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Rigid Box and 3D Schrodinger equation

  1. Oct 15, 2012 #1
    1. The problem statement, all variables and given/known data
    An electron is confined within a three-dimentional cubic region the size of an atom where L = 200 pm.
    a) write a wave equation for the electron
    b) wirte a general wave function for the possbile states of the electorn. List any quantum numbers and their possible values.
    c) calculate the energy of the four lowest states
    d) calculate the energies and wavelength of photons created during transitions between these states

    2. Relevant equations
    VcWiK.jpg
    and A = (2/L)^(3/2)

    3. The attempt at a solution
    The note above is from my lecture note. I think they probably work for this question.

    a, c)
    7nWFc.jpg
    b) I am confused here.
    d) Bn5Pe.jpg
     
    Last edited: Oct 15, 2012
  2. jcsd
  3. Oct 15, 2012 #2

    Simon Bridge

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    I think
    (a) means the Schrodinger equation ... it will have to be peicewise.
    (b) is the wave-function you have written ... if this is long answers you'll have to justify doing ##\psi_{k}(x,y,z)=\psi_l(x)\psi_m(y)\psi_n(z)##
    (c) ... look for states k=l+m+n with lowest energy.
    (d) ... once you have (c) this is just subtraction

    I think your reasoning in the later part is OK - from what I can make out - I'd prefer you to show more thinking for the first parts since it just looks like you are copying from lectures. You should try to show that you have understood the lecture to get full marks, and this usually means you have to write sentences as well as equations.
     
  4. Oct 15, 2012 #3
    Hi Simon, thanks for helping me again.

    wave equation is the schrodinger equation? and wave function is the solution?
     
  5. Oct 15, 2012 #4

    Simon Bridge

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    How else do we account for the first two questions.

    The (time independent) SE is a special case of the Helmholtz equation which is the time-independent part of a wave equation.
    (Therefore, the statistics described by solutions to the SE will behave like waves.)

    I suspect the answer for the wave equation should be the time-dependent SE ... I'd start by writing out $$\left ( \nabla^2+V(\vec{r})-i\hbar\frac{\partial}{\partial t}\right )\Psi(\vec{r},t)=0$$... then get more explicit for ##V##.

    The next question is asking for ##\Psi(\vec{r},t)## ...

    See why I think your prof is seeing if you have understood the lectures?
     
  6. Oct 16, 2012 #5
    Hi Simon, I'll come back and think more after my coming exam.
     
  7. Oct 17, 2012 #6

    Simon Bridge

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