Particle in infinite potential well

In summary, the conversation discusses the difference in probability distributions at two different times and the potential energy of a particle in a potential well suddenly disappearing. The solution involves using a Fourier transform to expand the wave function, and there is some disagreement about whether the energy of the particle changes over time.
  • #1
unscientific
1,734
13

Homework Statement



29xxisp.png


Sketch the difference of probability distributions at the two times. Does the energy change with time?

The potential well suddenly disappears, what is the form of the wavefunction?

Homework Equations


The Attempt at a Solution



Part (a)

At t = 0, the probability distribution is:
[tex]|\phi|^2 = \frac{1}{a} \left[ sin^2(\frac{2\pi x}{a}) + 2 sin (\frac{2\pi x}{a})sin(\frac{4\pi x}{a}) + sin^2 (\frac{4\pi x}{a}) \right][/tex]

At ##t = \frac{\pi}{6\omega}##:
[tex]|\phi|^2 = \frac{1}{a} \left[ sin^2(\frac{2\pi x}{a}) + sin^2 (\frac{4\pi x}{a}) \right] [/tex]

Thus the difference is simply ##\frac{2}{a} sin(\frac{2\pi x}{a}) sin (\frac{4\pi x}{a})##:

33acnkn.png


I think the energy of the particle changes with time, as simply using the TDSE ##-\frac{\hbar^2}{2m} \frac{\partial^2 \phi}{\partial x^2} = E \phi## doesn't remove the ##exp(i\omega t)## terms.Part (b)

As the potential disappears, the particle becomes free. I think the general form is a linear combination of the n=2 and n=1 states.

For a free particle, ##u_{(x)} = A e^{i\frac{px}{\hbar}} = e^{ikx}##.

Thus, the form is:
[tex]\phi = Ae^{i(2kx - \omega t)} + Be^{i(4kx - \omega t)}[/tex]

Not sure if I'm right.
 
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  • #2
Part b is not right. You have to expand the wave function using a Fourier transform. In your solution the particle jumps from a confined solution to a couple of plane waves that extend to infinity. That doesn't make sense. The Fourier transform will include all sorts of wavelengths - not just 2.
 
  • #3
dauto said:
Part b is not right. You have to expand the wave function using a Fourier transform. In your solution the particle jumps from a confined solution to a couple of plane waves that extend to infinity. That doesn't make sense. The Fourier transform will include all sorts of wavelengths - not just 2.

So the solution would be:

[tex]\phi = \sum_p a_p e^{i(\frac{xp}{\hbar} - \omega t)}[/tex]

Does this look right?
 
  • #4
The solution seems a little short for 5 marks, in my opinion
 
  • #5
Well you should use a Fourier transform as opposed to a Fourier expansion because the particles are free at the end of the problem. That means you should have an integral instead of a sum.
 
  • #6
unscientific said:
I think the energy of the particle changes with time, as simply using the TDSE ##-\frac{\hbar^2}{2m} \frac{\partial^2 \phi}{\partial x^2} = E \phi## doesn't remove the ##exp(i\omega t)## terms.

unscientific - I don't understand why you think the energy changes.

That equation looks like the TISE, not the TDSE.
 

What is a particle in an infinite potential well?

A particle in an infinite potential well is a theoretical model used in quantum mechanics to describe the behavior of a particle confined within a finite space with infinitely high potential energy barriers at its edges.

What are the properties of a particle in an infinite potential well?

The properties of a particle in an infinite potential well include discrete energy levels, a probability distribution that follows the shape of a standing wave, and a quantized position within the well.

How does the energy of a particle in an infinite potential well relate to its position?

The energy of a particle in an infinite potential well is directly related to its position. As the particle moves closer to the center of the well, its energy increases, and as it moves towards the edges, its energy decreases.

What is the significance of the infinite potential well model in quantum mechanics?

The infinite potential well model is significant because it provides a simplified representation of confined particles, allowing for the calculation of energy levels and probability distributions. It also serves as the basis for more complex systems, such as the quantum harmonic oscillator.

How does the behavior of a particle in an infinite potential well differ from a classical particle?

A classical particle can exist at any position within a given space and has a continuous energy spectrum. In contrast, a particle in an infinite potential well is confined to discrete energy levels and has a quantized position within the well.

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