# Quantum Theory of the Damped Harmonic Oscillator

The quantum theory of the damped harmonic oscillator has been considered a simple model for a dissipative system, usually coupled to another oscillator that can absorb energy or to a continuous heat bath [1–3]. This Demonstration treats a quantum damped oscillator as an isolated nonconservative system, which is represented by a time-dependent Schrödinger equation. It is conjectured that spontaneous transition to a lower state will occur when the energy is reduced to that of the lower state, and this recurs sequentially, down to the ground state, which asymptotically disappears as the energy approaches zero. Within this model, the obtained result is an exact solution of the time-dependent Schrödinger equation.
As derived in the Details below, the time-dependent wavefunction is given by , ,
where is an Hermite polynomial, and ,
where is the natural frequency of the undamped oscillator and is the damping constant. Atomic units are used. The real part of the expectation value of the Hamiltonian is assumed for the time-dependent energy, which gives .
The graphic shows the probability density and energy as functions of for . The inset shows the energy, as downward transitions occur, asymptotically decreasing to 0.

### DETAILS

The classical damped harmonic oscillator is described by the well-known equation ,
where is the mass of the oscillating particle, is the Hooke’s law force constant and is a damping constant. It is convenient to define the modified damping constant and the natural frequency of the undamped oscillator ,
which reduces the first equation to .
The solution of the equation of motion can be expressed in the form .
For the quantum analog of the damped oscillator, we propose the Hamiltonian ,
which is non-Hermitian but accounts nicely for the decay of the system. The corresponding time-dependent Schrödinger equation for is given by ,
with the solutions given above.
References
 P. Caldirola, "Forze Non Conservative Nella Meccanica Quantistica," Il Nuovo Cimento, 18(9), 1941 pp. 393–400. doi:10.1007/BF02960144.
 E. Kanai, "On the Quantization of the Dissipative Systems," Progress of Theoretical Physics, 3(4), 1948 pp. 440–442. doi:10.1143/ptp/3.4.440.
 C.-I. Um and K.-H. Yeon, "Quantum Theory of the Harmonic Oscillator in Nonconservative Systems," Journal of the Korean Physical Society, 41(5), 2002 pp. 594–616.

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