How Does the Wavefunction Behave at an Infinite Potential Barrier?

[ballzac]

Just a quick question. Solving the Schrödinger equation (time-independent) for an infinite potential barrier, I end up with two wavefunctions.

In region I,

$$V(x)=0$$
$$\Rightarrow\psi(x)=Acos(\frac{\sqrt{2mE}}{\hbar}x)+Bsin(\frac{\sqrt{2mE}}{\hbar}x)$$

In region II,

$$V(x)=\infty$$
$$\Rightarrow\psi(x)=Ce^\infty+De^{-\infty}=\infty+0$$

Clearly in the infinite potential region, the wavefunction must equal zero, so it is clear that the term with constant factor C must vanish. I interpret this as meaning that the D term is the wave that penetrates the barrier (in this case it does not because $$e^{-\infty}=0$$), and the C term is a wave coming from the right, that does not exist and therefore $$C=0$$. If I am right in assuming this, then how can one prove mathematically, that $$C=0$$, without resorting to non-mathematical common sense? If I'm wrong, then what is the explanation?

Thanks in advance, and sorry if there are any errors in the above...made a lot of typos the first time around :$

 

[malawi_glenn]

The probability to penetrate an infinite high potential barrier is 0, so wavefunction vanishes everywhere in that region.

Also, try to get you whole wave function continuous at the boundary.. you'll see that only C = 0 works.

 

[ballzac]

MG,

The first line of your reply is basically what I was getting at, but I neglected to mention probability, which is of course the motivation for assuming there is no wave function in that region.

Secondly, are you saying that
$$Ce^{\infty}+De^{-\infty}$$
must equal zero or infinity depending on choices of C and D,
but the wave function in region I (which must equal the above at the barrier) can only be infinite if A and/or B are infinite, and they are both finite so the above must be zero?

If we can assume that the potential can be infinite, could we not also take the wave function to be infinite? I'm guessing the answer is no, but not sure why.

Also, as I have not used these forums much, I'm not sure of the etiquette. I'm thinking I should have posted this in the homework section, but as I am finished uni for the year and have just been mucking around with QM among other things, it never occurred to me to post this kind of thing in the homework section. Please forgive me if I've erred. :)

 

[malawi_glenn]

You must know that the wavefunction should also be normalized and that an infinite value of the wavefunction means that the probability for finding particle at a particular point is... (infinite) which is unphysical.

No problem, you are atleast showing you works and thoughts, very good! keep it up.

 

[ballzac]

You must know that the wavefunction should also be normalized and that an infinite value of the wavefunction means that the probability for finding particle at a particular point is... (infinite) which is unphysical.

D'uh. Of course. It's easy to overlook the obvious when learning. Thanks :)