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2.1 Wave propagation
Slides: Video 2.1.1 Plane waves and
interferenceText reference: Quantum Mechanics
for Scientists and EngineersSection B.4
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Wave propagation
Plane waves and interferen
Quantum mechanics for scientists and engineers Da
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Wave equation in 3 dimensions
Generalizing to 3 dimensions, the waveequation becomes
where
or equivalently, with unit vectors the corresponding directions
22
2 21 0
c t
2 2 2
22 2 2
x y z
2
ˆ ˆ ˆ ˆ ˆ ˆ x y z x y
x y z x y z
ˆ ˆ ˆ, , andx y z
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Plane wave solutions
We can check that a monochromatic (one freque“plane wave” of the form
where andis a solution when
First note that
exp i t k r
/k c
ˆ ˆ ˆexp exp x y i t i k x k y
x y z
k r x y z
ˆ ˆ ˆ x y z r x y z
ˆ ˆ ˆ exp x y z x y
i k k k i k x k y x y z
expi i t k k r
ˆ ˆ x y k k k k x y
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Plane wave solutions
So
2 exp expi t i t k r k r
expi i t k k r
exp x y z
i k k k i x y z
2 2 2 exp x y z
k k k i k r
2
expk i t k r
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Plane wave solutions
Since
then, with
and choosing
So is indeed a solution for avector direction k provided
2
2
2 2
2 2 2
exp1exp
/ exp
i t i t
c t
k c i t
k rk r
k r
2
2
2 exp expi t i t
t
k r k r
2 2exp expi t k i k r k r
exp i t
k r /k c
2 2 exp k k i k
/k c
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Wave interferenc
One solution is
the plane wave w
wavevector k1
1k
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Wave interferenc
One solution is
the plane wave w
wavevector k1
Another solution is
the plane wave wwavevector k2
2k
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Wave interferenc
One solution is
the plane wave w
wavevector k1
Another solution is
the plane wave wwavevector k2
Because the wave elinear
the sum is also a
showing interfe
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