Particle in a box in cartesian coordinates

[gfd43tg]

Homework Statement

Homework Equations

The Attempt at a Solution

a) The schrödinger equation
$$i \hbar \frac {\partial \Psi}{\partial t} = - \frac {\hbar^{2}}{2m} \nabla^{2} \psi + V \psi $$
For the case $0 \le x,y,z \le a$, $V = 0$
$$i \hbar \frac {\partial \Psi}{\partial t} = - \frac {\hbar^{2}}{2m} \Big [ \frac {\partial^{2} \psi}{\partial x^{2}} + \frac {\partial^{2} \psi}{\partial y^{2}}+ \frac {\partial^{2} \psi}{\partial z^{2}} \Big ]$$
But with the sum of the second derivates, is it possible to set each of these equal to the time derivative separately? If so, then why?

 

[gfd43tg]

Okay I have made some progress on this problem, however I am curious, is it the case that $n_{x} = n_{y} = n_{z}$? If they are allowed to be different numbers, I suspect that is what allows for degenerate energies.

 

[PeroK]

Re your first question. If you have separate functions of each variable, then consider holding every spatial variable constant. This shows that the time function must be constant. In general:

$T(t) = F(x, y, z) \ \Rightarrow \ T(t) = C = F(x, y, z)$

And:

$X(x) + Y(y) + Z(z) = C \ \Rightarrow \ X(x) = C_x, \ Y(y) = C_y, \ Z(z) = C_z$ where $C_x + C_y + C_z = C$

But, there's no reason that $C_x = C_y = C_z = C/3$

 

[gfd43tg]

Is there any easy way to find a given $E_{n}$? Or do you need to do every combination to get $E_{14}$?

 

[BvU]

Not too many possibilities to get 14 as the sum of three squares ...

 

[gfd43tg]

That is not what is meant by $E_{14}$, they mean the 14th highest energy level. Not that the sum of the squares is 14

 

[BvU]

From the problem statement wording I'm inclined to say you are right.
Usually, we call them E_N with E_N = N E₀ -- so I got sidetracked.

But then why Griffiths thinks E14 is so interesting is a mystery to me. Let us know if you find something !

 

[PeroK]

Is there any easy way to find a given $E_{n}$? Or do you need to do every combination to get $E_{14}$?

There's a whole sequence here of the number of ways that each integer can be expressed as a sum of 3 non-zero squares:

https://oeis.org/A025427

https://oeis.org/A025427/b025427.txt

For example:

66 is the first integer that can be expressed in 3 different ways; 129 in 4 different ways; 194 in 5 different ways; 209 in 6 ways.