Some Differences Between Quantum and Classical Waves

 We take as our starting point the free particle classical action which may be written as A= -Et+px if t and x are treated as independent variables. It is possible to find two relationships linking v (velocity) to  x and t from dA=0. The first is the classical wave result vp = E while the second is p=Ev where E=mo/sqrt(1-vv)  (c=1). Furthermore one may write dA/dx partial = p ((1a)) and dA/dt partial = -E. ((1b)) This may be generalized to two eigenvalue differential equations, but this immediately begs the question: Why do this? We argue that Lorentz invariant form -Et+px represents scaling of both t and x. For example, p scales x and we argue there should be physical relevance to this. This physical relevance results in the wavelength proportional to 1/p. In other words, -Et+px represents more than velocity. 

     The treatment of the scaling for a quantum wave and classical wave differ, we argue, in that the former keeps E,t and p,x separate while a classical wave does not. In other words pv=E applies to a classical wave which is described by cos(-Et+px). In other words, one takes ((1a)) and ((1b)) which separate E,t and p,x into two independent equations and takes account of scaling by combining the two (to include both x and t)  i.e. creating the classical wave equation:  1/vv d/dt d/dt partial W(x,t) = d/dx d/dx partial W(x,t). 

    We argue that quantum mechanics on the other hand keeps ((1a)) and ((1b)) separate. The velocity (relativistic)  p=Ev is not a wave velocity. A solution to ((1a)) is  exp(ipx) and to ((1b)) exp(-iEt). These are usually combined for a free particle to give exp(-iEt+ipx) and a wave is described with a phase velocity pv(phase)=E. We prefer to keep exp(-iEt) and exp(ipx) separate . For example, for two slit interference one does not make use of phase velocity or exp(-iEt). For a bound state one uses p=Ev, but a single free particle is represented by exp(ipx) in W(x)=wavefunction = Sum over p a(p)exp(ipx). exp(-iEn t) appears after one has found En, using the exp(ipx)’s in the sense that one does not include an exp(- i pp/2m t) for each exp(ipx).

    We argue that by separating E,t and p,x, the latter leads to a scheme for which there is “no time”. It is true that the particle moves at a speed x/t, but p=m1v1=m2v2 etc so p does not uniquely define velocity, rather it defines an impulse hit 2p = Integral F dt . Thus the scaling px which leads to an x, p only wave exp(ipx) is linked to an interaction. If the particle is moving freely and not interacting, then one has classical determinism i.e. v=constant = x/t. As soon as an interaction occurs, one changes from variables x,t which are linked to x, P conditional (x/p) and there is uncertainty in the position of x within a wavelength. The phase in exp(ipx) is only dependent on x not on the time the particle reaches x.

 exp(ipx) thus represents a probability and one must consider a sum of different possible outcomes (as in a two slit experiment) in order to describe an interaction. This is very different from the local interaction of classical mechanics. Nevertheless for a bound state one makes use of the two formalisms i.e. KE(x)= Sum over p pp/2m a(p)exp(ipx) / W(x) = KEclassical(x). 

   We try to discuss some of these ideas in this note.