Collective and Stochastic Motion in the Time-Dependent Schrodinger Equation

Addendum Feb. 4, 2021

In this note, a potential of .5kxx + f(t)x is considered together with a product wavefunction Wa(y,t)exp(i y db/dt) where y=x-b(t). Product wavefunctions when inserted into the Schrodinger equation lead to two equations. The first may be chosen to depend only on one product e.g. Wa. In this case, it is a harmonic oscillator Schrodinger equation with an extra c(t)Wa term which may be handled by a pure time phase i.e. exp(i Integral (0,t) c(t1)dt1)). The second equation incorporates the potentail f(t)x and involves a coupling term -1/m dWa/dy d/dy exp(i y db/dt). This term, however, is exactly canceled by a term from i d/dt (partial) Wa(y-b(t)) exp(i y db/dt) so the coupling is eliminated. Thus, one has a second differential equation in b(t) linked to the potential f(t)x or f(t)y- f(t)b(t)  which is completely uncoupled. Thus, there are two decoupled types of motion. Harmonic motion in y=x-b(t) and collective motion related to b(t) linked directly to f(t).

Addendum Feb. 3, 2021

The term  -1/2m d/dy d/dy W1 in ((6)) is db/dt db/dt (1/2m) and is incorporated in c(t)Wo in equation ((5)). Also i d/dt (partial) {exp(i(x-b/m)db/dt) W(x-b/m)) = -i/m dW/dy db/dt +(1/m)db/dtdb/dt  -yd/dt db/dt as seen in the equation directly below ((6)) on the LHS. An extra pure time piece is added i.e. +(1/m)db/dtdb/dt.

The “time-independent” Schrodinger equation appears to be directly related to stochastic motion, we argue. If there is no potential, the wavefunction exp(ipx) is statistical, indicating periodic motion with a norm of one for all x. If a time-independent potential is added, it may be decomposed as V(x)=Sum over k Vk exp(ikx). Thus, time is present not just in exp(-iEt), but in the stochastic behaviour of the potential which is only V(x) on average. At each time, one does not know which Vk acts. Thus, the time-independent Schrodinger equation could perhaps be called the purely stochastic Schrodinger equation.

    It is possible to have a time dependent Schrodinger equation with a time dependent potential. 

Sometimes the time in the potential is hidden by a variable change as the equation involves partial derivatives. We try to argue that this potential contributes to collective motion. Time is already present in V(x), but in a stochastic manner. Explicit time (or a time related variable change) means a lack of stochasticity as one follows the potential in a predictable manner in time. It is possible that collective motion may be associated with a reference frame. For example, one may remove center-of-mass motion and consider only internal motion i.e. the quantum stochastic internal behaviour.

   In this note, we briefly examine two examples. The first is a free quantum particle viewed from a constantly accelerating frame as discussed in (1). In such a frame, the free particle appears as a particle in a gravitational potential +mgx1, but x1= x-X(t) so X(t) suggests collective motion. One may argue that x represents stochastic motion, but this is not necessarily the case because one may have collective potential energy linked to collective kinetic energy. In other words, id/dt partial W may yield a term which cancels  mgx indicating that it does not act as Sum Vk exp(ikx).

  The second example involves a quantum oscillator with an extra xf(t) potential. We argue that this leads to collective motion as the problem may be recast in terms of a variable y=x- b(t) and solved as a “time-independent oscillator” in y as discussed in (2). Again, there exists the notion of a kind of collective motion or extra frame which leads to a phase factor exp(iy d/dt partial b(t)) as the stochastic quantum behaviour defines the physical density and the phase term exp(i phase) disappears from the density).

 In such a case, W(y) the time-independent oscillator solution may be written as:  Sum over p  a(p) exp(ipy), but y contains t so the statistical nature of exp(ipy) is somewhat altered frome exp(ipx) i.e. there is an extra time related phase exp(i p f(t)) which has the appearance of a kind of collective motion as all p values are in synch with the same f(t) which may be sin(wt) for example.