barriers and boundary conditions
TRANSCRIPT
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3.1 Particles and barriers
Slides: Video 3.1.5 Barriers and
boundary conditionsText reference: Quantum Mechanics
for Scientists and EngineersSection 2.8
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Particles and barriers
Barriers and boundary condi
Quantum mechanics for scientists and engineers Da
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For our Schrödinger equation
if we presume that E , V and are finite
then must be finite also, so
If there was a jump in
then would be infinite at that po
Boundary conditions
22
22
d z
V z z E zm dz
2 2/d dz
/d dz 2 2
/d dz
/d dz must be continuous
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Boundary conditions
Alsomust be finite
otherwise could be infinitebeing the limit of a difference
involving infinite quantitiesFor to be finite
/d dz
2 2/d dz
/d dz
must be continuous
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Boundary conditions
Now that we have these two boundaryconditions
we can proceed to solve problems with finit“heights” of boundaries
/d dz must be continuous
must be continuous
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Infinitely thick barrier
Suppose we have a barrier of heightV owith potential 0 to the left of the
barrier
A quantum mechanical wave isincident from the left
The energy E of this wave ispositivei.e.,
E n e r g y
V
z
0 E
0
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Infinitely thick barrier
We allow for reflection from the barrierinto the region on the left
Using the general solution on the leftwith complex exponential waves
where, as beforeC exp(ikz) is the incident wave, going right
Dexp(-ikz) is the reflected wave, going left
E n e r g y
V
z
exp expleft
z C ikz D ikz
2
2 /k mE
0
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Infinitely thick barrier
Presume thati.e., the incident wave energy is less
than the barrier heightInside the barrier, the wave equation
is
i.e., mathematically
E n e r g y
V
z
o E V
22
22 o
d zV z E z
m dz
2
2 2
2o
d z m
V E zdz
0
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Infinitely thick barrier
The general solution of
for the wave on the right is
whereWe presume
otherwise the wave increasesexponentially to the right for ever
E n e r g y
V
0
z
2
2 22 o
d z mV E z
dz
exp expright
z F z G z
2
2 /om V E
0F
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Infinitely thick barrier
Hence the wave on the rightinside the barrier, is
with
This solution proposes that the waveinside the barrier is not zeroInstead, it falls off exponentially
E n e r g y
V
0
z
expright
z G z
22 /o
m V E
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Infinitely thick barrier
Using the boundary conditionswe complete the solution
On the left, we haveOn the right we have
Continuity of the wavefunctionat gives
Continuity of the wavefunctionderivative at givesi.e.,
exp left z C ikz exp
right z G z
0 z C D G
0 z i
C D Gk
ikC ikD G
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Infinitely thick barrier
Adding
gives
Equivalently
so we have found the amplitude G of the wave in t
in terms of the amplitude C of the incident wave
C D G i
C D Gk
2 1 i
C G k
22 k
G C k i
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Infinitely thick barrier
Subtracting
gives
Since we found
then
so we have found the amplitude D of the reflected
terms of the amplitude C of the incident wave
C D G i
C D Gk
2 1 i
D G
k
2 /G k k i C k i
D C k i
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Infinitely thick barrier
We have now formally solvedthe problem
the wave on the left iswith
the wave on the right is
with
where
exp left z C ikz k i
D C
k i
exp
right z G z
2k
G C k i
2 m 22 /k mE
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Reflection at an infinitely thick barrier
Note that
so
so
so the barrier is 100% reflectingthough there is a phase shift on r
an effect with no classical analo
k i D C
k i
D k i
C k i
2
1 D k i k i
C k i k i
f
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Wavefunction at a barrier
We plot the real part of thewavefunction
for an electronenergy 1 eV
at a barrierheight 2 eV
Notethe wavefunctionand its derivative
are both continuous
2 0
z (nm)
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W f i b i
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Wavefunction at a barrier
Note the exponential decaylength
2 0
z (nm)
0.2 nm
1/ 0.2 2nm Å
W f ti t b i
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Wavefunction at a barrier
As we in
energythe expdecay
P b bilit d it t b i
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2 0
z (nm)
Probability density at a barrier
Probability densityWith
then
falling by 1/e in1/ 2 0.1 1nm Å
expright
z z
2
exp 2right z z
2
right z
0.1 nm
P b bilit d it t b i
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Probability density at a barrier
As we inenergythe exp
decay
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