The harmonic oscillator as a tutorial introduction to quantum mechanics

Martin Devaud, Thierry Hocquet

Published Feb 1, 2018• 0 citations

Abstract

Stemming from the similar linearities of the Schrödinger equation in quantum mechanics on the one hand and of the harmonic oscillations in classical mechanics on the other hand, the idea that any N-degree-of-freedom harmonic oscillator (HON) is formally equivalent to a N-level quantum system is put forward. It is shown that the complex dynamic variables α introduced by R. J. Glauber can be regarded as the components of a state vector belonging to some N-dimension complex Hilbert space, and whose time-evolution is ruled by a Schrödinger-like equation. In case the classical HON is parametrically excited, the unitarity of the time-evolution of the associated quantum system is related to the Ehrenfest adiabaticity of the parametric excitation.

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The harmonic oscillator as a tutorial introduction to quantum mechanics

Martin Devaud, Thierry Hocquet

Published Feb 1, 2018• 0 citations

OpenAlex

Abstract

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The quantum dynamics of arbitrary

N

-level systems, including dissipative systems, are modeled exactly here by the dynamics of classical coupled oscillators. A one-to-one correspondence is established between the quantum states and the positions of the oscillators. Quantum coherence, expectation values, and measurement probabilities for system observables can therefore be realized from the corresponding classical states. Although the well-known equivalence [SU(2), SO(3) homomorphism] of two-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitrary

N

-level systems, the representation of quantum dynamics by a system of coupled harmonic oscillators in one physical dimension

is

general for any

N

. The time evolution of an

N

-level system [generated by a complex element of the SU(

N

) group], is first represented as the rotation of a real state vector in (unphysical) hyperspace, as previously known for density matrix states and also extended here to include Schr\"odinger states. The resulting rotor in

n

Euclidean dimensions [the rotation group SO(

n

)] is then mapped directly to

n

oscillators in one physical dimension, which significantly reduces the level of abstraction required to visualize quantum dynamics compared to vector models or generalized Bloch spheres in higher dimensions. The number of such oscillators needed to represent

N

-level systems scales as

{N}^{2}

for the density matrix formalism but increases only linearly with

N

for Schr\"odinger states. Values for the classical coupling constants are readily derived from the system Hamiltonian, allowing construction of classical mechanical systems that provide insight into the dynamics of abstract quantum systems (new dynamical invariants) as well as a metric for characterizing the interface between quantum and classical mechanics. A distinctive attribute of the quantum-classical connection as presented here is the necessity for both positive and negative couplings and, in the case of dissipative systems, antisymmetric couplings.

\mathit{x}

\hat{v}

) , with the commutation relation [

\mathit{x}

\hat{v}

] =

\mathit{i}\hbar

\mathit{m}

, and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion

\mathit{K}

(

\mathit{x, v, t}

) of the classical system, The comparison with the quantization in the space (

\mathit{x, p}

) is done with the usual Schrödinger’s equation for the Hamiltonian

\mathit{H}

(

\mathit{x, p, t}

), and with the commutation relation [

\mathit{x}

\hat{p}

] =

\mathit{i}\hbar

. It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (

\mathit{x}

\hat{v}

) quantization has fewer oscillations than the (

\mathit{x}

\hat{p}

) quantization, the average energy of the system is higher in (

\mathit{x}

\hat{p}

) quantization than on the (

\mathit{x}

\hat{v}

) quantization, and the Boltzmann- Shannon entropy on the (

\mathit{x}

\hat{p}

) quantization is higher than on the (

\mathit{x}

\hat{v}

) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.