A proof for a given formula of a normalization constant phi4

[mike232]

Homework Statement

Show that phi_n will find the proper phi_4. IE: show that it gives the correct normalization constant.

Richard Liboff...chapter 7

Homework Equations

A_n = (2^n * n! * pi^1/2)^-1/2

The Attempt at a Solution

I don't know where to start really. I tried some things with < phi | phi > = 1 but honestly stumped.

 

[Simon Bridge]

$\phi_n$ finds the proper $\phi_4$
... what is $\phi_n$?
... what does it mean to say "find the proper $\phi_4$"?
whatever - what is the normalization constant and how would you tell if it was correct?

 

[mike232]

I am given that general formula for the normalization constant phi_n. I am asked to show that this gives correct normalization for phi_4

That is all I have.

 

[Simon Bridge]

Is $\phi_n$ the normalization constant?
Then what is that $A_n$ that you wrote above?

That cannot be all you have - you also need the definition of normalization - or the formula for the normalization constant makes no sense.
What is it that is being normalized by the normalization constant?
Do you have no textbook nor class notes?

 

[Simon Bridge]

Aside: a hand with equations...

A_n = (2^n * n! * pi^1/2)^-1/2

Type out
A_n = (2^n n! \pi^{1/2})^{^-1/2}
... between double-hash signs to get: $A_n = (2^n n! \pi^{1/2})^{^-1/2}$, and between double-dollar signs to get: $$A_n = (2^n n! \pi^{1/2})^{^-1/2}$$
Greek letters are a backslash followed by the name so \phi \Phi gets you $\phi \; \Phi$ ... special functions are a backslash followed by the name of the function etc. So you can do \sin\theta to get $\sin\theta$ ... the rest is pretty intuitive.

 

[mike232]

I enclosed the image, sorry if I haven't been very good at conveying my question.

 

[Simon Bridge]

OK so what is the general formula for $\phi_n$?
How would you be able to tell if $\phi_n$ is normalized?
What is it that is being normalized by the normalization constant?
Do you have no textbook nor class notes?

 

[mike232]

I don't have anything for /phi other than the question. So I don't know to check to see if it normalized when I don't have an equation. Now I do know that my norm constant should be found by normalizing some function like... < /phi | /phi >
But I do have notes and the whole Internet, I can't really figure out how I'm supposed to start this. Or I wouldn't have given in and asked for help.

Do I have to try and work it backwards maybe? Like try and use the norm constant to try and build my eigenfunction from that?

 

[Simon Bridge]

What is the section of work you have just completed, or you gave just started?

Basically, $\phi_n = A_n \varphi_n$ where $\varphi_n$ is the solution to the shrodinger equation for the potential. Since $A_n$ are all real amd positive, $$\int_\infty \varphi_n = \frac{1}{A_n^2}$$ ... notice how you cannot use this to get the wavefunction from the normalization condition?

Afaict the problem, as stated, cannot be done without additional information, probably supplied in class.
Maybe younare expected to recognize the type of situation that would have normalizations of that form?
If you are still stumped, consult classmates... maybe they saw something you didn't.

 

[mike232]

The professor just assigned a couple 9 10 questions from a book we never really use. That does make sense about working back, it was just a shot in the dark.
My professor has never assigned straight "normal" work. It's always abstract questions like this, so it can be more like a graduate level course.

So I'm expected to get a general solution, or I guess in this case I think I need to show that for any \phi and knowing A_n I can find some kind of pattern. I honestly don't know, and I have asked my classmates and this is a general issue of no knowing what really to do.

But thanks for the help.

 

[Simon Bridge]

Its not, strictly, an abstract question, and you are asked for a specific result.
Do you at least know what the subject of Liboff ch7 is?
Next step is to ask the proff. Right now the correct answer is "not enough information".

 

[mike232]

The chapter is one dimensional systems bound and unbound states

Eigenfunctions of the harmonic oscillator hamiltonian

 

[Simon Bridge]

Since the section is about the harmonic oscillator, could the $A_n$ be the normalization terms for the harmonic oscillator wavefunctions?
Go look.