1000 Solved Problems in Modern Physics

(Grace) #1

124 2 Quantum Mechanics – I


2.68 (a) Letf=eiA; thenf†=


(

eiA

)†

=e−iA
Thereforef†f=e−iAeiA= 1
ThuseiAis unitary.
(b) (a) Momentum (b) Parity

2.69 (a) exp(iσxθ)= 1 +iσxθ+(iσxθ)^2 /2!+(iσxθ)^3 /3!+···
=(1−θ^2 /2!+θ^4 /4!...)+iσx(θ−θ^3 /3!+θ^5 /5!....)
=cosθ+iσxsinθ
(where we have used the identityσx^2 =1)


(b)


φ∗(dψ/dx)dx=φ∗ψ−


(dφ∗/dx)ψdx

Butφ∗ψ= 0
Hence


φ∗

(


dx

)

dx=

∫(

−dφ


dx

)

ψdx=


−(dφ/dx)†ψdx
Therefore

(d
dx

)†

=−d/dx
2.70 (a) [x,Px]ψ=xPxψ−Pxxψ


=x

(

−i


∂x

)

ψ+i


∂x

(xψ)

=−ix

∂ψ
∂x

+ix

∂ψ
∂x

+iψ
=iψ
∴[x,Px]=i
(b) [x^2 ,Px]ψ=x^2 (−i∂ψ/∂x+i


∂x

(x^2 ψ)

=−ix^2 ∂ψ/∂x+ix^2 ∂ψ/∂x+i(2x)ψ
= 2 ixψ
Therefore [x^2 ,px]= 2 ix

2.71 By definition a transformationAis said to be linear if for any constant (possi-
bly complex)λ
A(λX)=λAX
And if for any two vectorsxandy
A(x+y)=Ax+Ay
IfHis a hermitian operator
(x,Hλy)=(Hx,λy)=λ(Hx,y)=λ(x,Hy)=(x,λHy)
OrHλy=λHy
for anyy. Furthermore
(z,H(x+y))=(Hz,x+y)=(Hz,x)+(Hz,y)
=(z,Hx)+(z,Hy)=(z,Hx+Hy)
∴ H(x+y)=Hx+Hy


2.72 Consider the equation



∂x

(ψ∗ψ)=ψ∗

∂ψ
∂x


dψ∗
dx

(1)
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