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Quantum physics
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class notes for Quantum Physics
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quantum mechanics
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SHRÖDINGER EQUATIONRecall 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
1
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