Electron in 1D Box: classical or quantum at different temps

[psyklon]

Hi, I'm working on a problem that requires me to calculate thermal energy (kT) at different temperatures and compare those values to the lowest state energy of a particle in box (1D) of varying lengths.

I've calculated the ground-state energies of the electron in all of these different sized boxes. I have also evaluated kT for several temperatures ranging from near-zero K to 1000 K. Now I have to compare these values to determine if each system behaves either quantum mechanically or classically, and I'm not really sure how to do that.

My thought process at the moment is that at the lower temperatures the lack of thermal energy means that the electron will remain in n=1, and so will behave quantum mechanically. However, at sufficiently high temperatures it will behave classically. So my thought is that if the n=1 energy is higher than the thermal energy, the electron will be in the QM realm. But if the n=1 energy is lower than the thermal energy, there is a chance that the electron will behave classically, as n will be going towards infinity.

My next step, I think, is calculating the speed of the electron at each of the different temperatures and finding the deBroglie wavelength, then comparing this to the length of the box.

So I suppose my question is: is my train of thought correct so far? If so, how do I calculate the total energy of the electron as a result of the increased temperature?

 

[mfb]

Your approach is good and sufficient to give an answer. The classical approach works as soon as the energy levels are so close you don't see the differences any more. The thermal energy should correspond to a very large n (10, 100, something like that) to ignore quantum mechanics.

My next step, I think, is calculating the speed of the electron at each of the different temperatures and finding the deBroglie wavelength, then comparing this to the length of the box.

That won't tell you anything new because it compares the same things.