Particle confined in 3D box - quantum states

[greg997]

Homework Statement

Find quantum numbers for each of three possible quantum states

Relevant Equations

E= (( h^2)/(8mD) ) (nx^2_ny^2, nz^2)

Hi Everyone.
I hope someone can point me in right direction. I am struggling to work this out . If it was 1d confinement the calculated n number would be the energy level. So for example n= 3, means that quantum number is n= 3 and there is 3 possible quantum states. Is that correct?

With 3D box i am getting confused what values nx , ny, nx can have for the E given.

 

[anuttarasammyak]

I asked Wolfram as shown

There seems no integers to satisfy the relation. In neighbor, (2,3,27) satisfies
$$a^2+b^2+c^2=582$$ and (2,2,24),(6,8,22) satisfies
$$a^2+b^2+c^2=584$$ where I excluded (0,10,22) which includes physically prohibited 0.

 

[greg997]

Thank you for that. What would be the conclusion, interpretation of such solution? - No such energy levels exists?

 

[BvU]

I am surprised that, with 1 digit of precision in the given variables, the search is limited to $\sum n^2 = 583\quad$

$\$

 

[anuttarasammyak]

Thank you for that. What would be the conclusion, interpretation of such solution? - No such energy levels exists?

In a narrow sense of mathematics, you are right. But in physics almost all the numbers in calculation is approximate. I am afraid that thinking of integer 583 just is not practical.

 

[Steve4Physics]

If it was 1d confinement the calculated n number would be the energy level. So for example n= 3, means that quantum number is n= 3 and there is 3 possible quantum states. Is that correct?

Not quite. Given a particular value of energy, if you found n=3 that would mean the system is in the n=3 state. There is only one such state for a simple '1D particle in a box' and there are an infinite number of other states (each with its own unique energy value).

With 3D box i am getting confused what values nx , ny, nx can have for the E given.
View attachment 344371

The box is about the size of a (large) atom.
The mass is about $10^{15}$ times bigger than the mass of a large atom.
The value of energy is remakably small.
So I’m wondering if you have the correct data/units?

And, as others have pointed out, an energy of $5 \times 10^{-37}$ J is precise to only 1 significant figure. This would mean the energy is between $4.5 \times 10^{-37}$ J and $5.5 \times 10^{-37}$ J. Similarly for other values.