1000 Solved Problems in Modern Physics

(Grace) #1

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)
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