Transcript Chapter 2 - Physics & Astronomy

Light
and
Matter
Wave mechanics
Tim Freegarde
School of Physics & Astronomy
University of Southampton
Wave mechanics
• response to action of neighbour
• delayed reaction
f x, t   f x  x, t  t 
e.g.
f x, t   f vt  x 
• waves are bulk motions, in which the displacement is a delayed response to the
neighbouring displacements
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Gravitational waves
• delay may be due to propagation speed of force (retarded potentials)
m1
a
m1m2
F G 2
r

m2
• vertical component of force
m1G
F t   m2 3 at  r c 
r
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Wave mechanics
• waves result when the motion at a given position is a delayed response to the
motion at neighbouring points
• derivatives with respect to time and position are related by the physics of the
system, which lets us write differential equations
e.g.
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2 y

y
2
2
2

v

y

v
p
p
2
2
t
x
• in certain circumstances, a wave may propagate without distortion:
f x, t   f v pt  x   f  x, t 
 f v pt  r   f  r, t 
• a surface of constant phase, 
with the phase velocity, v p
x, t , is known as a wavefront, and propagates
• the solutions depend upon whether the system shows linearity or dispersion
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Linearity and superpositions
• if the system is linear, then the wave equation may be split into separate
equations for superposed components;
i.e., if y1 and y2 are wave solutions, then so is any superposition of them
• if sinusoidal solutions are allowed, then the wave shape at any time may be
written as a superposition of sinusoidal components
yx    ak coskx  k 
Fourier analysis
k
• complex coefficients allow waves which are complex exponentials:
cos kx  i sin kx  exp ikx
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Dispersion
• linear systems may show dispersion – that is, the wave speed varies with
frequency
• if sinusoidal solutions are allowed, then the wave shape may still be written
as a superposition of sinusoidal components
• dispersion causes the components to drift in phase as the wave propagates
• the wave may no longer be written as
f x, t   f v p t  x 
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Dispersion
• 2
10sinusoidal
sinusoidalcomponents:
components:
• spreading of wavepacket
vg  2v p
• this illustration corresponds to the wavepacket evolution of a quantum
mechanical particle, described by the Schrödinger equation
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Plane wave solutions to wave equations
• linear, non-dispersive
exp it  kx
sin t  kx
f vt  x 
• linear, dispersive
exp it  kx
sin t  kx
• non-linear
solitons:
f vt  x 
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Alternative solutions
• show that spherical waves of the form
exp  it  kr
 r , t  
r
are valid solutions to the Schrödinger equation of a free particle

 i   2
t
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Wave mechanical operators
• an operator is a recipe for determining an observable from a wave function
e.g. an operator
Ô could yield the parameter o from the wave ykx  t 
o  Oˆ  y kx  t 
• for convenience, to avoid the observable depending upon the magnitude of the
wavefunction, we instead define the general operator
i.e.
Oˆ  ykx  t 
o
ykx  t 
o ykx  t   Oˆ  ykx  t 
• the square brackets are commonly omitted
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Messenger Lecture
• Richard P. Feynman (1918-1988)
Nobel prize 1965
• Messenger series of lectures, Cornell University, 1964
• Lecture 6: ‘Probability and Uncertainty – the quantum mechanical view of nature’
• see the later series of Douglas Robb memorial lectures (1979) online at
http://www.vega.org.uk/series/lectures/feynman/
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