When reading about the quantum harmonic oscillator, you often hear about it in reference to bosons. You can define ladder operators with commutation relations, and when acting on Fock states can occupy the ground state energy level with as many bosons as you want. I see how the situation is the same when you occupy the HO levels with fermions, because the energy when adding a fermion to the next energy level is the same as if you occupied the ground state with another boson. However, wouldn't it be possible to use anti-commuting CAPs on the energy states? What if we had an electron in a quantum dot confined by a parabolic potential, is this still bosonic?
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The harmonic oscillator is important because it's an approximate solution to nearly every system with a minimum of potential energy. – rob Jun 04 '22 at 20:34
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To focus your thinking, you might consider just one fermion operator, in isolation of all others, and try to imagine an oscillation. – Cosmas Zachos Jun 04 '22 at 20:46