Lecture 5: Spin
Lessons from non-relativistic quantum mechanics:
• A = Ψ|A|Ψ expected value
• [ˆrj, ˆpk] = i¯h ⇒ [ˆLx, ˆLy] = i¯hˆLz, [ˆLy, ˆLz] = i¯hˆLx, [ˆLz, ˆLx] = i¯hˆLy commutators
• ˆrjψ = rj · ψ, ˆpjψ = −i¯h ∂
∂rj
ψ operators of position and momentum
• ˆHψ = i¯h ∂
∂t , operator of energy, equation of motion
• free particle: ˆH = (ˆp2
x + ˆp2
y + ˆp2
z )/2m p2/2m is kinetic energy
• electric field: ˆH = (ˆp2
x + ˆp2
y + ˆp2
z )/2m + QV ⇒ ˆH − QV = (ˆp2
x + ˆp2
y + ˆp2
z )/2m
• electromagnetic field: ˆH − QV = ((ˆpx − QAx)2 + (ˆpy − QAy)2 + (ˆpz − QAz)2)/2m,
B = × A
Lessons from relativistic quantum mechanics:
t =
t0
1 − v2/c2
m =
m0
1 − v2/c2
Et = mc2 =
m0c2
1 − v2/c2
, (1)
E2
t − c2p2
x − c2p2
y − c2p2
z = m2
0c4 ⇒ E2
t − c2p2
x − c2p2
y − c2p2
z − m2
0c4 = 0 (2)
¯h2 ∂2
∂t2
− c2¯h2 ∂2
∂z2
− c2¯h2 ∂2
∂x2
− c2¯h2 ∂2
∂y2
+ (m0c2)2 Ψ = 0 (3)
not equation of motion!
i¯h
∂
∂t
ˆγ0 + ic¯h
∂
∂x
ˆγ1 + ic¯h
∂
∂y
ˆγ2 + ic¯h
∂
∂z
ˆγ3 − m0c2ˆ1 Ψ = 0, (4)
equation of motion
electromagnetic field: ˆH − QV = ((ˆpx − QAx)2 + (ˆpy − QAy)2 + (ˆpz − QAz)2)/2m,
B = × A
i¯h
∂
∂t
− QV i¯h
∂
∂t
− QV ˆ1Ψ = i¯h
∂
∂t
− QV
2
Ψ =

c2 i¯h
∂
∂x
+ QAx
2
ˆγ0ˆγ1ˆγ0ˆγ1 + c2 i¯h
∂
∂y
+ QAy
2
ˆγ0ˆγ2ˆγ0ˆγ2 + c2 i¯h
∂
∂z
+ QAz
2
ˆγ0ˆγ3ˆγ0ˆγ3

 Ψ
+m2
0c4ˆγ0ˆγ0Ψ
−m0c3 i¯h
∂
∂x
+ QAx ˆγ0ˆγ1ˆγ0 + i¯h
∂
∂y
+ QAy ˆγ0ˆγ2ˆγ0 + i¯h
∂
∂z
+ QAz ˆγ0ˆγ3ˆγ0 Ψ
−m0c3 i¯h
∂
∂x
+ QAx ˆγ0ˆγ0ˆγ1 + i¯h
∂
∂y
+ QAy ˆγ0ˆγ0ˆγ2 + i¯h
∂
∂z
+ QAz ˆγ0ˆγ0ˆγ3 Ψ
+c2 i¯h
∂
∂x
+ QAx i¯h
∂
∂y
+ QAy ˆγ0ˆγ1ˆγ0ˆγ2 + i¯h
∂
∂y
+ QAy i¯h
∂
∂x
+ QAx ˆγ0ˆγ2ˆγ0ˆγ1 Ψ
+c2 i¯h
∂
∂y
+ QAy i¯h
∂
∂z
+ QAz ˆγ0ˆγ2ˆγ0ˆγ3 + i¯h
∂
∂z
+ QAz i¯h
∂
∂y
+ QAy ˆγ0ˆγ3ˆγ0ˆγ2 Ψ
+c2 i¯h
∂
∂z
+ QAz i¯h
∂
∂x
+ QAx ˆγ0ˆγ3ˆγ0ˆγ1 + i¯h
∂
∂x
+ QAx i¯h
∂
∂z
+ QAz ˆγ0ˆγ1ˆγ0ˆγ3 Ψ (5)
HOMEWORK!
i¯h
∂
∂t
− QV
2 ˆ1 ˆ0
ˆ0 ˆ1
Ψ = (6)

c2 i¯h
∂
∂x
+ QAx
2
+ c2 i¯h
∂
∂y
+ QAy
2
+ c2 i¯h
∂
∂z
+ QAz
2
+ m2
0c4


ˆ1 ˆ0
ˆ0 ˆ1
Ψ (7)
−c2¯hQ Bx
ˆσ1 ˆ0
ˆ0 ˆσ1 + By
ˆσ2 ˆ0
ˆ0 ˆσ2 + Bz
ˆσ3 ˆ0
ˆ0 ˆσ3 Ψ (8)
Approximation for low speed:
(m0c2)2 = E2
t − c2p2 = (m0c2 + E)2 − c2p2 = (m0c2)2 + 2E(m0c2) + E2 − c2p2 (9)
free particle: E is kinetic energy, at low speed: E m0c2
(m0c2)2 ≈ (m0c2)2 + 2E(m0c2) − c2p2 (10)
E =
p2
2m0
(11)
Approximation for low speed:
ˆH ≈
1
2m0

