This Quantum World/Feynman route/Schroedinger at last

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Schrödinger at last[edit | edit source]

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: