Quantum Field Theory I

58 CHAPTER 2. FREE SCALAR QUANTUM FIELD
Scalar Field as Harmonic Oscillators
The linear harmonic oscillator is just a special case of the already discussed
example, namely a particle in the potential U(x). One can therefore quantize
theLHOjust asin theabovegeneralexample(with the particularchoiceU(x) =
mω 2 x 2 /2), but this is not the only possibility.
Let us recall that search of the solution of the LHO in the QM (i.e. the
eigenvalues and eigenvectors of the Hamiltonian) is simplified considerably by
introduction of the operators a and a + . Analogous quantities can be introduced
already at the classical level 12 simply as
√ mω
a = x
2 +p i
√
2mω
√ mω
a + = x
2 −p i
√
2mω
The point now is that the canonical quantization can be performed in terms of
the variables a and a +
L = mẋ2
2
− mω2 x 2
2
H = p2
2m + mω2 x 2
2
= ωa + a or H = ω 2 (a+ a+aa + )
↓
{a,a + } = i ȧ = −iωa ȧ + = iωa +
H = ωa + a or H = ω 2 (a+ a+aa + )
[a,a + ] = 1 ȧ = −iωa ȧ + = iωa +
↓
H = space spanned by |0〉,|1〉,...
a|n〉 = |n−1〉
a + |n〉 = |n+1〉
Note that we have returned back to the convention = c = 1 and refrained
from writing the hat above operators. We have considered two (out of many
possible) Hamiltonians equivalent at the classical level, but non-equivalent at
the quantum level (the standard QM choice being H = ω(a + a+1/2)). The
basis |n〉 is orthogonal, but not orthonormal.
12 The complex linear combinations of x(t) and p(t) are not as artificial as they may appear
at the first sight. It is quite common to write the solution of the classical equation of motion
forLHO inthe complex formas x(t) = 1 2 (Ae−iωt +Be iωt ) and p(t) = − imω
2 (Ae−iωt −Be iωt ).
Both x(t) and p(t) are in general complex, but if one starts with real quantities, then B = A ∗ ,
and they remain real forever. The a(t) is just a rescaled Ae −iωt : a(t) = √ mω/2Ae −iωt .