Continuity of Function and Derivative at boundary.

[Prologue]

I am reading both Griffiths and Gasiorowicz and I can't get either of them to tell me why the continuity of the derivative of the natural log of the amplitude

$$\frac{d(ln(u(x)))}{dx}=\frac{1}{u(x)}\frac{du(x)}{dx}$$

or put a different way

$$\frac{1}{u(a^{-})}\frac{du(a^{-})}{dx}= \frac{1}{u(a^{+})}\frac{du(a^{+})}{dx}$$

at a boundary a implies that

$$u(a^{-})=u(a^{+})$$

and

$$\frac{du(a^{-})}{dx}=\frac{du(a^{+})}{dx}$$Why is this true?
It looks to me that there still is some ambiguity (as in the top equation is necessary but not sufficient) because

$$\frac{1}{u(a^{-})}\frac{du(a^{-})}{dx}= \frac{1}{u(a^{+})}\frac{du(a^{+})}{dx}$$

can be reconfigured to

$$\frac{u(a^{+})}{u(a^{-})}=\frac{\frac{du(a^{+})}{dx}}{\frac{du(a^{-})}{dx}}$$

Which just means that the ratios have to be equal. But say for instance that $$u(a^{+})$$ is twice as big as $$u(a^{-})$$ so that the ratio is 2. Then all that means is the ratio of the slopes at a has to be 2. So, I don't see where they are limited to be identical.

 

[Avodyne]

It isn't true.

What is true is that continuity of both u and du/dx can be derived from the Schrödinger equation. And then, since the overall normalization is not fixed by the Schrödinger equation, it's equivalent to just consider continuity of the log derivative.

 

[Prologue]

It isn't true.

What is true is that continuity of both u and du/dx can be derived from the Schrödinger equation. And then, since the overall normalization is not fixed by the Schrödinger equation, it's equivalent to just consider continuity of the log derivative.

Thank you! I appreciate the reply. I am glad to hear that I wasn't missing something in the math. I understand that Schrödinger's equation forces the continuity of Psi and it's derivative. But there still is a gap in my understanding, I do not know what is meant by "since the overall normalization is not fixed by the Schrödinger equation" and then how the log derivative follows from that. Can you expand on that just a little more?

Is the "normalization" that you are referring to the normalization of |Psi|^2 over all space?

 

[Avodyne]

Oops, what I said isn't quite true either.

You need both conditions to get both the energy eigenvalue and the eigenfunction. But if we are interested in only the eigenvalue (or we want to get that first), then taking the ratio removes the unknown info about the eigenfunction.

And yes, by normalization I mean the integral of psi^2.

 

[Prologue]

So the problem for me lies in this: It is our task to match two different functions and their derivatives at the boundary because Schrödinger's equation tells us it has to be that way. If we then just find two solutions in different regions that meet at a boundary then they must be continuous as well as their derivatives because the Schrödinger equation tells us it must be that way. But the book says that having the very first equation (composite equation) be true necessarily means the other two equations are true and I just don't see how that happens.

In other words, we need to find two solutions that fit into Schrödinger's constraints at the boundary. These solutions to start with have arbitrary constants all over the place so they can take on any finite value at the boundary. Then we apparently use the combined equation to simplify the finding out of these constants. Instead of doing two constraints we apply just one that apparently contains both. I don't understand how that combined equation can contain both for the reasons given in the first post.

Can you explain exactly how the normalization combined with this composite equation guarantees that we will find the constants that give us identical boundaries?
*Another view.*

Suppose we do the normalization integral, then we will get a sum of products/etc of constants that have to equal 1. The let's say we pick these constants so that they purposely solve the normalization equation, but also so that they do not satisfy the boundary rules that the Schrödinger equation insists on. Is it possible to do this? If not why are we guaranteed that we can't do this?

 

[Avodyne]

But the book says that having the very first equation (composite equation) be true necessarily means the other two equations are true and I just don't see how that happens.

You don't see how that happens because the book is wrong.

What is true is that continuity of both u and du/dx can be derived from the Schrödinger equation. And then, since the overall normalization is not fixed by the Schrödinger equation, it's equivalent to just consider continuity of the log derivative.

Oops, I did it again! This is only correct in special situations, where, before matching, you know the wave function in each region up to overall normalization. This happens if the potential is even, and you therefore know that the energy eigenfunctions are even or odd.

But, more generally, you don't know this, and then you do indeed have to separately match the wave function and its derivative at each boundary.