Energy Eingenstates of 2 particles in a potential 'box'

[mooeypoo]

Final is coming up, and I'm going over all my homework.

 

I'm trying to understand this part, though, and i'm not sure I get it.

Question: Two non interacting particles of mass m each are inside a cubical box of side a and rigid walls (potential 0 inside and infinity outside).

a) Write the states for the lowest 3 values of the energy if the particles are distinguishable. What is the degeneracy of each level?

So, this is an infinite square well in three dimentions, with two particles:

 

Energy eigenstates are represented with:

 

[math]E_n=\frac{\hbar^2 \pi^2}{2ma^2}(n_{x_1}^2+n_{y_1}^2+n_{z_1}^2 + n_{x_2}^2+n_{y_2}^2+n_{z_2}^2 )[/math]

 

In an infinite square well with one particle, I can start with an eigenstate [math]E_{100}=E_{010}=E_{001}[/math] - that is, I can have n=0.

 

In the answer sheet, though, the eigenstate are in a table, like this:

 

[math]E=\frac{\hbar^2 \pi^2}{2ma^2}(3 + 3)[/math]

Which is

[math]|111>_{1}|111>_{2}[/math]

(Degeneracy = 1)

 

[math]E=\frac{\hbar^2 \pi^2}{2ma^2}(6 + 3)[/math]

Which is

[math]|211>_{1}|111>_{2}[/math]

[math]|121>_{1}|111>_{2}[/math]

[math]|112>_{1}|111>_{2}[/math]

[math]|111>_{1}|211>_{2}[/math]

[math]|111>_{1}|121>_{2}[/math]

[math]|111>_{1}|112>_{2}[/math]

(Degeneracy = 6)

 

 

But why is this eigenstate wrong? Shouldn't that be the first energy level??

[math]E=\frac{\hbar^2 \pi^2}{2ma^2}(1 + 1)[/math]

Which is

[math]|001>_{1}|001>_{2}[/math]

[math]|001>_{1}|010>_{2}[/math]

[math]|001>_{1}|100>_{2}[/math]

[math]|100>_{1}|100>_{2}[/math]

[math]|100>_{1}|010>_{2}[/math]

[math]|100>_{1}|001>_{2}[/math]

[math]|010>_{1}|001>_{2}[/math]

[math]|010>_{1}|010>_{2}[/math]

[math]|010>_{1}|100>_{2}[/math]

(Degeneracy = 9)

 

I have a problem with part 2, two (the question then goes to ask what are the three values of the lowest energies if the particles are bosons, then fermions [i know one's symmetric and one isn't]), but I'll wait 'till I get the first part before I move on.

 

Thanks!

 

~moo

[timo]

The wave function of the 3D infinite square well potential factorizes into three factors each solving the 1D problem (note that this is a reminder, I assume you to know that), i.e f(x,y,z) = fx(x)*fy(y)*fz(z). Each of the functions fx, fy and fz have to solve the 1D problem. The lowest energy state for this 1D seems to be labeled n=1 in your problem; there is no n=0 solution (if there was then the lowest state would be |000>/|000>).

 

You can also think of it in a less abstract manner and in terms of the square of the wave function: The wave function needs to have a zero value on all of the six walls of the potential, not only at those parallel to one of the spacial directions.

 

EDIT: In short: |100>, |010>, |001> are not solutions for the single particle.

[mooeypoo]

The wave function of the 3D infinite square well potential factorizes into three factors each solving the 1D problem (note that this is a reminder, I assume you to know that), i.e f(x,y,z) = fx(x)*fy(y)*fz(z). Each of the functions fx, fy and fz have to solve the 1D problem. The lowest energy state for this 1D seems to be labeled n=1 in your problem; there is no n=0 solution (if there was then the lowest state would be |000>/|000>).

 

Right! I did this problem in 3D with a single particle, and the available energy levels did have 0 in them (|001> , |010>, |100> was the first).

 

How do I know that in this case the levels start from 111 ?? My initial answer had 0 in it.. is that wrong? I'm confused. How am I supposed to know in advance?

 

You can also think of it in a less abstract manner and in terms of the square of the wave function: The wave function needs to have a zero value on all of the six walls of the potential, not only at those parallel to one of the spacial directions.

I don't understand ..

 

EDIT: In short: |100>, |010>, |001> are not solutions for the single particle.

... why not?

 

Here's another problem I solved from an earlier homework (that's why I'm confused about the current one, they're VERY similar, only with another particle)

A particle of mass m is confined inside a rectangular box of sides a, a and 2a. Find the energy and deceneracy of the lowest three energy levels.

Since [math]\Psi=X(x)Y(y)Z(z)[/math] like you said, the wave equation is:

[math]\Psi = \frac{2}{a}\frac{1}{\sqrt{a}}sin(\frac{n_x \pi x}{a})sin(\frac{n_y \pi y}{a})sin(\frac{n_z \pi x}{2a})[/math]

 

And the Energy:

 

[math]E_n = \frac{\hbar^2 \pi^2}{2ma^2} \big( n_x^2 + n_y^2 + \frac{n_z^2}{4} \big)[/math]

 

And the lowest three states (according to the answer sheet too):

 

[math]E_{001}=\frac{\hbar^2 \pi}{8ma^2}[/math]

(the answer sheet writes the above as E_000, but that's surely a mistake)

 

And

[math]E_{100}=E_{010}=\frac{\hbar^2 \pi}{2ma^2}[/math]

 

And

[math]E_{101}=E_{011}=\frac{5\hbar^2 \pi}{8ma^2}[/math]

 

 

So, in this case we COULD have n=0 state! this is more or less the same situation as above (a box) only with another particle. What am I missing? Why is this case okay with n=0 and the first case with 2 particles not okay with n=0?

Edited May 19, 2010 by mooeypoo