Particle confined in 3D box - quantum states

  • #1
greg997
109
2
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.
1.jpg
 
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  • #2
I asked Wolfram as shown
1714694120082.png

There seems no integers to satisfy the relation. In neighbor, (2,3,27) satisfies
[tex]a^2+b^2+c^2=582[/tex] and (2,2,24),(6,8,22) satisfies
[tex]a^2+b^2+c^2=584[/tex] where I excluded (0,10,22) which includes physically prohibited 0.
 
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  • #3
Thank you for that. What would be the conclusion, interpretation of such solution? - No such energy levels exists?
 
  • #4
I am surprised that, with 1 digit of precision in the given variables, the search is limited to ##\sum n^2 = 583\quad ## :rolleyes:

##\ ##
 
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  • #5
greg997 said:
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.
 
  • #6
greg997 said:
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).

greg997 said:
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.
 
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