# Wave function requirements

1. Mar 13, 2005

### Spinny

Hi, I have a question about the mathematical requirements of a wave function in a potential that is infinite at $$x \leq 0$$. (At the other side it goes towards infinity at $$x = \infty$$.) Now, given a wave function in this potential that is zero for $$x = 0$$ and $$x = \infty$$. Does it matter what that wavefunction is at $$x = -\infty$$? I mean, I just figured you would have a wave function there that's zero all the way. Why will a wave function that goes to $$-\infty$$ at $$x = -\infty$$ not fit in the (time independent) Schrödinger equation, whereas one that goes to zero at $$-\infty$$ does? After all when we're normalizing it, we're just integrating from 0 to $$\infty$$ and doesn't really need to bother with it at negative x values. Or is that just some mathematical requirement that is independent of the physical properties? Can someone enlighten me, please?

2. Mar 13, 2005

### dextercioby

"Solve the unidimensional SE for one particle in the the potential field:

$$U(x)=\left\{\begin{array}{c}+\infty,\mbox{for} \ x\in(-\infty,0]\\0,\mbox{for} \ x\in (0,+\infty)\end{array}\right$$

,because you didn't say anything about the potential in the positive semiaxis...

Daniel.

3. Mar 13, 2005

### Spinny

The potential is the harmonic oscillator on the positive semiaxis. The problem is what are the mathematical requirements for the wave function. Let's say you have a function $$\psi(x)$$, then what are the mathematical requirements that function need to meet in order to be a wavefunction for that potential?

4. Mar 13, 2005

### dextercioby

Physical states are described by normalizable wavefunctions...

In your case,on the negative semiaxis the wave function is zero and on the positive semiaxis is a Hermite polynomial.So i'd say this is normalizable.

Then comes the continuity of the wavefunction.Both 0 & Hermite Polynomials are continuous,however,at the point 0,the continuity must be enforced.

The first derivative issue is rather tricky.U may wanna consult a book how to deal with infinite potentials & the conditions imposed on the wavefunction.

Daniel.

5. Mar 14, 2005