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The infinite square well

  • #1
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Homework Statement



As part of my homework, I am solving the TISE for the infinite square well model.

The potential is zero for |x| =< a and infinite otherwise.

Homework Equations





The Attempt at a Solution



For |x| >= a, the wavefunction is zero.

For |x| =< a, there are three possible cases:
1. E > 0
2. E = 0
3. E < 0

for the following TISE:

[tex]\frac{d^{2}u}{dx^{2}} + \frac{2mE}{hcross^{2}}u = 0[/tex].

For E > 0, the solutions are sinusoidal.

For E = 0, u = A + Bx.

For E < 0, the solutions are exponentials.

The problem is the only solution is sinusoidal. What have I done wrong?
 

Answers and Replies

  • #2
vela
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Why is that a problem?
 
  • #3
ideasrule
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For |x| >= a, the wavefunction is zero.

For |x| =< a, there are three possible cases:
1. E > 0
2. E = 0
3. E < 0
The energy of the particle can never be lower than the minimum potential energy. In other words, E+V_min > 0 for every stationary state. That's why you can exclude cases 2 and 3.
 
  • #4
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The energy of the particle can never be lower than the minimum potential energy. In other words, E+V_min > 0 for every stationary state.
Or did you mean E - V_min > 0 for every stationary state?

I am wondering why the energy of the particle can never be lower than the minimum potential energy.

Please help me out!
 
  • #5
vela
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Try satisfying the boundary conditions in the second and third cases. You'll find you can't except when the wave function vanishes. Only the first case allows non-trivial solutions.
 

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