# Wavefunction of a particle

1. Oct 26, 2003

### einai

Quantum question again....

What's the wave function in coordinate space &Psi;x0(x') of a particle (in 1-D) located at a certain position x0? What about the wave function &Phi;x0(p') in momentum space? Now, consider the totality of these wave functions for different values of x0. Do they form an orthonormal set?

The only thing I know is that if I know &Psi;x0(x'), I can Fourier transform it to &Phi;x0(p')? But what's &Psi;x0(x')?

2. Oct 26, 2003

### HallsofIvy

Staff Emeritus
The wave function of a particle AT a specific x0 is the Dirac delta function &delta;(x-x0). Of course, the wave function in momentum space is the fourier transform: the constant function. If you know the position exactly, then you have no information at all about the momentum.

3. Oct 26, 2003

### einai

Thank you! That makes a lot of sense. However, I'm not sure whether I understand this part of the question -

Now, consider the totality of these wave functions for different values of x0. Do they form an orthonormal set?

Does it mean whether all the delta functions at different x0 are orthogonal or not?

4. Oct 27, 2003

### HallsofIvy

Staff Emeritus
What does "orthogonal" mean for these functions?

(The delta function is not a true function. It is a "generalized function" or "distribution". But, the same concepts apply.)

5. Oct 28, 2003

### einai

I think it means if they're the same function, the product should be integrated to one, otherwise it's zero?

Hm.....I multiplied 2 wavefunction and integrate them. It gave me another delta function.

6. Oct 31, 2003

### arcnets

einai,
I think the trick is that you use substitution.
You got
&int &delta (x-x0) &delta (x-x1) dx.
Now, let y = x-x0.
Then, dx = dy, and
&delta (x-x0) = &delta (y), and
&delta (x-x1) = &delta (y+x0-x1).
So,
&int &delta (x-x0) &delta (x-x1) dx
=
&int &delta (y) &delta(y+x0-x1) dy
=
&delta (x0-x1).
You probably already got that, and it's the orthogonality relation that you want.

Last edited: Oct 31, 2003
7. Oct 31, 2003

### arcnets

How can I make these math symbols show?

8. Nov 1, 2003

### einai

You need to put ; after &delta :D.

And thanks for answering my question. I did get the same thing, but I wasn't sure whether that implies orthonormality. Now I know, since I got the solution from the prof. It does imply orthonormality, and I got it right! :)

9. Nov 1, 2003

### arcnets

Just testing...
&int; &delta; (x-x0) &delta; (x-x1) dx = ?
Now, let y = x-x0.
Then, dx = dy, and
&delta; (x-x0) = &delta; (y), and
&delta; (x-x1) = &delta; (y+x0-x1).
So, &int; &delta; (x-x0) &delta; (x-x1) dx
= &int; &delta; (y) &delta; (y+x0-x1) dy
= &delta; (x0-x1).

Fine! Learned something!

Last edited: Nov 1, 2003
10. Nov 1, 2003

### einai

Looks much better :)!

I didn't use substitution. I just treated
&int;&delta; (x-x0) &delta; (x-x1) dx
as &int;f(x) &delta;(x-x1) dx

so when I integrate it, it gives f(x) -> f(x1) = &delta; (x0-x1).

I'm not sure whether this is a correct method, although it does give me the answer :D.