Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I 2D LHO

  1. Dec 12, 2017 #1
    A question I have faced in exam to calculate ground state energy
    Given Hamiltonian
    1/2m(px2+py2)+1/4mw2(5x^2+5y^2+6xy)
    ground state energy has to be obtained
    Its clear that the Hamiltonian is a 2D LHO Hamiltonian but what for the term 3/4(x+y)2
     
  2. jcsd
  3. Dec 13, 2017 #2

    mfb

    User Avatar
    2017 Award

    Staff: Mentor

    Where does the last term come from?

    It can be interesting to introduce mew variables for the sum and difference of x and y.
     
  4. Dec 13, 2017 #3
    I have broken the terms and got hamiltonian of a 2D LHO plus that term
     
  5. Dec 13, 2017 #4

    mfb

    User Avatar
    2017 Award

    Staff: Mentor

    It would help if you could show the work instead of describing it like that.
     
  6. Dec 13, 2017 #5
    H=1/2m(px2+py2)+mw2/2(x2+y2)+3/4(x2+y2+2xy)
    where first two terms are of 2D LHO ,and there is the last term
    The ground state energy is to be calculated
     
  7. Dec 13, 2017 #6

    DrDu

    User Avatar
    Science Advisor

    The kinetic term is invariant under an orthogonal transform of the variables x and y. Can you use this to bring the potential term to a more standard form?
     
  8. Dec 13, 2017 #7
    Mr Dr Du will you please give some kind of hints to solve this problem
     
  9. Dec 13, 2017 #8
    For the term containing xy if I directly calculate the expectation value using the ground state wave function the answer is coming as 3mw2/8 which actually dimensionally doesn't match
    Since <Φ0(x)Φ0(y)I3mw2xy/2IΦ0(x)Φ0(y)>
    expectation value of x and y is 1/2​

    Is there any discrepancy regarding this
     
    Last edited by a moderator: Dec 13, 2017
  10. Dec 13, 2017 #9

    mfb

    User Avatar
    2017 Award

    Staff: Mentor

    The post was a hint. My first post said basically the same, just a bit more direct. Did you try that?
    You didn't find the ground state wave function yet.
    The function is symmetric with respect of x and y, the expectation value for both is zero. They are correlated, the expectation value of xy doesn't have to be zero. But you are not yet at the step where you can calculate these things.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted