Results are out! Find what you need...fast. Get quick advice or join the chat
x

Unlock these great extras with your FREE membership

  • One-on-one advice about results day and Clearing
  • Free access to our personal statement wizard
  • Customise TSR to suit how you want to use it

Showing two 1D box potentials merge into a 2D

Announcements Posted on
Rate your uni — help us build a league table based on real student views 19-08-2015
  1. Offline

    ReputationRep:
    Click here for question


    So I showed

    U_1(x) = \begin{Bmatrix} 0, \ 0<x<a \\ 0, \ ow

    is represented by the eigenfunction

    \phi_n(x) = \sqrt{\dfrac{2}{a}} sin \dfrac{n \pi x}{a}, \ n=1,2,3...

    with eigenenergies

    E_n = \dfrac{\hbar^2 n^2 \pi ^2}{2ma^2}

    Then to show that the 2D box

    U_2(x) = \begin{Bmatrix} 0, \ 0<x<a, \ 0<y<a \\ 0, \ ow

    is represented by the eigenfunction

    \psi_{{n_x},{n_y}}(x,y) = \phi_{n_x}(x) \phi_{n_y}(y) = \dfrac{2}{a}sin \dfrac{n_x \pi x}{a} sin \dfrac{n_y \pi y}{a}

    with eigenenergies

    E_{{n_x},{n_y}} = E_{n_x}+E_{n_y} = \dfrac{\hbar^2 \pi^2}{2ma^2}(n_x^2 + n_y^2)

    Is it sufficient to simply state that by separating the variables, and letting \psi_{{n_x},{n_y}} = f(x)g(y), one can separate the 2D box into two independent 1D boxes?

    Just to clarify here, I'm not explicitly asked to separate the 2D box, but to show that two 1D boxes will combine into a 2D box... so is it okay to just say that, "since we can seperate the 2D box eigenfunction into two 1D box eigenfunctions, it is clear that the 2D eigenfunction is a combination of two 1D eigenfunctions."
  2. Offline

    ReputationRep:
    Yeah, just assume you can always separate variables.

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?
  2. this can't be left blank
    this email is already registered. Forgotten your password?
  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. By joining you agree to our Ts and Cs, privacy policy and site rules

  2. Slide to join now Processing…

Updated: May 11, 2012
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

New on TSR

Rate your uni

Help build a new league table

Poll
How do you read?
Study resources
Quick reply
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.