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3 Quantum Mechanics – II
Table 3.1 Dynamic quantities and operators
Physical Quantity
Operator
Position
r
R
Momentum
P
−i∇
Kinetic energy
T
−
2
2μ
∇
2
Potential energy
V
V (r)
Angular momentum square
L
2
l(l + 1)
2
z-component of angular momentum
L z
−i
∂
∂φ
Expectation values of dynamical variables and operators
An arbitrary function of r has the expectation value
< f (r)>=
ψ
∗ f (r)ψ dτ
(3.7)
The expectation value of P
< P >=
ψ
∗
i
∇ψ
dτ
(3.8)
The expectation value of the kinetic energy
< T>=
ψ
∗
−
2
2μ
∇
2
ψ
dτ
(3.9)
Pauli spin matrices
σ x =
0 1
1 0
, σ y =
0 −i
i 0
, σ z =
1 0
0 −1
(3.10)
σ
2
x = σ
2
y = σ
2
z = 1
(3.11a)
σ x σ y = iσ z , σ y σ z = iσ x , σ z σ x = iσ y
(3.11b)
These matrices are both Hermetian and unitary. Further, any two Pauli matrices
anticommute
σ x σ y + σ y σ x = 0, etc.
(3.11c)