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The harmonic oscillator as a quantum field
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The harmonic oscillator has two different interpretations:
(i) It can be viewed as a single particle in 1+1 dimensions (which is
how it is introduced in QM), where it describes a particle in
1-dimensional space R^1 with states in L^2(R^1), bound in an external
field and oscillating in time.
(ii) It can be viewed as a free field in 1+0-dimensions, where it
describes an arbitrary number of noninteracting particles in
0-dimensional R^0={0} with states in C^1=L^2(R^0).
Mathematically, it is precisely the same - physically, the
interpretation is radically different! Here we look at the field
interpretation. In a field interpretation, Single-particle QM is
1+0-dimensional field theory.
For the harmonic oscillator, a^* creates one particle at a fixed
(unmentioned) time. Since there is no space, there is no momentum to
distinguish single particle states.
|0>=|> is the vacuum (ground state),
|1>=a^*|0> is the 1-particle state,
|2>=a^*|1> the 2-particle state, etc..
The frequency omega of the harmonic oscillator is the particle mass
(setting c=1 and hbar=1): m=omega, and the N-particle state has the
mass E_N=N*omega of N particles. The Hamiltonian is H=omega a^*a.
In higher dimension, each particle comes together with its quantum
numbers (for a scalar field just the momentum p); thus there is one
creation operator a^*(p) for each allowed quantum number p, and there
are many 1-particle states |p>=a^*(p)|>, and even more 2-particle
states |p',p>=a^*(p')|p>.
Thus the notation is a bit different. To make the analogy perfect,
one should assign the 1-particle state in dimension 0 a zero momentum
and write a^*(0) for a^*, |> for the vacuum, |0> for the 1-particle
state, |0,0> for the 2-particle state, etc., and the Hamiltonian as
H = sum_{p in R^{0}} m a^*(p)a(p),
which indeed equals omega a^*a, since R^{0} contains only one element.