Intro To Schrodringer Equation

The Schrödinger equation 

In the framework of classical mechanics, a particle with definite momentum p and energy E can be associated with a plane wave

 ψ=e^i(k.x-ωt)

In non-relativistic systems, conservation of energy can be written

H=T+V

  Where T is the kinetic energy and V is the potential energy. For a given state the Hamiltonian is the total energy hence H=E. A particle of mass m and momentum p has non-relativistic kinetic energy

T=p^2/2m

 We also introduce the energy and momentum operators

E=ih∂/∂t p=-h∇

If we take natural units so that h=1 we arrive at the Schrödinger equation acting on a free particle , ψ ,where V=0

i∂ψ/∂t+∇^2ψ/2m=0

Klein Gordon equation 

The Schrödinger equation works well for particles travelling at classical speeds. Once we approach relativistic velocities, the equation violates Lorentz covariance: that is to say the system varies as we perform a boost in a given coordinate system. To account for this we must now vary the Schrödinger equation. We start by considering the relativistic energy relation:

E^2=p^2+m^2

Where we have taken natural units c=1 . Using the operator relation we have

-∂^2Ψ/∂t^2+∇^2Ψ=m^2Ψ

If we introduce the D’Alembert operator:

□^2=∂μ∂μ

 becomes

(□^2+m^2)ϕ=0

This is the Klein-Gordon equation. Till now we have worked with the energy, E. In reality we have used the Hamiltonian operator where we have just simply stated that H=E. This in conjunction with the wave function gives us the eigenvalues of the allowed energies for the Klien-gordon given by:

E=±(p^2+m^2)^1/2

Unlike the Schrödinger equation we now have both positive and negative energy values. The problem arises due to the fact E<0 implies there exist indefinite negative energy states and hence a system can theoretically draw negative energy to function. The second problem is the fact negative energies relate to negative probabilities. The problem cannot be ignored as we must consider all energy states to fully describe the system.