132 3 Quantum Mechanics – II
Table 3.1Dynamic quantities and operators
Physical Quantity Operator
Position r R
Momentum P −i∇
Kinetic energy T −
^2
2 μ
∇^2
Potential energy VV(r)
Angular momentum square L^2 l(l+1)^2
z-component of angular momentum Lz −i
∂
∂φ
Expectation values of dynamical variables and operators
An arbitrary function of r has the expectation value
<f(r)>=
∫
ψ∗f(r)ψdτ (3.7)
The expectation value of P
=
∫
ψ∗
(
i
∇ψ
)
dτ (3.8)
The expectation value of the kinetic energy
=
∫
ψ∗
(
−
^2
2 μ
∇^2 ψ
)
dτ (3.9)
Pauli spin matrices
σx=
(
01
10
)
,σy=
(
0 −i
i 0
)
,σz=
(
10
0 − 1
)
(3.10)
σx^2 =σy^2 =σz^2 = 1 (3.11a)
σxσy=iσz,σyσz=iσx,σzσx=iσy (3.11b)
These matrices are both Hermetian and unitary. Further, any two Pauli matrices
anticommute
σxσy+σyσx= 0 ,etc. (3.11c)