2.1.1_slides Plane Waves and Interference

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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



Wave propagation

Plane waves and interferen

Quantum mechanics for scientists and engineers Da



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



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




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




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



Wave interferenc

One solution is

the plane wave w

wavevector k1

1k



Wave interferenc

One solution is

the plane wave w

wavevector k1

Another solution is

the plane wave wwavevector k2

2k



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

2.1.1_slides Plane Waves and Interference

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