Crossing nodes of wavefunction

In summary, the conversation discusses the concept of a stationary state in quantum physics and how it differs from classical physics. The question raised is about the zero probability density in certain positions and how the particle can move across these positions. The answer is that in a stationary state, nothing "moves" in the classical sense and the notion of trajectory is different in quantum physics. The concept of interference is also discussed in relation to this question.
  • #1
DeShark
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Hi all, a couple of weeks ago, I was reading a book (Eisberg and Resnick) in which one of the questions asked was:

In the n=3 state, the probability density function for a particle in a box is zero at two positions between the walls of the box. How then can the particle ever move across these positions?

Basically, the book doesn't give the answer and I don't know it. I also can't work it out, despite the past two weeks of wracking my brain over this problem. Can anyone offer a resolution? Thank you!
 
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  • #2
First of all, in a stationary state, nothing "moves" in the sense of "changes".
Next, the notion of trajectory has a totally different meaning in quantum physics than it has in classical physics.

As such, you shouldn't picture a stationary state (described by the wavefunction you mention) as a kind of density of "presence" of a classical particle moving around on a certain trajectory, which is what is implicitly assumed in this question.
 
  • #3
vanesch said:
First of all, in a stationary state, nothing "moves" in the sense of "changes".
Next, the notion of trajectory has a totally different meaning in quantum physics than it has in classical physics.

As such, you shouldn't picture a stationary state (described by the wavefunction you mention) as a kind of density of "presence" of a classical particle moving around on a certain trajectory, which is what is implicitly assumed in this question.

So, the purpose of the question is to make me realize what a stupid question it is..?
It seems very bizarre that there would be these locations in the potential where there is zero probability density. Is this merely because I have no intuition for quantum situations like this? Is it anything at all to do with the particle interfering with itself, thus in a way (I know this isn't quite the best way of looking at it) creating the standing waves found as solutions? In the same way that electrons, unobserved, passing through young's slits "interfere" with each other and create minima of the wave function...

I read somewhere on here that it is this same "interference" which causes probability densities for electons around nuclei to have high values here and low values there.
 
  • #4
DeShark said:
In the same way that electrons, unobserved, passing through young's slits "interfere" with each other and create minima of the wave function...

The electrons do not interfere with each other. You get an interference pattern even if you shoot one electron through the apparatus at a time, for a long enough period of time that you get a large number of total hits on the "screen" or whatever detector you're using.
 
  • #5
DeShark said:
Hi all, a couple of weeks ago, I was reading a book (Eisberg and Resnick) in which one of the questions asked was: In the n=3 state, the probability density function for a particle in a box is zero at two positions between the walls of the box. How then can the particle ever move across these positions? Can anyone offer a resolution? Thank you!
An example: particle in the one dimensional well.
1. If we don't change the well in time, the situation is stationary and particle doesn't "move".
2. If we change the well or change the potential in time, then the "zero's" can move or even "zeros" could come to an end (psi function may by complex function in general case).

For example when we change the length of the well specifically in time we can get as a result 1 zero function (cooling) or 3 zero function (heating). "Zeros" will be borned or killed at the boundaries of the well.

Zero probability means nothing. We can have FLOW of probability in time from the left and from the right and they may kill each other.
 
  • #6
DeShark said:
So, the purpose of the question is to make me realize what a stupid question it is..?
It seems very bizarre that there would be these locations in the potential where there is zero probability density. Is this merely because I have no intuition for quantum situations like this? Is it anything at all to do with the particle interfering with itself, thus in a way (I know this isn't quite the best way of looking at it) creating the standing waves found as solutions?

I'd say so, yes.
 

FAQ: Crossing nodes of wavefunction

What are crossing nodes of wavefunction?

Crossing nodes of wavefunction refer to the points where the wavefunction of a particle or system crosses the x-axis. These nodes represent regions where the probability of finding the particle is zero.

How do crossing nodes affect the behavior of particles?

Crossing nodes can affect the behavior of particles by limiting the regions where they can exist. For example, in quantum mechanics, particles are only allowed to exist in regions where the wavefunction is non-zero, so crossing nodes can restrict the possible states and movements of particles.

Are crossing nodes of wavefunction always present?

No, crossing nodes of wavefunction are not always present. In some cases, the wavefunction may not have any crossing nodes, while in others, it may have multiple crossing nodes.

How are crossing nodes related to energy levels?

Crossing nodes of wavefunction are related to energy levels in that the number and placement of crossing nodes can determine the energy levels of a particle or system. For example, in a particle in a box system, the number of crossing nodes in the wavefunction can determine the allowed energy levels of the particle.

Can crossing nodes be manipulated or controlled?

Yes, crossing nodes can be manipulated or controlled to some extent. In quantum mechanics, manipulating the potential energy of a system can change the number and placement of crossing nodes in the wavefunction, thereby altering the behavior and properties of the particles in that system.

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