Questions:
• What are the boundary conditions for tunneling problem?
• How do you figure out amplitude of wave function in
different regions in tunneling problem?
• How do you determine probability of tunneling through a
barrier?
Today: Answer these question for step potential:
HW after spring break: You answer it for barrier potential
(tunneling problem):
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1
Big Picture:
• So far we’ve talked a lot about wave functions
bound in potential wells:
Finite Square Well/
Non-Rigid Box/
Electron in Wire
Coulomb Potential/
Hydrogen Atom/
Hydrogen-like Atom
Infinite Square Well/
Rigid Box/
Electron in wire with work
function  thermal energy
E
E
E
+
• Here, generally looking at “energy eigenstates”
– fixed energy levels.
-
-
2
Big Picture:
• Can also look at free electrons
moving through space or
interacting with potentials:
E
– Plane wave spread through space:
 ( x, t )  A exp( ikx  iEt / )
– Wave packet localized in space:
 ( x, t )   An exp( ik n x  iE nt / )
n
• In these cases (tunneling,
reflection/transmission from step,
etc.) just pick some initial state,
and see how it changes in time.
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3
Tunneling
• Particle can “tunnel” through barrier even though it
doesn’t have enough energy.
• Visualize with wave packets – show sim:
http://phet.colorado.edu/new/simulations/sims.php?sim=QuantumTunneling
• Applications: alpha decay, molecular bonding,
scanning tunneling microscopes, single electron
tunneling transistors, electrons getting from one wire
to another in your house wiring, corona discharge, etc.
• So far we’ve talked about how to determine general
solutions in certain regions, wavelengths, decay
constants, etc.
• Next step: How do you determine amplitudes of
waves, probability of reflection and transmission?
– Boundary conditions!
4
• Start with an easier problem…
• An electron is traveling through a very long
wire, approaching the end of the wire:
e-
• The potential energy of this electron as a
function of position is given by:
0
V ( x)  
V0
x0
x0
5
e-
If the total energy E of the electron is GREATER
than the work function of the metal, V0, when the
electron reaches the end of the wire, it will…
A.
B.
C.
D.
E.
stop.
be reflected back.
exit the wire and keep moving to the right.
either be reflected or transmitted with some probability.
dance around and sing, “I love quantum mechanics!” 6
At this point you may be saying…
• “You said it would either be reflected or
transmitted with some probability, but in the
simulation it looks like part of it is reflected and
part of it is transmitted. What’s up with that?”
• Can anyone answer this question?
• Wave function splits into two pieces –
transmitted part and reflected part – but when
you make a measurement, you always find a
whole electron in one place or the other.
• Will talk about this more on Friday.
7
•Electron will either be
reflected or transmitted with
some probability.
•Why? How do we know?
•Solve Schro. equation
Solve time-independent Schrodinger equation:
 2 d 2 ( x)

 V ( x) ( x)  E ( x) where  ( x, t )   ( x) exp( iEt / )
2
2m dx
Region I:
 2 d 2 ( x)

 E ( x)
2
2m dx
d 2 ( x)
2mE
  2  ( x)
2
dx

k1 2
 1 ( x)  A exp( ik1 x)  B exp( ik1 x)
1 ( x, t )  A exp( i(k1 x  Et / ))
 B exp( i(k1 x  Et / ))
Region II:
 2 d 2 ( x)

 V0 ( x)  E ( x)
2
2m dx
2m( E  V0 )
d 2 ( x)

