Harmonic Oscillators and Coherent States
The harmonic oscillator is the general approximation for the dynamics of small fluctuations around a minimum of a potential. This is the reason why harmonic oscillators are very important model systems both in mechanics and in quantum mechanics. In additi
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Harmonic Oscillators and Coherent States
The harmonic oscillator is the general approximation for the dynamics of small fluctuations around a minimum of a potential. This is the reason why harmonic oscillators are very important model systems both in mechanics and in quantum mechanics. In addition there is another reason why we have to discuss the quantum harmonic oscillator in detail. For the discussion of quantum mechanical reactions between particles later on, we have to go beyond ordinary quantum mechanics and use a technique called second quantization or canonical quantum field theory. The techniques of second quantization are based on linear superpositions of infinitely many oscillators. Therefore it is important to have a very good understanding of oscillator eigenstates and of the calculational techniques involved with oscillation operators.
6.1 Basic Aspects of Harmonic Oscillators The classical motion of a particle in the three-dimensional isotropic potential V (x) =
m 2 2 ω x 2
(6.1)
without external driving forces is described by the classical solution P sin(ωt), mω p(t) = P cos(ωt) − mωX sin(ωt), x(t) = X cos(ωt) +
(6.2)
˙ where X = x(0), P = mx(0) are the values of location and momentum of the particle at time t = 0. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_6
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6 Harmonic Oscillators and Coherent States
The corresponding Schrödinger equation is d ih¯ |ψ(t) = dt
p2 m 2 2 + ω x |ψ(t) 2m 2
(6.3)
or after substitution of energy-time Fourier transformation (5.35), E|ψ(E) =
p2 m 2 2 + ω x |ψ(E). 2m 2
(6.4)
The corresponding differential equation in x representation
h¯ 2 m 2 2 Ex|ψ(E) = − + ω x x|ψ(E) 2m 2
(6.5)
can be decomposed into three one-dimensional problems through separation of the spatial variables. The separation ansatz x|ψ(E) =
3
xi |ψi (Ei )
(6.6)
i=1
yields E=
3
Ei ,
(6.7)
i=1
where the three energy values Ei and wave functions xi |ψ(Ei ) have to satisfy the one-dimensional equation Ex|ψ(E) = −
h¯ 2 d 2 m x|ψ(E) + ω2 x 2 x|ψ(E). 2m dx 2 2
(6.8)
Indeed, the results of Sect. 5.5 imply that the solutions of the three-dimensional Eq. (6.5) can always be found in the separated form (6.6) and (6.7).
6.2 Solution of the Harmonic Oscillator by the Operator Method The one-dimensional oscillator equation (6.8) is in representation free notation E|ψ(E) =
m p2 + ω2 x2 |ψ(E). 2m 2
(6.9)
6.2 Solution of the Harmonic Oscillator by the Operator Method
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There exists a powerful and elegant method to solve equation (6.9) through a transformation from the self-adjoint operators x and p to mutually adjoint operators a and a + . The substitutions √ √ 1 p 1 p + , a =√ , (6.10) a=√ mωx + i √ mωx − i √ mω mω 2h¯ 2h¯ yield the commutation relation [a, a + ] = 1, the inverse transformation
h¯ a + a+ , x= 2mω
p = −i
(6.11)
mωh¯ a − a+
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