If a wavefunction can only collapse onto a few eigenstates

[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

 

[chroot]

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

 

[kehler]

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?

 

[chroot]

Are you talking about pictures showing the spatial probability distribution of an electron in an orbital? Like these?

http://en.wikipedia.org/wiki/File:Electron_orbitals.svg

- Warren

 

[kehler]

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.

 

[chroot]

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

- Warren

 

[kehler]

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

 

[chroot]

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

 

[alxm]

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

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.

 

[dx]

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.