Calculate the ground state energy of a helium atom in a box of1cm on a side.
Solution: these are constants that we will see again andagain. I set c = 1.
In the ground state = 3, the helium mass is about 4 times theproton 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 roomtemperature energy of .025 ev. You can quickly verify that this isalso the spacing between the ground and first excited state. We cancalculate the associated classical velocity
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Remember c = 1 but this is still quite slow (it’s about amicrometer per second) in the grand scheme. I think it’s importantto do back of the envelope calculations like this to buildintuition about the magnitudes of things in a theory
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2 : repeat the above calculation for diatomic oxygen in a 1mon a side box. Find the classical velocity of the particle.
3: Take the oxygen molecule in the one centimeter box to deepspace where the temperature is 3K. Assume the particle is now at 3Kas well. Determine its classical speed and find an estimate (useequipartition of energy arguments) for the particles quantumstate.
