In classical mechanics the operative variable are the position and velocity of particles. In quantum mechanics particles do not have a position and velocity, instead there they probability density functions. This is taken to mean that a particle is somehow smeared over a region of space. At least this is the Copenhagen Interpretation of quantum theory which was promoted by Niels Bohr. What is promoted here is that what has been called the probability density function for a system of particles is a time average of their positions and is the probability that the system is in a particular state at a randomly chosen instant of time. To illustrate this consider a rapidly rotating propeller or fan. The propeller or fan appears to be a circular disk. This disk corresponds to the probability density function. The propeller or fan still has a definite position and rate of rotation.

Consider a point particle of mass m and a point P some distance from it. The gravitational force vector F on a unit mass at P is given by

F = −(Gm/r²)(R/r) = −GmR/r³

where r is the distance from the particle to the point P and R is the vector from the particle to P. (R/r) is the unit vector in the direction of R. G is the gravitational constant.

If the point is moving then

F(t) = −GmR(t)/r(t)³

Suppose now that the particle's motion is periodic with a period T. The time average of the force at P is

F = −(1/T)∫₀^T(GmR/r³)dt

Let the path length along the particle's trajectory be represented by s(t) with s(0)=0 and s(T)=S. Now consider a change of the variable of integration from dt to ds. This involves dt being replaced by ds/v(s) where v(s) is particle's velocity ds/dt. Thus

F = −(1/T)∫₀^S(GmR/r³)(ds/v(s))

The quantity 1/v(s) can be construed as something in the nature of a probability distribution function, but it has to be normalized so that its total value over the trajectory is 1. This is achieved by defining

p(s) = (1/v(s)/[∫₀^S(dz/v(z))]

However ∫₀^S(dz/v(z)) is easily evaluated by a change of variable to dt=dz/v(z). This reveals that ∫₀^S(dz/v(z)) is equal simply to T. Thus

p(s) = (1/v(s))/T

This means that

F = −∫₀^S(GmR(s)/r(s)³)(1/(Tv(s)))ds = −G∫₀^S(mp(s)R(s)/r(s)³)ds

The term mp(s) in the integrand indicates that the total mass of m is being distributed around the trajectory of the particle. The above expression is exactly the same as what would arise for a wire in the shape of the particle's trajectory with a mass density of mp(s) at each point along the wire.

A System of Point Particles

Consider n particles with periodic trajectories s_i(t) for i=1 to n. With all of the preceding analysis indexed by i the time averaged force at a point P is

F = −GΣ_i=1ⁿ∫₀^S_i(m_ip_i(s)R_i(s)/r_i(s)³)ds

Again this is the expression that would arise for a set of n wires shaped as the particles' trajectories with the masses distributed along the wires according to p_i(s).

Generalizations

First consider instead of a point particle a line segment dz lying along the path segment ds. The mass m of the line segment is equal to μdz where μ is a linear mass density. This results in the same formula for the time average force as with the point particle. When the line segment is of a finite length for z=0 to z=Z the force at any instant is given by

F = −∫₀^Z(Gμ(z))(R/r³)dz

and the time average of the force when the center of mass of the line segment travels through a trajectory s(t) of period T

F = −(1/T)∫₀^T∫₀^Z(Gμ(z))(R/r³)dzdt

Again a change in the variable of integration from dt to ds with dt=ds/v(s) results in

F = −∫₀^S∫₀^Z(Gμ(z)p(s))(R/r³)dzds

which is just the same as the formula that would arise for the band that would result from carrying the line segment along the trajectory s(t).

The further generalization of the proposition requires considering an infinitesimal region of dx by dy by dz with the mass density μ(x, y, z). which is maintained over the trajectory of its motion. The velocity may differ at different values of x, y and z and give rise to a p(s) being a function of the form p(s, x, y, z).