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]

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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21 replies
1. A. Neumaier says:
hilbert2

I tested the nonlinear Schrödinger equation that I mentioned in the insight as one way to produce effective damping in QM, and the correct way to form the damped oscillator TDSE seems to be
##ihbar frac{partial Psi}{partial t} = -frac{hbar^2}{2m}frac{partial^2 Psi}{partial x^2} + frac{1}{2}kx^2Psi + alpha Imleft(int_{-infty}^{x}Psi (x',t)dx'right)Psi##,

But this is no longer a differential equation but an integro-differential equation with a memory term that depends on an infinitely long past. You cannot even solve it when you initially only have information at time $t=0$, as in most physical problems. So I don't understand how you can evolve your state at time $t=0$.

This might be ok for an academic computational project. But no engineer would be using such equations to study dissipative problems in fluid flow, say. Your approach is very ad hoc and specialized – it does not even cover such well-studied dissipative processes as the Navier-Stokes equations!

2. hilbert2 says:

The integration is only over ##x'##, and the variable ##t## in the integrand is set to the constant value of present time, so there is no memory. Therefore the solution can be approximately calculated by updating the potential ##V(x)## on each time step to correspond to the wavefunction ##psi (x)## at that moment.

EDIT: Here's a video about fluid mechanics simulations where a nonlinear Schrödinger equation produces results equivalent to Navier-Stokes: .

3. A. Neumaier says:
hilbert2

the variable t in the integrand is set to the constant value of present time, so there is no memory.

Sorry for having misread your formula. Still, it is no longer a partial differential equation but a nonlocal dynamics that is very uncommon in practice.

4. Robert M says:

Energy dissipation in turbulent flow is certainly a problem of great interest to scientists, engineers, and mathematicians, but it is not the first path I would choose to study "damped motion." One natural analog of damped oscillator problems, which have generally been rendered artificially linear, that are studied in introductory mechanics would be to study problems that can be made artificially linear in fluid mechanics, whether compressible or not, or, say, magnetohydrodynamics, where Ohm's Law provides a dissipative knob that can be turned. Conflating the mechanics of linear viscous dissipation with the complicated mechanics of turbulent flow seems unhelpful as pedagogy.

Introduction of particle kinetics is another unnecessary distraction. Linear or nonlinear, deriving continuum equations by taking moments of the Boltzmann equation is tangential to the subject at hand. Standard approximations used by engineers, mathematicians, and even plasma physicists take care of the thermodynamics, which can also be made linear for studying, say, damped acoustic waves, so non-equilibrium thermodynamics would be interesting mostly if that and not "damped motion" were the subject at hand.

I'm not a solid mechanics wizard, so I don't know the details of how structural engineers do structural damping or how damping is modeled in seismic mechanics, but those also seem like better paths than wandering off into turbulent flow, nonequilibrium thermodynamics, particle kinetics, or other advanced topics.

I'm confident that there must be a need for studying "damped motion" or its analog in the context of quantum mechanics. I just don't know what it is.

5. hilbert2 says:
Robert M

I'm confident that there must be a need for studying "damped motion" or its analog in the context of quantum mechanics. I just don't know what it is.

The infrared and microwave spectral line broadening in solid and liquid phase samples can be modelled by using some kind of effective damping to represent the interaction of the rotating or vibrating molecules with neighboring atoms and molecules. This can be done by calculating the spectrum as a Fourier transform of an autocorrelation function ##left<psi(t_0 )|psi (t)right>##, as described here: http://home.uchicago.edu/~tokmakoff/TDQMS/Notes/6.1.-6.3_Absorption_Lineshape_2-25-09.pdf .

6. Robert M says:
hilbert2

The infrared and microwave spectral line broadening in solid and liquid phase samples can be modelled by using some kind of effective damping to represent the interaction of the rotating or vibrating molecules with neighboring atoms and molecules. This can be done by calculating the spectrum as a Fourier transform of an autocorrelation function ##left<psi(t_0 )|psi (t)right>##, as described here: http://home.uchicago.edu/~tokmakoff/TDQMS/Notes/6.1.-6.3_Absorption_Lineshape_2-25-09.pdf .

Thanks. It occurred to me after I had posted my comment that there must be examples in solid state physics where it is generally thought to be appropriate to speak of damped motion, the first likely example of which that occurred to me was phonon damping. If I understand your offered article correctly, the "damping" is really relaxation from one state to another, with whatever energy is lost being carried off as radiation. I believe that's a fairly general characteristic of "damping" in quantum mechanics that distinguishes it from classical mechanics, where energy simply "disappears" (or is converted to heat).

7. Robert M says:
hilbert2

The infrared and microwave spectral line broadening in solid and liquid phase samples can be modelled by using some kind of effective damping to represent the interaction of the rotating or vibrating molecules with neighboring atoms and molecules. This can be done by calculating the spectrum as a Fourier transform of an autocorrelation function ##left<psi(t_0 )|psi (t)right>##, as described here: http://home.uchicago.edu/~tokmakoff/TDQMS/Notes/6.1.-6.3_Absorption_Lineshape_2-25-09.pdf .

Line-broadening may, after all, be a better way to characterize the topic than anything to do with (say) opposing motion, as it generalizes naturally (as you have shown) to quantum mechanics. For many applications, the "Q" of resonant processes is more important for calculation than intuitive notions of frictional losses. A focus on line width also sidesteps a messy detour into thermodynamics and a necessarily complicated discussion of where energy is "lost" to.

8. Dr.D says:

For holonomic systems, the inclusion of dissipation terms (and driving/exciting terms) in the equations of motion is not difficult if the system is formulated by means of Hamilton's Principle. This topic is developed at some length in Dynamics of Mechanical and Electromechanical Systems by Crandall, Karnopp, Kurtz, and Pridmore-Brown (McGraw-Hill, 1968). Provided the functional form of the frictional force can be expressed, it is rather automatic to include such terms in a virtual work term to be included in the variational indicator.

9. hilbert2 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). That seems to be based on monitoring a vibration and applying a force that's in anti-resonance with the motion. Here's a publication about a related thing: http://www.dspe.nl/files/MechatronicsPaper.pdf . This seems to be similar to my own idea of a damping term that is proportional to a momentum expectation value.