1000 Solved Problems in Modern Physics

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)