 ( x)
2
2
dx

k2 2
 2 ( x)  C exp( ik 2 x)  D exp( ik 2 x)
2 ( x, t )  C exp( i(k2 x  Et / ))
 D exp( i(k2 x  Et / ))
8
Note that general solutions are plane waves:
1 ( x, t )  A exp( i(k1 x  Et / ))  B exp( i(k1 x  Et / ))
2 ( x, t )  C exp( i(k2 x  Et / ))  D exp( i(k2 x  Et / ))
•
Wave packets are more
physical…
• But it’s easier to solve for
plane waves!
• And you can always add up a bunch of
plane waves to get a wave packet.
9
In region I, the general solution is:
 1 ( x)  A exp( ik1 x)  B exp( ik1 x)
1 ( x, t )   ( x) exp( iEt / )  A exp( i(k1 x  Et / ))  B exp( i(k1 x  Et / ))
What do the A and B terms represent physically?
A. A is the kinetic energy, B is the potential
energy.
B. A is a wave traveling to the right, B is a wave
traveling to the left.
C. A is a wave traveling to the left, B is a wave
traveling to the right.
D. A and B are both standing waves.
E. These terms have no physical meaning.
10
• If Ψ(x,t) = Aexp(i(kx-ωt)), which way is this wave
moving?
• Recall from HW: exp(iθ)=cos(θ)+isin(θ), so:
– Real(Ψ) = Acos(kx-ωt)
– Imaginary(Ψ) = Asin(kx-ωt)
• Real & imaginary parts of Ψ just differ by phase.
Usually we just plot the real part:
Real(Ψ(x,t=0))
Textbook is sloppy: sometimes
says ψ(x) when it means Re(ψ(x))
A
x
• Increase t a little, Ψ same as a little bit to the left.
Ψ(x,t+Δt) = Ψ(x-Δx,t).
11
Wave moves to right.
A
e-
C
B
D
Region I:
d 2 ( x)
2mE
  2  ( x)
2
dx

k1 2
 1 ( x)  A exp( ik1 x)  B exp( ik1 x)
Wave traveling right Wave traveling left
Region II:
2m( E  V0 )
d 2 ( x)

 ( x)
2
2
dx

k2 2
 2 ( x)  C exp( ik 2 x)  D exp( ik 2 x)
Wave traveling right Wave traveling left
What do each of these waves represent?
a)
b)
c)
d)
e)
A = reflected, B = transmitted, C = incoming, D = incoming from right
A = transmitted, B = reflected, C = incoming, D = incoming from right
A = incoming, B = reflected, C = transmitted, D = incoming from right
A = incoming, B = transmitted, C = reflected, D = incoming from right
12
A = incoming from right, B = reflected, C = transmitted, D = incoming
e- A
C
B
D
Region I:
 1 ( x)  A exp( ik1 x)  B exp( ik1 x)
0
Region II:
 2 ( x)  C exp( ik 2 x)  D exp( ik 2 x)
Wave traveling right Wave traveling left
Wave traveling right Wave traveling left
Incoming
Transmitted
Reflected
Incoming
from Right
Use initial/boundary conditions to determine constants:
Initial Conditions: Electron coming in from left  D = 0
Boundary Conditions: BC1.  1 (0)  2 (0)
A+B=C
d 1 (0) d 2 (0)  ik (A – B) = ik C
BC 2.

1
1
13
dx
dx
e- A
C
B
Region I:
 1 ( x)  A exp( ik1 x)  B exp( ik1 x)
Region II:
 2 ( x)  C exp( ik 2 x)
Wave traveling right Wave traveling left
Wave traveling right
Incoming
Transmitted
Reflected
If an electron comes in with an amplitude A, what’s the probability
that it’s reflected? What’s the probability that it’s transmitted?
a)
b)
c)
d)
e)
P(reflection) = B, P(transmission) = 1 – B
P(reflection) = B/A, P(transmission) = 1 – B/A
P(reflection) = B2, P(transmission) = 1 – B2
P(reflection) = B2/A2, P(transmission) = 1 – B2/A2
P(reflection) = |B|2/|A|2, P(transmission) =1 – |B|2/|A|2
14
e- A
C
B
Region I:
 1 ( x)  A exp( ik1 x)  B exp( ik1 x)
Region II:
 2 ( x)  C exp( ik 2 x)
Wave traveling right Wave traveling left
Wave traveling right
Incoming
Transmitted
Reflected
If an electron comes in with an amplitude A, what’s the probability
that it’s reflected? What’s the probability that it’s transmitted?
P(reflection) = “Reflection Coefficient”
= R = |B|2/|A|2,
P(transmission) = “Transmission Coefficient”
= T = 1 – |B|2/|A|2
15
To find R and T, use boundary conditions:
Wave Functions:
Boundary Conditions:
BC 1.  1 (0)  2 (0)
d 1 (0) d 2 (0)
BC 2.

