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Quantum physics, particle in a circle

  1. May 2, 2010 #1
    Quantum physics, I belive Lz is the term for angular momentum.
    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.
    Last edited: May 2, 2010
  2. jcsd
  3. May 2, 2010 #2
    This is quite similar to the particle in a box problem. Also, your last integral is incorrect as you suspected, [itex]N_k^2\int_0^{2\pi}d\phi = 2\pi N_k^2[/itex]. Finding N_k should not be very hard :-)
    Last edited: May 2, 2010
  4. May 2, 2010 #3
    Aha! It makes sence :p . I thought the integral was from -inf to inf like normal. That [tex]N=\sqrt{\frac{1}{2\pi}}[/tex] makes more sence, it even fits what my book says, I think.

    Then I guess there is only c) left, and the problem is that I am pretty certain the solution is that psi is 2pi, and k is integrers, and then how the heck do I express quantified angular momentum and the kinetic energy out of this? And what is degeneration (freely translated from anotjher language)-value for E? Can't recall even hearing about it :(
  5. May 2, 2010 #4
    Angular momentum:
    I believe you should have shown in b) that the stationary wavefunction is a eigenfunction for the angular momentum operator. Your eigenvalue should contain your quantization number k.

    Use the hamiltonian operator on the stationary wavefunction, and you should find a expression for the energy containing your quantization number k. Again, in the solution to b) you should already have this.

    First understand that a degenerate state has two or more distinct solutions to the Schrödinger equation with the same energy E. This is a one-dimensional problem, so you should ask yourself if two distinct solutions with the same energy is possible or not?
  6. May 2, 2010 #5
    There is, however, a small problem with b) . It's the calculations, I am just assuming that the task is easy, which is why I haven't really bothered with the derivations to a big degree (the mathematical problems isn't to hard for me). Is it fair to assume that V(\psi)=0? Or am I supposed to, when putting in to Schrödinger gonna end up with some constant C and a expression on form C\Psi=V\Psi, where [tex]C=\frac{\hbar ^2 N_k k^2}{2M}(\frac{1}{R^2}-1)=V[/tex], but as far as I am concerned, the symbols inside the () are unsubtractable. My calculations:
    [tex]E\Psi = \frac{-\hbar ^2}{2M}\frac{\partial ^2 \Psi}{\partial \psi ^2}+V\Psi[/tex]
    [tex]EN_ke^{-ik\psi} = \frac{-\hbar ^2}{2M}\frac{\partial ^2 N_ke^{-ik\psi}}{\partial \psi ^2}+VN_ke^{-ik\psi}[/tex]
    Just removing the Psi from here, and putting in the thingy for Ek I found in b) , and by someplace, the R^2 comes in and ruins eveerything :(
    [tex]-\frac{\hbar ^2}{2MR^2}(-k^2)=-\frac{\hbar ^2}{2M}(-k^2)+V[/tex], now, V is the function (am I supposed tok now this or what?).
    And to show that Lz is an eigenfunction thingy, I ended up with [tex]Lzf=\lambda Lz=-k\hbar \Psi[/tex], which makes not to much sence from what I have gathered, unless if it's something vital (Again) I am missing :(
  7. May 2, 2010 #6
    The particle is moving in a circle. Think about the gravitational potential in a circular orbit centered on a mass, is it not the same everywhere? And if the potential is constant, with reference to what? You can safely set V = 0.

    Use your Hamiltonian (derive twice, set H\psi = E\psi, divide by \psi and lo and behold, you have an expression for E!). Edit: Also, regarding degeneracy; can you, by choosing k wisely, find two wavefunctions with the same energy? Are they linearly independent?

    Angular momentum:
    Apart from the incorrect sign (-1 goes against i^2), you have Lz\psi = k\hbar\psi. Now divide by \psi and you have a very nice expression for the angular momentum...
    Last edited: May 2, 2010
  8. May 2, 2010 #7
    Oh, that's not to bad. I really feel like a physisist when it comes to having a mess in my papers and writing.
    And since it's the squared of k, the positive and negative version will provide the same energi. And, aaarhh, linearly independency, I hate that stuff. If you mean you can express the two different k's energies in two different ways? No I'm unable to do that :(

    While I have you here :p
    I also have another question.
    'We now assume that there are two particles and these are 1/2-fermions (free translation by me). We ignore how they work on each other (guessing ignoring gravity n stuff). '
    Formulate Paulis Exlusionsprinciple (translated by me): This one is simple, that two particles can't be in same state and spin etc. Basically unable to have same four quantum numbers at the same time.
    Then the task is 'Assume that the two particles are in different angular momentum-state, k1 and k2, where k1 differs from k2. Write down the possible two-particle-wavefunctions. Spesify both room and spin-part. Normalize the room part, write down eigenvalue of the state'.
    I am lost here, I can't find any place where the spin is related to angular momentum here, and confused about the two-particle-wavefunction-thingy,
    Last edited: May 2, 2010
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