Solution of Schrödinger Equation

Free Particle

A particle (electron), alone in the universe.

Particle mass m and a fixed total energy E.

Alone no force on the particle.

constant potential energy everywhere

U(x, y, z) = constant, say zero

Let universe = one Dimensional only x variation

Time independent Schrödinger equation  

which can be written as , a one dimensional differential equation.Where or or E has a parabolic dependence on k. General solution for the simple equation can be written as where are unknown constants.

Therefore, total wave function (space and time dependence)

Compare this with a time-harmonic electromagnetic wave in free-space where

k = constant of propagation =

Hence wave function of free particle consists of a traveling wave. If particle moves in+x then and

Where

Normalizing, = constant for all values of x.

Thus the probability of finding the particle in any dx is equal/same

If we take the universe to be infinite Probability = 0

If we take the universe to be finite but large

no such difficulty

Now momentum operator =

Therefore, expected value of momentum

Same as classical. Therefore, DeBroglie relationship