This Quantum World/Feynman route/Schroedinger at last

Schrödinger at lastEdit

The Schrödinger equation is non-relativistic. We obtain the non-relativistic version of the electromagnetic action differential,

 

by expanding the root and ignoring all but the first two terms:

 

This is obviously justified if   which defines the non-relativistic regime.

Writing the potential part of   as   makes it clear that in most non-relativistic situations the effects represented by the vector potential   are small compared to those represented by the scalar potential   If we ignore them (or assume that   vanishes), and if we include the charge   in the definition of   (or assume that  ), we obtain

 

for the action associated with a spacetime path  

Because the first term is the same for all paths from   to   it has no effect on the differences between the phases of the amplitudes associated with different paths. By dropping it we change neither the classical phenomena (inasmuch as the extremal path remains the same) nor the quantum phenomena (inasmuch as interference effects only depend on those differences). Thus

 

We now introduce the so-called wave function   as the amplitude of finding our particle at   if the appropriate measurement is made at time     accordingly, is the amplitude of finding the particle first at   (at time  ) and then at   (at time  ). Integrating over   we obtain the amplitude of finding the particle at   (at time  ), provided that Rule B applies. The wave function thus satisfies the equation

 

We again simplify our task by pretending that space is one-dimensional. We further assume that   and   differ by an infinitesimal interval   Since   is infinitesimal, there is only one path leading from   to   We can therefore forget about the path integral except for a normalization factor   implicit in the integration measure   and make the following substitutions:

 

This gives us

 

We obtain a further simplification if we introduce   and integrate over   instead of   (The integration "boundaries"   and   are the same for both   and  ) We now have that

 

Since we are interested in the limit   we expand all terms to first order in   To which power in   should we expand? As   increases, the phase   increases at an infinite rate (in the limit  ) unless   is of the same order as   In this limit, higher-order contributions to the integral cancel out. Thus the left-hand side expands to

 

while   expands to

 

The following integrals need to be evaluated:

 

The results are

 

Putting Humpty Dumpty back together again yields

 

The factor of   must be the same on both sides, so   which reduces Humpty Dumpty to

 

Multiplying by   and taking the limit   (which is trivial since   has dropped out), we arrive at the Schrödinger equation for a particle with one degree of freedom subject to a potential  :

 

Trumpets please! The transition to three dimensions is straightforward: