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Lecture 12
Matter waves (Quantum mechanics).
 Aims:
Massive particles as dispersive waves.
Phase and group velocity
Evanescent waves (tunneling).
Schrödinger equation.
Time-dependent equation;
Time-independent equation.
Interpretation of the wave function.
Heisenberg’s Uncertainty Principle.
Applications:
Reflection/transmission at potential
step.
1-D potential well
Energy quantisation
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Matter waves
 de Broglie (1924)
 Proposed wavelike properties for particles
(since light showed particle properties)
 de Broglie wavelength
l = h/p
(or p  k )
 Davisson and Germer (1927)
 Electron diffraction. Now the basis of the use of
LEED (Low Energy Electron Diffraction) as a
probe for surface structure.
 At 100eV, K.E.=p2/2m is such that
p=5.4x10-24 Kg m s-1.
l=1.2x10-10m, i.e. of order of atomic dimensions.
d sin   l
 GP Thompson (1927) directed 15keV electrons
through a thin metal foil. Again discrete
scattering angles were observed.
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LEED and diffraction
 Modern electron diffraction
 Results from the (111) surface of a Nickel
crystal:
Left panel: native Ni(111)
Right panel: with adsorbed hydrogen
Ni(111)-(2x2)H
Shadow of sample
Additional spots
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Dispersion relation
 Dispersion relation
 By analogy with light:
p  k
E  w
( N .B.   h / 2 )
 For a particle of mass m,
E  KE  PE  p 2 2m  V
 2k 2
w 
V
2m
9.1
Dispersion relation for massive paerticles
“Free electron parabola”
 Arbitrary constant V implies freqency and phase
velocity (w/k) of a massive particle are not
uniquely defined.
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Group velocity
 Group velocity
 follows from the dispersion relation:
2
dw 2 k


dk
2m
k p
vg 
 v
m m
Classical velocity
of the particle
 The same is true relativistically.
 Classiclly forbidden region.
 Classically, particles cannot enter regions
where their total energy, E, is less than V.
 Under such circumstances, eq [9.1] gives
2m
k 2  2  E  V   a 2
i.e. k  ia

where a is real
 The wave becomes evanescent.
 The wave penetrates into the classically
forbidden region but the amplitude decays
exponentially.
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Schrödinger equation
 “Derivation”
 Start from expression for the energy of a nonrelativistic particle
E  p 2 2m  V
9.2
 Take a plane wave (travelling in +ve x-direction)
Y x, t   Yoei kx wt 
(note we use the Q.M. “convention” of kx-wt)
 2 Y  x, t 
p2
2
  k Y  x, t    2 Y  x, t 
2
x

Y  x, t 
iE
 iwY  x, t    Y  x, t 
t

Inserting into [9.2] and multiplying by Y gives:
Y  x, t 
 2  2 Y  x, t 
i

 VY  x, t 
2
t
2m x
9.3
 Time dependent Schrödinger equation
 It is a linear equation so we can superpose
solutions
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Time independent equation
 V=V(x)  V(x,t).
 Potential is a function of position only.
 Separate variables
Y  x, t   y  x T t 
dT
 2 d 2y
iy

T 2  VyT  EyT
dt
2m dx
divide by yT
1
1 dT   2 d 2y
 E
i
 

V
y
y
T dt  2m dx 2

Function of x
Function of t
 E must be constant (cannot simultaneously be a
function of x and t).
 Integrate equation for t.
iEt / 
iwt
T  Ae
 Ae
 Equation for x is:
2 2
 d y
  E  V y  0
2
2m dx
9.4
 Time independent Schrödinger equation.
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Wave function
 Interpretation
 Born (1927): The probability that a particle with
wave function Y(x,t) will be found at a position
between x and x+dx at time t is given by
|Y(x,t)|2dx.
 Normalisation

2
 Yx, t 
dx  1
Particle must be somewhere
 Y itself is unobservable, only |Y|2 has physical
significance.
 Plane wave:
Y x, t   Yoei (kx wt )
YY   Yo2
complex form of Y is necessary (not a
mathematical convenience) to ensure uniform
probability.
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Heisenberg’s Uncertainty Principle
 Localised wavepacket
 Probability of finding particle localised in space.
 MUST be a corresponding spread in the Fourier
Transform.
 Wavepacket of width Dx corresponds to
Dk~2/Dx.
DkDx  2

DpDx  h
 It is impossible to measure the position and
momentum of a particle with arbitrary precision
simultaneously.
 Since particles with mass are dispersive waves,
the packet will spread with time and the particle
becomes less well localised.
 Uncertainty in k means we loose knowledge of
the particle’s position as time goes on.
 Computer animation.
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Applications: simple systems
 Potential steps
 Scattering
 Potential barriers
 Tunelling
radioactive decay
 Potential wells
 Stationary states
atoms (crude model)
electrons in metals (surprisingly good
model)
 Boundary conditions at a potential discontinuity
 (1) The wavefunction, Y, is continuous
 (2) The gradient , dY/dx, is continuous
 (See notes and QM course for justification)
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Reflection/transmission
 Finite potential step.

x0
V 0
v  Vo
2 2m
k1  2 E
2 2m
k1  2  E  Vo 


 Boundary conditions at x=0.
Y continuous
A+r=t
dy/dx continuous
ik1A-ik1r=ik2t
k1  k2
r
k1  k2
2k1 A
t
k1  k2
 Vo<0. Classical: accelerate, (no reflection)
Q.M.: Some reflection; phase change of .
 0<Vo<E. Classical: decelerate, (no reflection)
Q.M.: Some reflection.
 Vo>E. Classically forbidden, all reflect.
Q.M.: y a e-ax penetrates barrier. i.e. finite
probability for being in classically forbidden
region, though all reflect.
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Potential barrier
 Potential barrier height, Vo; width a.

 Full solution given in Q.M. course. Here we
consider two distinct situations:
 Vo<E Classically no reflection;
Q.M. some reflection. Except when k2a=n,
when all particles transmitted (c.f. l/4
coupler, but with phase change of  at one of
the boundaries).
 Vo>E Classically no transmission;
Q.M. gives some transmission
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1-D, infinite potential-well
 Particle in a box:
 V(x) = 0 if 0< x <l
V(x) =  if x< 0 or x>l.
 Probability of particle
being outside box is
zero. Total reflection
at the barrier, gives a
standing wave.
 y x 
A sin kx
k  n / l
A  2/l
Normalisation
Quantisation condition
 Corresponding energies are:
2 2
 2k 2
2 
E
n
2m
2ml 2
Quantum number, n
The end
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