Question

Calculate the ground state energy of a helium atom in a box of 1cm on a side.

Solution: these are constants that we will see again and again. I set c = 1.

In the ground state = 3, the helium mass is about 4 times the proton mass, and using , the energy is roughly

E = [4(10^6)10(3)]/[2(4)(10^9)(10^16)] ~ 1.5 x 10^(-18) ev.

This is a small energy, especially when compared to room temperature energy of .025 ev. You can quickly verify that this is also the spacing between the ground and first excited state. We can calculate the associated classical velocity

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Remember c = 1 but this is still quite slow (it’s about a micrometer per second) in the grand scheme. I think it’s important to do back of the envelope calculations like this to build intuition about the magnitudes of things in a theory

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2 : repeat the above calculation for diatomic oxygen in a 1m on a side box. Find the classical velocity of the particle.

3: Take the oxygen molecule in the one centimeter box to deep space where the temperature is 3K. Assume the particle is now at 3K as well. Determine its classical speed and find an estimate (use equipartition of energy arguments) for the particles quantum state.

Answer 1