exp(ikx-iEt), Wave-equation and Energy-Momentum Relationships

    A wave equation already exists in classical physics (for example for a vibrating string) with a solution of the form exp(ikx-iwt) where k=2*3.14/wavelength and w=2*3.14 frequency. K and w are not simply linked to momentum and energy, but the wave is physically visible.

 The wavelike function exp(ikx-iEt) also appears in solutions for the photon, the relativistic Klein Gordon equation (and the Dirac too) as well as in the Schrodinger equation. This solution separates  x and k and E and t so -id/dx exp(ikx-iEt) = k exp(ikx-iEt) and id/dt exp(ikx-iEt) = E exp(ikx-iEt). If a classical equation exists linking energy and momentum (like EE=pp+momo or E=.5mvv), one may convert it into a differential equation with independent fluctuations in x and t, but linked through the energy relationship. This retains the classical information, but introduces a wave solution.

   For the Schrodinger equation with a potential, exp(-iEt + i Integral (0,x) dx1 v(x1)m is a solution which is equivalent to exp(i Integral (0,t) L dt1) where L is the Lagrangian. In this case, -id/dx exp(i action)  = p(x) exp(i action) and id/dt exp(i action) = E exp(i action). This also holds for the relativistic Lagrangian -mo sqrt(1-vv). In the case of an electromagnetic potential v dot A,  one may write Integral dt1 v dot A = Integral (0,x) dx1 dot A and so A behaves exactly like p(x). 

     For the case of light, exp(ikx-iEt) follows as a solution of a wave equation for either the electric or magnetic fields (which follows from Maxwell’s equations). The equation:

d/dt (.5eoEl El  + .5/uo BB) + grad dot El x B = 0 (for no source) also follows from Maxwell’s equations. It is an equation relating energy density to momentum flux and also allows for an exp(ikx-iEt) solution if El and B have the same magnitude (c=1) and are perpendicular to each other and the motion of the photon.