# 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]

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21 replies
1. Mentz114 says:

I'm glad this subject is being discussed. The world would not function the way it does without friction, viscosity and various kinds of resistance.
In fact it is very difficult to eliminate these effects and any realistic modelling must include them.
But most of the physical modelling ignores dissipation and leads to strange and even ridiculous conclusions.

If a marble is set moving in a bowl it will keep moving forever without some kind of dissipative effect – in fact one could say that the experiment has no outcome unless there is friction. Dissipation creates definite outcomes and without including it in models one soon meets conundra like the 'measurement problem'.

2. A. Neumaier says:

The way described in this Insight article is not the correct way to include dissipation into the description of physical systems; it works well only in a few toy examples. I'd like suggest that the author retracts the article and makes himself better informed before reposting an improved version.

For a system of ##n## oscillators with ##n##-dimensional position vector ##q##, the correct dissipative version of the conservative equation for nonlinear (Lagrangian or hamiltonian) oscillators dynamics ##Mddot q+nabla V(q)=0## is given by ##Mddot q+C(q,dot q)dot q +nabla V(q)=0##. Here $M$ is the diagonal mass matrix whose diagonal entries are the oscillator masses, ##V(q)## is the potential in which the oscillators move, and ##C(q)## is a positive definite damping matrix that determines the detailed friction behavior. it is easily seen that the total energy ##E=frac12 Mdot q^2+ V(q)## is strictly decreasing as long as the velocity is nonzero, matching the experimental characteristics of friction. The case where ##C## is constant and ##V(q)_frac12 q^TKq## with a constant stiffness matrix ##K## leads to the linear dynamical system ##Mddot q+Cdot q +Kq=0##, which is the basis of most engineering calculations for friction in oscillating mechanical structures. More complex dissipative systems need a more complicated form of these equations, generalizing them to a Hamiltonian or Lagrangian framework.

In the quantum case, things are also a bit more complicated, though for different reasons. The correct high-level description of dissipative processes is given by a so-called Lindblad equation, a generalization of the quantum Liouville equation for conservative systems.

3. hilbert2 says:

I've done some elasticity calculations where we had to use a strain rate tensor to describe viscous friction in a deformable object. Simply adding a force proportional to velocity to the equations of motion of the volume elements of the object didn't work, as it would also damp the translational or rotational motion of the object as whole, too, even if it was supposed to be in vacuum.

4. A. Neumaier says:
hilbert2

I've done some elasticity calculations where we had to use a strain rate tensor to describe viscous friction in a deformable object. Simply adding a force proportional to velocity to the equations of motion of the volume elements of the object didn't work, as it would also damp the translational or rotational motion of the object as whole, too, even if it was supposed to be in vacuum.

Well, your article only describe ODEs, and I gave the most typical way to include dissipation, not the most general one. For PDEs such as in elasticity calculations, one needs schemes that also produce Navier-Stokes from Euler and can handle dissipative reaction-diffusion equations. None of these are of the form you described in your article. The book by Oettinger, Nonequilibrium Thermodynamics, might be a good starting point.

5. hilbert2 says:

If I make a velocity dependent potential energy ##V(x,dot{x})=frac{1}{2}kx^2 + beta x dot{x}## (implying that at any moment there's a linear potential that increases when going to the direction of motion), it seems that I get a velocity term in the equation of motion from the Lagrange equations. But then the generalized momentum ##p## seems to depend on ##x##, too, and the Hamiltonian function is not equal to the total energy, it seems to me. I admit I should have described this in more detail.

6. A. Neumaier says:
hilbert2

If I make a velocity dependent potential energy ##V(x,dot{x})=frac{1}{2}kx^2 + beta x dot{x}## (implying that at any moment there's a linear potential that increases when going to the direction of motion), it seems that I get a velocity term in the equation of motion from the Lagrange equations. But then the generalized momentum ##p## seems to depend on ##x##, too, and the Hamiltonian function is not equal to the total energy, it seems to me. I admit I should have described this in more detail.

From a variational principle for which the Hamiltonian is the energy, you never get dissipative equations. And if you change the Lagrangian such that it produces the right equations, and you don't introduce explicit time dependence (which would be unnatural in a system where there are no external forces) the Hamiltonian will typically be zeros. Thus the standard connection to the physics is lost.

The principles of Lagrange and Hamilton are taylored to the conservative case; this is why they are so prominent in the books. Dissipation is (in general, not in certain special cases) a much more complicated phenomenon. It is of thermodynamic origin (friction creates heat), and to get the correct equations one usually must go through a thermodynamic derivation from a microscopic description, or at least use equations of a form that are inspired from such a derivation.

7. hilbert2 says:
A. Neumaier

The principles of Lagrange and Hamilton are taylored to the conservative case; this is why they are so prominent in the books. Dissipation is (in general, not in certain special cases) a much more complicated phenomenon. It is of thermodynamic origin (friction creates heat), and to get the correct equations one usually must go through a thermodynamic derivation from a microscopic description, or at least use equations of a form that are inspired from such a derivation.

I'm familiar with articles like this, but haven't gone through the derivation myself. If there's too few oscillators in the heat bath, I guess there will also be Brownian motion-like fluctuation.

8. A. Neumaier says:

Yes, this is the thermodynamic treatment There is fluctuation no matter how many bath oscillators you have. One couples a system to a heat bath (in the most typical case) and then eliminates the bath, gets complicated equations, makes a Markov approximation to eliminate the memory, is left with a stochastic differential equation, and if the noise is so small that it can be neglected one ends up with a dissipative deterministic equation. See, e.g., Chapters 15 and 16 of my online book.

For fluid flow and elasticity, there is no external heat bath; dissipation is energy lost into the high frequency modes, and one must start with a microscopic multiparticle system or quantum field theory.

9. hilbert2 says:

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##,

where ##alpha## is a positive constant. The ##Im## means imaginary part. This equation seems to conserve norm and if an initial state ##Psi (x,0) = Ce^{-a(x-x_0 )^2}## is evolved with it, the time-dependent expectation value ##left<xright>## behaves like a damped classical oscillator.

I have a sample of a code (written in R) in my blog that calculates the evolution of a gaussian wavepacket in harmonic potential: https://physicscomputingblog.wordpr…solution-of-pdes-part-4-schrodinger-equation/ .

A version of the code that adds the nonlinear damping looks like this:

library(graphics)                                    #load the graphics library needed for plotting

lx <- 6.0                                                        #length of the computational domain
lt <- 5.0                                        #length of the simulation time interval
nx <- 300                                      #number of discrete lattice points
nt <- 200                                       #number of timesteps
dx <- lx/nx                                    #length of one discrete lattice cell
dt <- lt/nt                                                    #length of timestep

V = c(1:nx)                                    #potential energies at discrete points

for(j in c(1:nx)) {
V[j] = as.complex(2*(j*dx-3)*(j*dx-3))                        #Harmonic oscillator potential with k=4
}

kappa1 = (1i)*dt/(2*dx*dx)                            #an element needed for the matrices
kappa2 <- c(1:nx)                                #another element

for(j in c(1:nx)) {
kappa2[j] <- as.complex(kappa1*2*dx*dx*V[j])
}

psi = as.complex(c(1:nx))                                                        #array for the wave function values

for(j in c(1:nx)) {
psi[j] = as.complex(exp(-(j*dx-2)*(j*dx-2)))                    #Gaussian initial wavefunction, displaced from equilibrium
}

xaxis <- c(1:nx)*dx                                #the x values corresponding to the discrete lattice points

for(m in c(1:nt)) {

A = matrix(nrow=nx,ncol=nx)                                #matrix for forward time evolution
B = matrix(nrow=nx,ncol=nx)                                #matrix for backward time evolution

for(j in c(1:nx)) {
kappa2[j] <- as.complex(kappa1*2*dx*dx*V[j])
for(k in c(1:j)) {
kappa2[j] = kappa2[j]+2*kappa1*2*dx*dx*dx*Im(psi[k])  # Add a nonlinear integral damping term to the potential
}
}

for(j in c(1:nx)) {
for(k in c(1:nx)) {
A[j,k]=0
B[j,k]=0
if(j==k) {
A[j,k] = 1 + 2*kappa1 + kappa2[j]
B[j,k] = 1 - 2*kappa1 - kappa2[j]
}
if((j==k+1) || (j==k-1)) {
A[j,k] = -kappa1
B[j,k] = kappa1
}
}
}                                        #main time stepping loop

sol <- solve(A,B%*%psi)                         #solve the system of equations

for (l in c(1:nx)) {
psi[l] <- sol[l]
}

if(m %% 3 == 1) {                                                     #make plots of psi(x) on every third timestep
jpeg(file = paste("plot_",m,".jpg",sep=""))
plot(xaxis,abs(psi)^2, xlab="position (x)", ylab="Im(Psi)",ylim=c(-1.5,1.5))
title(paste("Abs(psi(x,t))^2 at t =",m*dt))
lines(xaxis,abs(psi)^2)
dev.off()
}
}

Just need to choose a short enough timestep and spatial step.