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Question:

A particle of mass m is in the symmetric well
$$ V(x) = \begin{cases}\infty & x < 0\\V_0 & 0 < x <a \\ 0 & a< x <2a \\ V_0 & 2a < x <3a \\ \infty & 3a < x \end{cases} $$ The ground state happens to have energy $E_1 = V_0 / 2 $.

  • (a) what conditions must the wave function satisfy at each of the potential step $ x = 0, x=a, x=2a, x=3a $.

  • (b) Sketch the corresponding ground state wave function $\psi_1(x)$. Indicate the energy $E_2$ and sketch the wave function $\psi_2(x)$ of the first excited state.

  • (c) Indicate the ground state energy and sketch the ground state wave function for. 1. a particle of mass $2m$ in the same potential. 2. a particle of mass $m/2$ in the same potential.


I have a mediocre idea on the conditions the wave function must satisfy at each step. But I not so sure if I miss anything.

At $x=0, x=3a$ $\psi(x)$ continuous at infinite step.

At $x=a, x=2a$ $\psi(x)$ and $\psi'(x)$ continuous at finite step.

In part b, I think I need to work out the its potential for different region as below but I not so sure what to do next.

$$ \psi(x) = \begin{cases}0 & x < 0\\Ce^{q(x-a)} + De^{-q(x-a)} & 0 < x <a \\ A\sin(kx) + B\cos(kx)& a< x <2a \\ Ce^{q(x-a)} + De^{-q(x-a)} & 2a < x <3a \\ 0& 3a < x \end{cases} $$

any guidance or solution would be great. Thanks

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    $\begingroup$ Welcome to the site! Please take a minute to read our guidelines for homework and exercise questions as well as check-my-work questions. We intend our questions to be potentially useful to a broader set of users than just the one asking, and we prefer conceptual questions over those just asking for a specific computation. $\endgroup$ – Emilio Pisanty Aug 28 '18 at 10:22
  • $\begingroup$ Have you tried using your continuity conditions from part a in part b? $\endgroup$ – Aaron Stevens Aug 28 '18 at 10:41
  • $\begingroup$ The question as phrased does not meet the guidelines but with a bit of work on the part of the OP it can be turned into a valuable question. Two classics on this topics are the books by French and Taylor Intro. to Quantum Physics and the book by Robert Resnik on Quantum Physics of Atoms, Molecules etc. $\endgroup$ – ZeroTheHero Aug 28 '18 at 15:41