Damped Motion in Classical and Quantum Mechanics

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This Insights article is about dissipative forces (friction, air resistance, viscosity…).

Damping by friction forces is one of the concepts that is encountered earliest by physics students. The reason for its importance is that almost all moving objects in everyday life are affected by friction and air resistance, which is why Newton’s 1st law is not intuitively obvious.[1] When motions are damped, it’s not easy to see that force is only required for the production of acceleration, not for maintaining a constant velocity. The science that studies friction between surfaces is called tribology.

In classical Newtonian mechanics, where one solves a problem by drawing a free body diagram of the bodies involved, the dissipative forces of friction and air resistance are usually modeled by adding to the total force a term that has a direction opposite to the direction of motion and magnitude that is proportional to some power of the speed of the object. [2] In the simplest possible case of motion of a single object in one dimension, this would mean that the equation of motion has a term

##F_d = -\beta \frac{dx(t)}{dt}##, or

##F_d = -\beta \left|\frac{dx(t)}{dt}\right|\frac{dx(t)}{dt}##,

or similar (the dimensions of parameter ##\beta## have to be chosen appropriately, of course).

In Lagrangian or Hamiltonian mechanics, where energy is chosen as the most important physical quantity instead of the force of Newton’s mechanics, it is a lot more difficult to create dissipative equations of motion, as frictional forces break the conservation of energy which is a built-in property of systems with a time-independent Hamiltonian. The same problem also appears in the theory of quantum mechanical damped motion, as quantum mechanics is formed on the basis of classical Hamiltonian mechanics. A physically correct treatment of friction would require a very large number of degrees of freedom, corresponding to the thermal motion of molecules, being weakly coupled to the macroscopic object that is subject to frictional forces. In that kind of a system, damping by transfer of energy from the macroscopic object to the microscopic degrees of freedom would be guaranteed by the 2nd law of thermodynamics, as long as the number of the degrees of freedom would be large enough for concepts of statistical mechanics to be applicable.[3]

Fortunately, there are some ways how a dissipative force can be artificially added to a harmonic oscillator or some other simple physical system in Hamiltonian mechanics. One of these is to take the Hamiltonian of the undamped system

##H(p,x) = \frac{p^2}{2m}+\frac{1}{2}kx^2## ,

where ##m## is the mass of the object and ##k## the Hooke’s spring constant, and make ##m## and ##k## time dependent exponentially growing quantities:

##m(t) = m_0 e^{at}##
##k(t) = k_0 e^{at}##

The reader may check that putting these forms of mass and spring constant in the Lagrangian or Hamiltonian function results in equations of motion that contain a velocity-dependent damping term. However, this approach will cause trouble if the system contains other particles/bodies that interact with the damped oscillator. Also, it is only a mathematical trick that produces the “correct” damped trajectories of motion, and has nothing to do with the actual physical mass or spring constant really changing in time.

In quantum mechanics, the Hamiltonian operator of a harmonic oscillator is, of course

##\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}k\hat{x}^2##,

and the time development of a wave function ##\psi (x,t)## is given by the time-dependent Schrödinger equation

##i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi (x,t)## .

If the initial state of the oscillator is given by a displaced Gaussian wave function

##\psi (x,0) = Ce^{-a(x-x_0 )^2}##,

the time evolution of the position expectation value ##\left<x\right>## has the form of a cosine function

##\left<x\right>(t) = x_0 \cos \omega t##,

just like a classical oscillator would move if released from rest at a non-equilibrium position. [4] To make a dissipative quantum harmonic oscillator, one can just make the value of the mass or the spring constant (or both) slightly complex, with a small imaginary part. Then, if the signs of the imaginary parts are chosen correctly, the time evolution of ##\left<x\right>## has the form of a “cosine with exponentially decaying amplitude”, at least in the limit of weak damping. The problem with this approach is that the time development will not be unitary then, so the wave function has to be normalized again by hand after numerically computing the time evolution for the desired interval. The same complex-values method also works for a classical oscillator (if the time variable or the mass/spring constant are given a suitable imaginary part), but then the position ##x(t)## will be a complex valued function and its real or imaginary part has to be interpreted as the actual physical position.

Other ways to make a dissipative quantum harmonic oscillator include adding a non linearity to the TDSE. One way to do this, which unfortunately requires giving up the locality of the theory, is to use an equation

##i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + \frac{1}{2}kx^2 \Psi + \alpha Im\left( \int_{-\infty}^{x}\Psi (x’,t)dx’\right)##,

where ##Im## means the imaginary part, or something that contains a nonlinear term dependent on the temporary momentum expectation value ##\left< p \right>##. Effective damping models can be used to model the IR or microwave spectral line broadening in solid and liquid samples, where the molecules interact with many neighboring ones (a heat bath).[5]

[1] https://www.khanacademy.org/science/physics/forces-newtons-laws/newtons-laws-of-motion/a/what-is-newtons-first-law

[2] http://physics.gantep.edu.tr/media/kunena/attachments/237/chap2-2-3-4.pdf

[3] https://web.stanford.edu/~peastman/statmech/friction.html

[4] http://www.weizmann.ac.il/chemphys/tannor/Book/ch3/har_numeric2.html

[5] http://home.uchicago.edu/~tokmakoff/TDQMS/Notes/6.1.-6.3_Absorption_Lineshape_2-25-09.pdf


Physicist from Finland, graduated B.Sc and M.Sc. from university of Jyväskylä, now working and doing PhD studies in a fire safety research group in Aalto university, Espoo, Finland.
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  1. Robert M
    Robert M says:

    I saw an announcement about a diploma thesis presentation (in the university I'm doing PhD studies in) today, which was on an interesting subject, namely active damping of oscillations (in contrast to passive friction forces).

    Active dampers are used in skyscrapers to minimize wind-induced vibrations and seismic shaking. An active damper was part of solving the problem of windows popping out of their frames in the I M Pei-designed Hancock Tower in Boston.

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