 i¯h
∂
∂x
+ QAx
2
+ i¯h
∂
∂y
+ QAy
2
+ i¯h
∂
∂z
+ QAz
2
+ QV

 0 1
1 0
(12)
−
¯hQ
2m0
Bx
0 1
1 0
+ By
0 −i
i 0
+ Bz
1 0
0 −1
. (13)
E = −µ · B = −(µxBx + µyBy + µzBz) (14)
ˆH = −
¯hQ
2m0
0 1
1 0
Bx +
¯hQ
2m0
0 −i
i 0
By +
¯hQ
2m0
1 0
0 −1
Bz (15)
ˆIxˆIy − ˆIy ˆIx = i¯hˆIz, ˆIy ˆIz − ˆIz ˆIy = i¯hˆIx, ˆIz ˆIx − ˆIxˆIz = i¯hˆIy. (16)
Guess:
γ =
Q
2m
⇒ ˆIx = ¯h
0 1
1 0
ˆIy = ¯h
0 −i
i 0
ˆIz = ¯h
1 0
0 −1
(17)
Check:
ˆIxˆIy − ˆIy ˆIx = ¯h2 0 1
1 0
0 −i
i 0
− ¯h2 0 −i
i 0
0 1
1 0
= (18)
¯h2 i 0
0 −i
−
−i 0
0 i
= 2i¯h2 1 0
0 −1
= 2i¯hˆIz (19)
ˆIxˆIy − ˆIy ˆIx = i¯hˆIz, ˆIy ˆIz − ˆIz ˆIy = i¯hˆIx, ˆIz ˆIx − ˆIxˆIz = i¯hˆIy. (20)
New guess:
γ =
Q
m
⇒ ˆIx =
¯h
2
0 1
1 0
ˆIy =
¯h
2
0 −i
i 0
ˆIz =
¯h
2
1 0
0 −1
(21)
Check:
ˆIxˆIy − ˆIy ˆIx =
¯h2
4
0 1
1 0
0 −i
i 0
−
¯h2
4
0 −i
i 0
0 1
1 0
= (22)
¯h2
4
i 0
0 −i
−
−i 0
0 i
= i
¯h2
2
1 0
0 −1
= i¯hˆIz (23)
OperatorEigenfunction = Eigenvalue · Eigenfunction (24)
¯h
2
1 0
0 −1
1
h3
ψ
0
=
¯h
2
1
h3
ψ
0
(25)
¯h
2
1 0
0 −1
1
h3
0
ψ
= −
¯h
2
1
h3
0
ψ
(26)
OperatorEigenfunction = Eigenvalue · Eigenfunction (27)
¯h
2
1 0
0 −1
1
h3
ψ
1
0
=
¯h
2
1
h3
ψ
1
0
(28)
¯h
2
1 0
0 −1
1
h3
ψ
0
1
= −
¯h
2
1
h3
ψ
0
1
(29)
OperatorEigenvector = Eigenvalue · Eigenvector (30)
¯h
2
1 0
0 −1
1
0
=
¯h
2
1
0
(31)
¯h
2
1 0
0 −1
0
1
= −
¯h
2
0
1
(32)
• If the particle is in state |α , the result of measuring Iz is always +¯h/2. The expected value is
Iz = α|Iz|α = 1 0
¯h
2
1 0
0 −1
1
0
= +
¯h
2
. (33)
• If the particle is in state |β , the result of measuring Iz is always −¯h/2. The expected value is
Iz = β|Iz|β = 0 1
¯h
2
1 0
0 −1
0
1
= −
¯h
2
. (34)
• Any state cα|α + cβ|β is possible, but the result of a single measurement of Iz is always +¯h/2 or
−¯h/2. However, the expected value of Iz is
Iz = α|Iz|β = c∗
α c∗
β
¯h
2
1 0
0 −1
cα
cβ
= (|cα|2 − |cβ|2)
¯h
2
. (35)
|Ψ = |α
−γ¯h
2
+γ¯h
2
1 2 3 4 5 6 7 8 9 10
µz = +γ¯h
2
µz
|Ψ = |β
−γ¯h
2
+γ¯h
2
1 2 3 4 5 6 7 8 9 10
µz = −γ¯h
2
µz
Measurement number
|Ψ = 1√
2
|α + 1√
2
|β
−γ¯h
2
+γ¯h
2
1 2 3 4 5 6 7 8 9 10
µz = 0
µz
Measurement number
|Ψ (t = 0) = |α ; ˆH = −γB0ˆIz = ω0ˆIz
0
1
Pα|Ψ (t = 0) = |β ; ˆH = −γB0ˆIz = ω0ˆIz
0
1
Pα
|Ψ (t = 0) = |α ; ˆH = −γB1ˆIx = ω1ˆIx
0
1 1/ω1
Pα
t
• The states described by basis functions which are eigenfunctions of the Hamiltonian do not evolve
(are stationary).
• It makes sense to draw energy level diagram for such states, with energy of each state given by
the corresponding eigenvalue of the Hamiltonian.
• Energy of the |α state is −¯hω0/2 and energy of the |β state is +¯hω0/2.
• The measurable quantity is the energy difference ¯hω0, corresponding to the angular frequency ω0.
• The states described by basis functions different from eigenfunctions of the Hamiltonian are not
stationary
• They oscillate between |α and |β with the angular frequency ω1, given by the difference of the
eigenvalues of the Hamiltonian (−¯hω1/2 and ¯hω1/2).
• It should be stressed that eigenstates of individual magnetic moments are not eigenstates of the
macroscopic ensembles of nuclear magnetic moments.
• Eigenstates of individual magnetic moments do not determine the possible result of measurement
of bulk magnetization.