If a wavefunction can only collapse onto a few eigenstates

1. Mar 11, 2009

kehler

I just started learning QM. I was wondering, if a wavefunction can only collapse onto a few eigenstates, how come the probability distribution graph is a usually continuous one? :S

2. Mar 11, 2009

chroot

Staff Emeritus
Re: Eigenstates

Imagine a probability space spanned by two eigenstates -- it's a 2D space, containing an infinite number of points. At each point in the space, there's a specific probability of collapsing onto each eigenstate. That's a continuous quantity.

- Warren

3. Mar 11, 2009

kehler

Re: Eigenstates

I don't quite get it :S. From my understanding, the probability distribution graph depicts the probability of where the particle will collapse. But you're saying that it actually represents the probability of a particle, currently at a particular position on the graph, collapsing onto an eigenstate?

4. Mar 11, 2009

chroot

Staff Emeritus
5. Mar 12, 2009

kehler

Re: Eigenstates

I was referring to a graph of the square of the wavefunction vs position. That's what the textbook that I'm reading (Griffiths) uses to depict the probability of where a particle associated with some wavefunction will collapse.. It's only taking 1-D into account I think.

6. Mar 12, 2009

chroot

Staff Emeritus
Re: Eigenstates

I have Griffiths... which page number? I'll pull it out.

- Warren

7. Mar 12, 2009

kehler

Re: Eigenstates

Just something like on page 3, fig 1.2 where it's a continuous graph..

8. Mar 12, 2009

chroot

Staff Emeritus
Re: Eigenstates

The wavefunction is a function of all space. If you give me any point in space, I can give you the value of the wavefunction there. Therefore, the wavefunction is continuous. The book hasn't even introduced eigenstates yet.

- Warren

9. Mar 12, 2009

alxm

Re: Eigenstates

The states are discrete, but the corresponding eigenfunctions aren't discrete in space. Consider the particle-in-the-1D-box example. Every wave function is continuous with a value at every point from 0 to L.

So obviously a state that's a superposition, a sum, of several eigenfunctions is also going to continuous and defined from 0 to L, and so is the absolute square of that superposition.

10. Mar 12, 2009

dx

Re: Eigenstates

Eingenstates of what? A particle in a box has a discrete energy basis, but the position basis is continuous. The diagrams of wavefunctions are usually drawn in position space, so they will be continuous.

Last edited: Mar 12, 2009