Comment on "Quantization of the damped harmonic oscillator" [Serhan et al, J. Math. Phys. 59, 082105 (2018)]

TL;DR

This paper critiques a recent work on quantizing the damped harmonic oscillator, emphasizing the importance of non-Hermiticity in the Hamiltonian as a key quantum signature of dissipation.

Contribution

It highlights the oversight of non-Hermiticity in the previous Hamiltonian construction, clarifying its role in representing dissipation in quantum systems.

Findings

The Hamiltonian in the criticized paper is non-Hermitian.

Real eigenvalues claimed in the paper are actually complex.

Non-Hermiticity is essential for capturing dissipation quantum mechanically.

Abstract

A recent paper [J. Math. Phys. {\bf 59}, 082105 (2018)] constructs a Hamiltonian for the (dissipative) damped harmonic oscillator. We point out that non-Hermiticity of this Hamiltonian has been ignored to find real discrete eigenvalues which are actually non-real. We emphasize that non-Hermiticity in Hamiltonian is crucial and it is a quantal signature of dissipation.

Equations

$$\ddot{q}+\lambda \dot{q}+{\omega }^{2}q=0.$$

$$H=\frac{p_y^2}{2m}+\frac{1}{2}m{\omega }^2{y}^2+\frac{\lambda y{p}_y}{2}.$$

$$E_n=(n+1/2)\hslash \sqrt{{\omega }^2-{\lambda }^2/4}$$

$$H=\frac{p_y^2}{2m}+\frac{1}{2}m{\omega }^2{y}^2+\frac{\lambda (y{p}_y+p_{y}y)}{4}+\frac{i\hslash \lambda }{4}.$$

$$H=\frac{(p_y+m\lambda y/2{)}^2}{2m}+\frac{1}{2}m({\omega }^2-{\lambda }^2/4)y^2+\frac{i\hslash \lambda }{4}.$$

$$e^{(im\lambda y^2/4\hslash )}[p_y+m\lambda y/2]e^{-(im\lambda y^2/4\hslash )}=p_y,$$

$$e^{(im\lambda y^2/4\hslash )}[p_y+m\lambda y/2{]}^n{e}^{-(im\lambda y^2/4\hslash )}=p_y^n.$$

$$\eta H{\eta }^{-1}=\frac{p_y^2}{2m}+\frac{1}{2}m({\omega }^2-{\lambda }^2/4)y^2+\frac{i\hslash \lambda }{4}.$$

$$E_n=(n+1/2)\hslash \sqrt{{\omega }^2-{\lambda }^2/4}+\frac{i\hslash \lambda }{4},$$

Full text

Heuristically, by making a Hamiltonian non-Hermitian one can account for dissipation of energy and matter. Presently interesting frameworks are being developed [1] wherein classical dissipative systems can be associated with non-Hermitian quantum Hamiltonians. The simplest model of dissipative systems is classical damped harmonic oscillator (DHO) whose equation of motion is

[TABLE]

Recently, this has been transformed to a Hamiltonian $H$ [2].

[TABLE]

We point out that the non-Hermiticity of $H$ due to $(yp_{y})^{\dagger}=yp_{y}-i\hbar$ has been ignored and real discrete eigenvalues

[TABLE]

have been obtained [2] by some N-U method [2]. Using $yp_{y}-p_{y}y=i\hbar$, $H$ can be re-written as

[TABLE]

This can be re-written as

[TABLE]

Consider the eigenvalue equation: $H\psi=E\psi\Rightarrow\eta H\eta^{-1}\eta\psi=E\eta\psi$ (see [3]). Let us choose $\eta=\exp(im\lambda y^{2}/4\hbar)$. We find that

[TABLE]

Consequently,

[TABLE]

Finally the quantization of (7) leads to

[TABLE]

which are complex and it is rightly so, as the Hamiltonian (2) is non-Hermitian. One needs to highlight that the Hamiltonian (2) for the damped harmonic oscillator (1) is crucially non-Hermitian, this will help in the development of quantization of dissipative systems.