Quantum physics, I belive Lz is the term for angular momentum.(adsbygoogle = window.adsbygoogle || []).push({});

I have a one dimensional system with a particle with mass M and it's moving a long a circle with radius R.

a)Use Lz=MvR to express the particle kinetic energi with help of Lz. Then use the substituion that [tex]L_z\rightarrow -\frac{i\hbar\partial}{partial\psi}[/tex] to show that [tex]H=\frac{-\hbar ^2}{2MR^2}\cdot\frac{\partial ^2}{\partial\psi ^2}[/tex], H is the hamilton-operator (I guess?).

Am I right if I say that [tex]E_k=\frac{1}{2}Mv^2=\frac{L_z ^2}{2MR^2}[/tex] and then I sumpli put in for Lz?

b) Show that [tex]\Psi _k(\psi )=N_k e^{ik\psi }[/tex] is a solution of the timeindependent Schrödinger-equation and at same time an eigenfunction for Lz. Decide the normalizationconstant Nk.

Am I supposed to put [tex]N_k e^{ik\psi }[/tex] into [tex]E\Psi (\psi)=-\frac{\hbar ^2}{2M}\frac{\partial ^2\Psi (\psi) }{\partial\psi ^2}+V(\psi )\Psi (\psi)[/tex]?

I don't got either E or V, do I? I have no clue what to do with this.

c) Since the particle is living on a circle, we need to identify the angles [tex]\psi[/tex] and [tex]\psi +2\pi[/tex] as same point in space. The wavefunction need to have same vaulue for these two angles. Define the mathematical requirement for wavefunction and show what condition this gives for allowed values of k, write down the quantified values for the particles angularmomentum Lz and energi Ek. What is the degenerasjongrade for Ek.

I am kind of stuck at b) and c). I understand that c) need a cosinus or sinusfunction, maybe if I write out [tex]e^{ik\psi }=\cos(k\psi )+i\sin(k\psi )[/tex]? Wont this force k to have only integrers values to make the values the same by adding 2pi. But how I get the angular momentum and energi I have no frigging clue about.

Help please :) . I must also add that the original text is not written in english, so it's translated by me (therefor it might be sloppy formulations but I did my best)

Edit: For b), I understand now that [tex]H\Psi = E\Psi[/tex] where [tex]E\Psi =-\frac{\hbar ^2}{2m}\cdot\frac{\partial ^2 \Psi}{\partial \psi ^2}+V(\psi )\Psi[/tex], and I guess that I end up with an expression along the forms of V=somethingsomething. But when it comes to normalize it, I honestly have no idea, other then that [tex]\int |\Psi |^2d\psi =1= \int N_k^2 e^{-ik\psi }\cdot e^{ik\psi}d\psi =N_k^2\int e^{ik\psi (-1+1)}d\psi =N_k^2=1[/tex]. Now, I have no idea if this makes sense or not, would be nice if it was one, but I belive it should not be 1.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Quantum physics, particle in a circle

**Physics Forums | Science Articles, Homework Help, Discussion**