1000 Solved Problems in Modern Physics

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∗

(

dψ

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

+ψ

dψ∗

dx

(1)