class notes

SHRÖDINGER EQUATION
Recall the topic, which you have studied last year, on matter waves (de Broglie’s
hypothesis and wave particle duality). It can be used to derive by Schrödinger equa-
tion, which is a wave equation governing the behaviour of matter waves.

A FREE PARTICLE IN ONE DIMENSION

Consider one dimensional non-relativistic motion of a free particle moving with
velocity v, momentum p and energy E. The particle will have momentum, p = mv,
and energy, E = 21 mv 2 . Since the particle is free, there is no potential energy and
energy is entirely kinetic). The energy-momentum relation is

p2
E= (1)
2m

According to the de Broglie hypothesis, a harmonic wave is associated with a free
particle. Energy and momentum of the wave in terms of propagation constant k and
angular frequency w are given by:

p = ~k and E = ~ω. (2)

So that by virtue of Eq. (1),
~2 k 2
~ω = (3)
2m
The possible forms for such a harmonic wave are ψ = a cos(kx − ωt) or ψ =
b sin(kx − ωt), or, in general, linear combination of the two, namely

ψ(x, t) = a cos(kx − ωt) + b sin(kx − ωt) (4)

where a and b are arbitrary constants. Let us try to obtain a differential equation
for ψ.

∂ψ
= k [−a sin(kx − ωt) + b cos(kx − ωt)] ,
∂x
∂ 2ψ
= −k 2 [a cos(kx − ωt) + b sin(kx − ωt)] = −k 2 ψ,
∂x2
∂ψ
= ω [−a sin(kx − ωt) + b cos(kx − ωt)] ,
∂t
∂ 2ψ
= −ω 2 [a cos(kx − ωt) + b sin(kx − ωt)] = −ω 2 ψ. (5)
∂t2


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