
r
2 1 2 1ψ
ψ
Note that descending order is unusual[ ]
( )
( )
r
c
c
r
r
ψ
r
ψ
=
↓ ↑:
↓ ↑c
c
Is then a pure spinorSchrödinger's Equation
• We start from:• Hit from left with with
• Insert the projector
• Let: – For s=1/2:
ψ
ψ
H
dt
d
i
h
=
sm
r
r
,
ψ
ψ
r
m
H
m
r
dt
d
i
h
r
,
s=
r
,
sψ
ψ
d
r
r
m
H
r
m
r
m
m
r
dt
d
i
s s s s m V s s′
′
′
′
′
=
∑ ∫
− = ′,
,
,
,
3r
r
r
r
h
[ ]
[ ] [ ]
[ ]
(
)
2
2R
V
I
M
P
H
=
+
r
[ ]
[ ]
= 1 0 0 1 I[ ]
[ ]
=
↓↓ ↓↑ ↑↓ ↑↑)
(
)
(
)
(
)
(
)
(
r
V
r
V
r
V
r
V
r
V
r
r
r
r
r
[ ]
( )
[ ]
( )
r
[ ]
[ ]
V
( )
r
[ ]
( )
r
M
r
dt
d
i
h
ψ
r
=
−
h
∇
2ψ
r
+
r
ψ
r
22
Example: Electron in a Uniform
Field
[ ]
( )
[ ]
( )
r
[ ]
[ ]
V
( )
r
[ ]
( )
r
M
r
dt
d
i
h
ψ
r
=
−
h
∇
2ψ
r
+
r
ψ
r
22
( )
( )
( )
( )
( )
( )
− + ∂ ∂ − ∇ − = ↓ ↑ ↓ ↑ ↓ ↑ r r B r r B i m r r dt d i B B e r r r r h r r h ψ ψ µ ψ ψ φ µ ψ ψ 1 0 0 1 2 0 0 2 2( )
( )
( )
( )
i
B
( )
r
B
( )
r
m
r
dt
d
i
r
B
r
B
i
m
r
dt
d
i
B B e B B er
r
h
r
h
r
r
h
r
h
↓ ↓ ↓ ↑ ↑ ↑−
∂
∂
−
∇
−
=
+
∂
∂
−
∇
−
=
ψ
µ
ψ
φ
µ
ψ
ψ
µ
ψ
φ
µ
ψ
0 0 2 2 0 0 2 22
2
• This is just a representation of two separate equations:
• We would have arrived at these same equations using Dirac notation, without ever mentioning ‘Spinors’
Pauli Matrices
• Here we see that we have recovered one of the Pauli Matrices:
• The other Pauli matrices are:
• Then in the basis of eigenstates of we have:
• If we only care about spin dynamics:
( )
( )
( )
( )
( )
( )
− + ∂ ∂ − ∇ − = ↓ ↑ ↓ ↑ ↓ ↑ r r B r r B i m r r dt d i B B e r r r r h r r h ψ ψ µ ψ ψ φ µ ψ ψ 1 0 0 1 2 0 0 2 2
−
=
1
0
0
1
zσ
=
0
1
1
0
xσ
−
=
0
0
i
i
yσ
zS
σ
r
h
r
2
=
S
[ ]
( )
B
i
[ ]
( )
r
M
r
dt
d
i
h
r
h
Bσ
zψ
r
φ
µ
ψ
+
∂
∂
−
+
∇
−
=
2 0 22
[ ]
B
[ ]
c
m
e
c
dt
d
i
er
r
⋅
=
σ
2
Two particles with spin
• How do we treat a system of twoparticles with masses M1 and M2, charges q1 and q2, and spins s1 and s2?
– Basis:
– Wavefunction:
– Hamiltonian w/out motional degrees of freedom:
– Hamiltonian w/ motional degrees of freedom: 2 2 2 1 1 1
,
s
,
m
s;
r
,
s
,
m
sr
r
r
ψ
ψ
s1,ms1,s2,ms2(
r
r
1,
r
r
2)
:
=
r
r
1,
s
1,
m
s1;
r
r
2,
s
2,
m
s2)
(
)
(
2 2 2 2 1 1 1 1S
B
R
M
q
R
B
S
M
q
H
=
−
r
⋅
r
r
−
r
⋅
r
r
(
)
(
(
)
)
(
)
(
)
2
1
)
(
)
(
)
(
2
1
2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1R
B
S
M
q
R
q
R
A
q
P
M
R
B
S
M
q
R
q
R
A
q
P
M
H
r
r
r
r
r
r
r
r
r
r
r
r
r
r
⋅
−
Φ
+
−
⋅
−
Φ
+
−
=
Example #1
• A spin _ particle is in the ↑ state with respect to the z-axis. What is the
probability of finding it in the ↓-state with respect to the x-axis?
• Let:
• In the basis, the operator for the x-component of spin is:
• By symmetry,
σ
x must have eigenvalues +1 and -1• The eigenvector corresponding to -1 is defined by: z
↑
=
ψ
{
↑
z,
↓
z}
=
0
1
1
0
xσ
x x x↓
=
−
↓
σ
Example #1 continued:
• This implies that:
=
0
1
1
0
xσ
σ
x↓
x=
−
↓
x x x z x z↓
=
−
↑
↓
↑
σ
x x x=
−
↓
↓
σ
x z↓
↓
−
=
(
z z)
x=
↑
−
↓
↓
2
1
1
−
=
2
1
2=
↑
↓
=
x zP
Example #2
• Two identical spin-1/2 particles are placed in a uniform magnetic field. Ignoring
motional degrees of freedom, what are the energy-levels and degeneracies of the
system? • States:
– Z-axis chosen along B-field • Hamiltonian:
• Basis states are already eigenstates:
