Interpreting These '2 Particles in a Box' Plots

In summary: The probability distribution of the position of one of the particles (found using that integral you wrote above) does not depend on whether they are bosons or fermions, and I suppose this is related to the entanglement of the two subsystems.
  • #1
tomdodd4598
138
13
Hey there,

I am familiar with the mathematics of multi-particle systems, but have now moved on to trying to plot them. I was a little ambitious and thought I'd attempt to somehow plot energy eigenfunctions of two particles in a 2-D box. Obviously I immediately ran into the issue that there are actually four coordinates in the wave function rather than two.

Looking for ideas, I quickly came along the Wikipedia article on identical particles, and found these two plots:
https://en.wikipedia.org/wiki/File:Symmetricwave2.png
https://en.wikipedia.org/wiki/File:Asymmetricwave2.png

All that's said about them are that they are the plot of the "(anti)symmetric wave function for a fermionic/bosonic 2-particle state in an infinite square well potential".

I thought that it may be a 1-D box, with the two axes being the positions of the two particles, but if this is indeed a plot of two particles in a 2-D box, what exactly is being plotted?

Thanks in advance.
 
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  • #2
tomdodd4598 said:
the Wikipedia article on identical particles

Link, please?
 
  • #3
tomdodd4598 said:
Obviously I immediately ran into the issue that there are actually four coordinates in the wave function rather than two.
You have the alternative of plotting the density function which is a function of 2 coordinates in 2D case.
tomdodd4598 said:
I thought that it may be a 1-D box, with the two axes being the positions of the two particles,
Yes, the two plots are for 1D infinite well potential, not 2D.
 
  • #4
PeterDonis said:
Link, please?
https://en.wikipedia.org/wiki/Identical_particles
blue_leaf77 said:
You have the alternative of plotting the density function which is a function of 2 coordinates in 2D case.
Really? Would I do that by adding the integrals of the probability density over x1, y1 and x2, y2?
blue_leaf77 said:
Yes, the two plots are for 1D infinite well potential, not 2D.
Yep, just confirmed this by recreating them. Thanks :)
 
  • #5
tomdodd4598 said:
Would I do that by adding the integrals of the probability density over x1, y1 and x2, y2?
The density function is the probability density of finding a particle in space. It's given by
$$
\rho(\mathbf r) = \int_{\mathbf r_2} \int_{\mathbf r_3} \ldots \int_{\mathbf r_N} |\psi(\mathbf r, \mathbf r_2, \mathbf r_3, \dots \mathbf r_N)|^2 \ d^3\mathbf r_2 \ d^3\mathbf r_3 \dots d^3\mathbf r_N
$$
for ##N## number of particles in 3D space. The adaptation to 2D case should be straightforward.
 
  • #6
Thanks - I think I've got it.
It took me a little while to get my head around the apparent fact that, if the particles are indistinguishable, the probability distribution for each particle doesn't depend on whether the particles are fermions or bosons. Wasn't expecting that for some reason, but I guess the reason for this is that, if you express the (anti)symmetric wave function as a sum of products of the one particle wave functions, the cross terms that appear when squaring the wave function (possible because the eigenfunctions are real) vanish, as the integral of a product of two orthogonal eigenfunctions is 0.

I guess this is because systems of multiple indistinguishable particles are entangled?
 
  • #7
tomdodd4598 said:
if you express the (anti)symmetric wave function as a sum of products of the one particle wave functions, the cross terms that appear when squaring the wave function (possible because the eigenfunctions are real) vanish, as the integral of a product of two orthogonal eigenfunctions is 0.
You mean when the wavefunction is of the form ##\psi(x_1,x_2) \propto u_a(x_1) u_b(x_2) - u_b(x_1) u_a(x_2)## (for fermions)?
That's not really the reason, the functions ##u(x)##'s appearing above do not have to be orthogonal for the wavefunction to be symmetric or antisymetric. In fact, the form of wavefunction above where ##u_n(x)## is an eigenfunction of the Hamiltonian only holds for non-interacting particle. For interacting particles, the wavefunction is a sum of terms of that form. The real reason why the probability of detecting a particle is the same for every particle is that because of the nature of the wavefunction itself, namely symmetric - ##\psi(x_1,x_2) = \psi(x_2,x_1)## - or antisymmetric ##\psi(x_1,x_2) = -\psi(x_2,x_1)##. If you take the modulus square of these functions, the two types of wavefunction are easily seen to have the property ##|\psi(x_1,x_2)|^2 = |\psi(x_2,x_1)|^2##.
 
  • #8
I understand, but what I mean is that the probability distribution of the position of one of the particles (found using that integral you wrote above) does not depend on whether they are bosons or fermions, and I suppose this is related to the entanglement of the two subsystems.
 
Last edited:

Related to Interpreting These '2 Particles in a Box' Plots

1. What is a "2 Particles in a Box" plot?

A "2 Particles in a Box" plot is a visual representation of the energy levels and wave functions of two particles confined in a box. The box is often used as a simplified model for quantum systems, with the particles representing particles such as electrons or atoms.

2. How is the energy level represented in these plots?

The energy level is represented by the horizontal lines in the plot. Each horizontal line corresponds to a different energy level, with the higher lines representing higher energy levels. The spacing between the lines also indicates the difference in energy between each level.

3. What do the vertical lines in the plot represent?

The vertical lines represent the probability of finding the particles at a specific position within the box. The taller the line, the higher the probability of finding the particles at that position. The position of the lines also corresponds to the position of the energy levels in the horizontal lines.

4. How can these plots be used to interpret quantum systems?

These plots can be used to understand the behavior and properties of quantum systems. The energy levels and wave functions can provide insights into the stability, reactivity, and other characteristics of the system. They can also be used to predict the behavior of the system under different conditions.

5. Are these plots only applicable to particles in a box?

No, these plots can also be used to represent other confined systems, such as molecules in a potential energy well. However, the simplified model of particles in a box is often used in introductory quantum mechanics courses to illustrate key concepts and principles.

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