Orthogonality of wave functions

[Plaetean]

Homework Statement

Homework Equations

The Attempt at a Solution

http://i.imgur.com/tktQBsp.jpg
[/B]
I assume that you need to prove that the integral of psi1*psi0 is 0, so I have written out the integral and attempted to solve using integration by parts, but whichever way I write out the integration by parts it doesn't seem to cancel down. Do I need to write the integral as a definite integral instead and evaluate it at limits? Or perhaps invoke some properties of waves instead that will allow me to say =0?

 

[RUber]

I would imagine you should have some boundary conditions at infinity due to physical conservation laws. What do you get if you assume ## \psi_0 (\pm \infty)=0##?

 

[Orodruin]

What boundary conditions must ##\psi## fulfil in order for it to be normalisable? (It really does not matter if these boundary conditions are in infinity or not.)

 

[Plaetean]

From the previous questions, we are told that psi0 is normalised, real and even. There's nothing in terms of boundary conditions given, which is why I'm having trouble getting any further than just writing out the integration by parts.

 

[vela]

Please show your actual work.

 

[Plaetean]

http://i.imgur.com/tktQBsp.jpg

This is all I have

 

[vela]

Move the integral from the righthand side to the lefthand side.

 

[Plaetean]

Move the integral from the righthand side to the lefthand side.

That just gives me 2A(integral)=APsi₀², which is a circular argument and just proves that the integral = the integral. I feel like I must be missing something really obvious and its driving me a little crazy.

 

[vela]

I'm not sure why you think it's showing the integral equals the integral. You have an equation that says the integral is equal to some other expression. You need to show or argue that that expression is equal to 0.

 

[Plaetean]

http://imgur.com/T4fQLqt Proving Psi₀²=0 seems even more difficult surely?

 

[Orodruin]

What boundary conditions do you usually put on psi?

 

[Plaetean]

What boundary conditions do you usually put on psi?

We've done infinite square well, square well, free particle and harmonic oscillator. However there's no info given here and I don't know how to prove it 'generally', so I think it just may be an issue with the way the question is worded. I'm finding the questions worth far more marks a lot easier than this one, so I don't think its intended to be this confusing. I'll speak to the tutor tomorrow.

edit: thank you everyone for the replies though, its very much appreciated

 

[Orodruin]

However there's no info given here and I don't know how to prove it 'generally', so I think it just may be an issue with the way the question is worded.

There is no issue. The boundary conditions are general if you want to have a wave function that is possible to normalize (which it has to be in order to be normalized).

 

[Dick]

http://imgur.com/T4fQLqt Proving Psi₀²=0 seems even more difficult surely?

You didn't write down limits in that image. Suggest you do so. You don't have to prove ##{\Psi_0}^2=0##.

 

[Plaetean]

I don't know what the limits would be, -infinity to +infinity? 0 to +infinity? How you would even evaluate the integral at any limits considering we aren't actually given the function?

 

[Orodruin]

What conditions would you get if the potential becomes infinite at some point? What conditions would you get if it does not? I suggest starting with the second case as it is more general - the former can be seen as a trivial extension.

 

[Matterwave]

I don't know what the limits would be, -infinity to +infinity? 0 to +infinity? How you would even evaluate the integral at any limits considering we aren't actually given the function?

You are not given the region in this problem, so you have to allow a general region (within physical reason, so you probably don't have to consider, e.g. disconnected sets). You should put in the limits:
$$2\int_a^b \psi_1(x)\psi_0(x) dx = A\left.\psi_0(x)^2\right|_a^b$$ Now you just have to figure out what ##\left.\psi_0(x)^2\right|_a^b## has to be in order for the wave function to be normalizable and continuous and differentiable. So you have to know what ##\psi_0(a)## and ##\psi_0(b)## are, these are the boundary conditions that people keep talking about.

This problem seems to be a bit hard if you have to also prove that all physical boundary conditions lead to the conclusion that ##\left.\psi_0(x)^2\right|_a^b=0##, but at least for all the boundary conditions I can think to impose, this condition is met.