Solving Quantum Rigid Rotator Problems - Sam

[Sam Johnson]

Hi! I have been doing some problems to prepare for my physics final and came across this series which I have not been able to solve and was hoping someone here might be able to help me out with.

The problem is that of a two-dimensional rigid rotator which rotates in the xy plane and has angular momentum Lz=-ih ∂/∂t

It is in cylindrical coordinates.


Given:

- hbar^2/(2I) d^2Φ(φ)/(dφ^2) = EΦ(φ)

and

dT(t)/(dt) = -iE(T(t))/hbar

Here E is the separation constant; also, Φ(φ)T(t)=Ψ(φ,t)

First of all we must solve the equation for the time dependence of the wave function just listed.

Second show that the separation constant is the total energy.

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One solution is Φ(φ)=e^imφ where m = sqrt[2IE]/hbar
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Third apply the condition of single-valuedness.

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The allowed values of energy are E=hbar^2m^2/(2I) when abs[m] = 0,1,2...
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Fourth normalize the funcions Φ(φ)=e^imφ found previously.


I SINCERERLY appreciate any help!
Sam

 

[valtorEN]

if Lz=(h/i)*(∂/∂φ), how can u write the energy in terms of the operator Lz?

E=-hbar/(2*I)d^2Φ(φ)/(dφ^2)=EΦ(φ)

 

[Sam Johnson]

All you have to do to find the energy in terms of L is:

E=.5Iω^2 +0potential = .5I(L/I)^2 = L^2/(2mr^2)

because L=Iω and I=mr^2.

Thanks,
Sam