If a wavefunction can only collapse onto a few eigenstates

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Discussion Overview

The discussion revolves around the nature of wavefunction collapse in quantum mechanics, specifically addressing the relationship between discrete eigenstates and the continuous probability distribution graphs typically associated with wavefunctions. Participants explore the implications of these concepts in the context of quantum mechanics learning.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how a wavefunction can collapse onto a few eigenstates while the probability distribution graph is usually continuous.
  • Another participant suggests that a probability space spanned by two eigenstates can contain an infinite number of points, leading to a continuous probability distribution.
  • A participant expresses confusion about the interpretation of the probability distribution graph, seeking clarification on whether it represents the probability of a particle collapsing onto an eigenstate from a specific position.
  • Discussion includes references to visual representations of electron orbitals and the square of the wavefunction versus position, indicating different interpretations of continuous graphs.
  • Participants note that while eigenstates may be discrete, the wavefunctions themselves are continuous functions defined over space.
  • One participant mentions the particle-in-a-box example to illustrate that superpositions of eigenfunctions remain continuous across defined intervals.
  • Another participant points out that while energy eigenstates are discrete, the position basis is continuous, which aligns with the typical representation of wavefunctions in position space.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of wavefunctions and probability distributions, indicating that multiple competing perspectives remain without a clear consensus.

Contextual Notes

Some participants reference specific figures and pages from a textbook (Griffiths) to support their points, but there is no agreement on the interpretation of these materials. The discussion also highlights the complexity of transitioning from discrete eigenstates to continuous wavefunctions.

Who May Find This Useful

Individuals interested in quantum mechanics, particularly those new to the subject, may find this discussion relevant as it addresses foundational concepts and common confusions regarding wavefunctions and probability distributions.

kehler
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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
 
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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
 


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?
 


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.
 


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

- Warren
 


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


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
 


kehler said:
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.
 
  • #10


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.
 
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