Particle in Ring (Griffiths)

Consider a particle with mass $\mu$ constrained to move on a circle (radius $r_0$). The wave function of the particle is

$\psi(\phi)=N/(3+\sin(2\phi))$

a) What is the normalization constant?

b) What is the expectation value of $L_z$

c) Find the expectation value of the energy.

2 D polar coordinates:

$x=r_0*\cos⁡(φ)$, $y=r_0*\sin⁡(φ)$
$dx=-r_0*\sin⁡(φ)*dφ,dy=+r_0*\cos⁡(φ)*dφ$

$∇^2 ψ=1/r*∂/∂r (r*∂ψ/∂r)+1/r^2 *(∂^2 ψ)/(∂φ^2 )=1/(r_0^2 )*(∂^2 ψ)/(∂φ^2 )$

Normalization constant:

$1=N*∫_0^2π dφ/(3+\sin⁡(2φ))=N*π/√2$,that is $N=√2/π$

b)
$Lz=x*p_y-y*p_z=-ih_bar*(x*∂/∂y-y*∂/∂x)=-2ih_bar*∂/∂φ$

$L_z=<ψ|(L_z ) ̂ |ψ>$

$|(L_z |) ̂ ψ> =(-2i\hbar)(-√2/π*(2*\cos⁡(2φ))/(\sin⁡(2φ)+3)^2 )$

$L_z=(4i\hbar)/π^2 *∫_0^{2π} 1/((3+\sin⁡(2φ))*(2*\cos⁡(2φ))/((3+\sin⁡(2φ)^2 )*dφ=0$

Particle is free to move. It means $V(r,φ)=0$. Hamiltonian is

$H=p^2/2m=-(\hbar^2)/2m*∇^2=-(\hbar^2)/2m*1/r_0^2 *∂^2/(∂φ^2 )$

$|H ̂ |ψ> =-(\hbar^2)/(2m r_0^2 )*√2/π*(2(6*\sin⁡(2φ)+\cos⁡(4φ)+3))/[\sin⁡(2φ)+3]^3$

$E=<ψ|H|ψ>= -(\hbar^2)/(2mr_0^2 )*4/π^2 ∫_0^2π (6 \sin⁡(2φ)+\cos⁡(4φ)+3)/(\sin⁡(2φ)+3)^4 *dφ=$

$=0.10413*(2\hbar^2)/(mr_0^2 π^2 )$

Reference