What are the possible solutions for the TISE in the infinite square well model?

[spaghetti3451]

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:

\frac{d^{2}u}{dx^{2}} + \frac{2mE}{hcross^{2}}u = 0.

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?

 

[vela]

Why is that a problem?

 

[ideasrule]

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.

 

[spaghetti3451]

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!

 

[vela]

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.