dx
dx
Write down the equations for Bound. Conds:
(1) A+B=C
(2) ik1(A–B) = ik2C
Solve for B and C in terms of A:
Note that you can’t
determine A from
B = A*(k1–k2)/(k1+k2)
boundary conditions. It’s
C = A*2k1/(k1+k2)
the initial condition – must
be given. But you don’t
Find R and T:
need it to find R & T.
2
2
2
2
R = |B| /|A| = (k1–k2) /(k1+k2)
16
T = 1 – |B|2/|A|2 = 4k1k2/(k1+k2)2
 1 ( x)  A exp( ik1x)  B exp( ik1x)
 2 ( x)  C exp( ik 2 x)
Once you have amplitudes,
can draw wave function:
Real(
)
17
If the total energy E of the electron is
LESS than the work function of the metal,
V0, when the electron reaches the end of
the wire, it will…
A.
B.
C.
D.
E.
stop.
be reflected back.
exit the wire and keep moving to the right.
either be reflected or transmitted with some probability.
dance around and sing, “I love quantum mechanics!” 18
Solve time-independent Schrodinger equation:
 2 d 2 ( x)

 V ( x) ( x)  E ( x) where  ( x, t )   ( x) exp( iEt / )
2
2m dx
Region I (same as before):
Region II:
 2 d 2 ( x)

 E ( x)
2
2m dx
d 2 ( x)
2mE
  2  ( x)
2
dx

k1 2
 1 ( x)  A exp( ik1 x)  B exp( ik1 x)
 2 d 2 ( x)

 V0 ( x)  E ( x)
2
2m dx
2m( E  V0 )
d 2 ( x)

 ( x)
2
2
dx

α2
 2 ( x)  C exp( x)  D exp(x)
1 ( x, t )  A exp( i(k1 x  Et / ))
 B exp( i(k1 x  Et / ))
2 ( x, t )  C exp( x  iEt / ) 19
 D exp(x  Et / )
e- A
C
B
D
Region I:
 1 ( x)  A exp( ik1 x)  B exp( ik1 x)
0
Region II:
 2 ( x)  C exp( x)  D exp(x)
Wave traveling right Wave traveling left
Exponential decay
Incoming
Note: no transmitted wave
appears in equations
Reflected
Use boundary conditions to determine constants:
Boundary Conditions: BC 0.  ( x)  0 as x  
2
BC 1.  1 (0)  2 (0)
Exponential growth
| B |2
R
1
2
|A|
Solve for B in terms of A
A+B=C
d 1 (0) d 2 (0)  ik(A – B) =20–αC
BC 2.

dx
dx
Once you have amplitudes,
can draw wave function:
Real(
)
Electron may go part of the
way into barrier, may be
reflected from inside barrier
rather than surface, but it
always gets reflected
eventually.
21
Tunneling Problem – in HW
Same as above but now 3 regions!
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Real(
)
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22
Tunneling Problem – in HW
• Find  in 3 regions:
 1 ( x)  A exp( ik1 x)  B exp( ik1 x)
 2 ( x)  C exp( x)  D exp(x)
 3 ( x)  F exp( ik1 x)  G exp( ik1 x)
• Solve for B,C,D,F,G in terms of A
• Find probability of transmission/reflection:
1


αw
 αw 

e e
• Final Result: T =
 e–2αw
1

 4 E (1 E ) 

V0
V0 
w = width of barrier
2m(V0  E)
α = 1/ = decay constant =

23