Why do exponential functions occur outside of finite walls?

[jjson775]

TL;DR

Why are exponential functions the solution to the time independent Schrödinger equation outside the walls of a finite well?

Between the walls of a finite well, the solution to the time independent Schrödinger equation is a combination of sines and cosines. Outside the walls where E - Uo is positive, the solutions are exponential functions. Why?

 

[PeroK]

Those are the solutions to the differential equation in each case. You can directly check they are solutions, even if you don't know how to solve the equations themselves.

 

[jjson775]

I understand that and have checked the solutions. The wording of the text is “Uo - E is positive, so the solutions are exponential functions“. Is this supposed to follow logically? If you drop the word “so”, I am OK.

 

[Vanadium 50]

If the particle is inside the well (which I think is U0 - E < 0 im your notation), the solutions are sines and cosines. If it is outside (which I think is U0 - E > 0 im your notation) the solutions are siinh and cosh, i.e. exponential.

As mentioned, you can verify these solutions even if you cannot solve for them,

 

[jjson775]

If the particle is inside the well (which I think is U0 - E < 0 im your notation), the solutions are sines and cosines. If it is outside (which I think is U0 - E > 0 im your notation) the solutions are siinh and cosh, i.e. exponential.

As mentioned, you can verify these solutions even if you cannot solve for them,

That’s what I said In my post. My reply to PeroK shows my point of confusion. Also, my post has a typo, should have been Uo - E, potential energy > mechanical energy.

 

[PeterDonis]

The wording of the text is “Uo - E is positive, so the solutions are exponential functions“. Is this supposed to follow logically?

Yes, by solving the relevant differential equations. If you are unable to solve them yourself, you will not be able to make the logical connection in question yourself. But you can still verify the connection by verifying that exponentials are solutions.

 

[jjson775]

Alright. You can’t tell by inspection that the solutions are exponential just because Uo > E is positive, but through solution of the differential equations. I have verified the solutions.

 

[PeterDonis]

You can’t tell by inspection that the solutions are exponential just because Uo > E is positive

Well, people who are familiar enough with the relevant differential equations can, because they already have more than enough experience in solving them. Any veteran of a college level differential equations course will understand what I mean.

 

[Vanadium 50]

I don't know what to tell you. It seems like you are saying that you can't do the math and can;t follow the words, but also don't want to trust what the people who do say. You've kind of painted yourself into a corner there.

 

[jjson775]

Well, people who are familiar enough with the relevant differential equations can, because they already have more than enough experience in solving them. Any veteran of a college level differential equations course will understand what I mean.

Your answer is what I was looking for. Thanks. I did take differential equations 60+ years ago but never used it professionally.