1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quantum Harmonic Oscillator

  1. May 22, 2007 #1
  2. jcsd
  3. May 22, 2007 #2

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

  4. May 22, 2007 #3

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

    and yeah, the thing i post is an alternate way to solve it, by using ladder operators. But you can find the solutions for the differential equations needed for solving the Shrödinger equation for this potential in almost any basic QM book.
     
  5. May 22, 2007 #4

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

  6. May 22, 2007 #5
    Ahhhhh, now this make since.

    I have a problem I am working on where there a 2N electrons (of mass m) that are free to move along the x-axis. The potential energy for each electron is U(x)=(1/2)kx^2, where k is a positive constant. I need to find the total enery of the system for a) integer angular momentum particls, and b) half-interger. (all magnetic and electric forces can be ignored.

    so for a), the particles would act like bosons and not be restricted by the exclusion princple, i.e. they would all sit in the same quantum state. That would give a total energy of E=(2N)(1/2)k. (the x^2 can be dropped because all particles will be in the lowest and same state), giving E=Nk. omega=(k/m(r))^(1/2), and I can solve this in terms of k. However, would m(r), the reduced mass, be [2Nm(e)m(n)]/[(2Nm(e)+m(n))]? My thinking is that all electrons would be in one state and can be viewed as a single point mass of 2Nm(e). Can I do that?

    As for part b with half-integer angular momentum, it would be just be E=(n+1/2)(h/2pi)w, which would become E=(n+1/2)(h/2pi)(k/m(r))^(1/2), correct?


    However, this does make alot more sense now.
     
  7. May 23, 2007 #6

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

  8. May 23, 2007 #7
    this is it word for word:"there are 2N electrons (of mass m) that are free to move along the x-axis. The potential energy for each electron is U(x)=(1/2)kx^2, where k is a positive constant. I need to find the total enery of the system for a) integer angular momentum particls, and b) half-interger. (all magnetic and electric forces can be ignored."

    I know that for part (b) i treat it like a quantum harmonic oscillator. However, I am unsure of what the reduced mass would be (although as I understand it electrons are still though of as point masses; therefore i can treate a group of 2N particles as a single mass of 2Nm(e).)
     
  9. May 23, 2007 #8

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

    I do not think I can help you any further =(

    do you by "angular momentum" mean intristinc angular momentum, spin ?
     
  10. May 23, 2007 #9
    ya. one is to think of the electrons has having there normal 1/2 spin, and the other is to look at them as whole integers, i.e. think of the elctrons as bosons and then as fermions.
     
  11. May 23, 2007 #10

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

    Well for the bosons, you just add them up 2N times, all will be in the ground state. 2N(0+1/2)h_bar*omega.

    For the fermions, you get this m_s quantum number (spin "up" or "down"), so there can only be two fermions for each n.

    So you get this sum

    E = 2 * sum{n= 0 to r}((n+1/2)*h_bar*omega)
    were r is N/2 - 1

    this should be right =)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Quantum Harmonic Oscillator
Loading...