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I am currently learning QFT, and after watching the wonderful lectures by Leonard Susskind (https://theoreticalminimum.com/courses/advanced-quantum-mechanics/2013/fall), I am still struggling to see the connection between multi-particle (Fock) states and harmonic oscillators. When constructing Fock states, prof. Susskind used the "particle in a box" model for individual particle states. In this model, the particle wave functions are the energy eigenstates (standing waves) of a particle in a box. A Fock state is written as a sequence of occupation numbers for each energy eigenstate (i.e. how many particles exist in each state). However, from other QFT lectures, I recall that adding a particle with a specific momentum corresponds to increasing the excitation number of a harmonic oscillator. This is quite different from the "particle in a box" model. What am I missing here? Is the "particle in a box" model just a simplification, and the actual states should be associated with harmonic oscillators?

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Fock space description and second quantization are not specific to harmonic oscillators - this is simply counting how many particles are in each state, whatever is the nature of the states. Creation/annihilation operators serve here to increase or reduce the number of particle in a state.

What often serves as a source of confusion is that for a one-particle oscillator (not in a Fock space!) one can introduce creation and annihilation operators that increase/reduce the excitation number. Moreover, when we quantize electromagnetic field, which is interpreted as a collection of oscillators, the excitation numbers are interpreted as the number of photons - quite literally becoming the creation and annihilation operators in the Fock space (note that the second quantization formalism applied to the electromagentic field is actually the first quantization of this field, sicne it is already a wave field).

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