Quantum Mechanics 102
Tunneling and its Applications
Review
• The probability of finding a particle in a particular
region within a particular time interval is found by
integrating the square of the wave function:
• P (x,t) =  |Y(x,t)|2 dx =  |c(x)|2 dx
• |c(x)|2 dx is called the “probability density; the
area under a curve of probability density yields the
probability the particle is in that region
• When a measurement is made, we say the wave
function “collapses” to a point, and a particle is
detected at some particular location
Particle in a box
c(x) = B sin (npx/a)
n=3
c(x)
n=2
|c(x)|2
certain wavelengths l = 2a/n are allowed
 Only certain momenta p = h/l = hn/2a are allowed
 Only certain energies E = p2/2m = h2n2/8ma2 are
allowed - energy is QUANTIZED
 Allowed energies depend on well width
 Only
“Real-World” Wells
• Solution has non-trivial form, but only certain
states (integer n) are solutions
• Each state has one allowed energy, so energy is
again quantized
• Energy depends on well width a (confinement
width)
|c(x)|2
n=2
n=1
x
Putting Several Wells Together
How do the electrons move between wells?
Quantum wells
• An electron is trapped since no empty energy
states exist on either side of the well
Escaping quantum wells
• Classically, an electron could gain thermal energy and
escape
• For a deep well, this is not very probable. Given by
Boltzmann factor.
EB  E A  k BT
Relative Probability  e
Escaping quantum wells
• Thanks to quantum mechanics, an electron has a non-zero
probability of appearing outside of the well
• This happens much more often than thermal escape if the
wells are close together.
Energy Bands in Solids
• Atomic energy levels become energy bands.
What if free electron encounters
barrier?
Do Today’s Activity
What Have You Seen?
• What happens when electron energy is less
than barrier height?
• What happens when electron energy is
greater than barrier height?
• What affects tunneling probability?
T  e–2kL
k = [8p2m(Epot – E)]½/h
A classical diode
• According to classical physics, to get to the holes on the
other side of the junction, the conduction electrons must
first gain enough energy to get to the conduction band on
the p-side
• This does not happen often once the energy
barrier gets large
• Applying a bias increases
the current by decreasing
the barrier
A tunnel diode
• According to quantum physics, electrons could tunnel
through to holes on the other side of the junction with
comparable energy to the electron
• This happens fairly often
• Applying a bias moves the
electrons out of the p-side
so more can tunnel in
Negative resistance
• As the bias is increased, however, the energy of the empty
states in the p-side decreases
• A tunneling electron would then end up in the band gap no allowed energy
So as the potential difference is increased, the current actually
decreases = negative R
No more negative resistance
• As bias continues to increase, it becomes easier for
conduction electrons on the n-side to surmount the energy
barrier with thermal energy
• So resistance becomes positive again
The tunneling transistor
• Only electrons with energies equal to the energy
state in the well will get through
The tunneling transistor
• As the potential difference increases, the energy levels on the
positive side are lowered toward the electron’s energy
• Once the energy state in the well equals the electron’s energy,
the electron can go through, and the current increases.
The tunneling transistor
• The current through the transistor increases as each successive
energy level reaches the electron’s energy, then decreases as the
energy level sinks below the electron’s energy
Quantum Entanglement
(Quantum Computing)
• Consider photons going through beam splitters
• NO way to predict whether photon will be
reflected or transmitted!
(Color of line is
NOT related to
actual color of
laser; all beams
have same
wavelength!)
Randomness Revisited
• If particle/probabilistic theory correct, half the
intensity always arrives in top detector, half in
bottom
• BUT, can move mirror so no light in bottom!
(Color of line is
NOT related to
actual color of
laser; all beams
have same
wavelength!)
Interference effects
• Laser light taking different paths interferes,
causing zero intensity at bottom detector
• EVEN IF INTENSITY SO LOW THAT ONE
PHOTON TRAVELS THROUGH AT A TIME
• What happens if I detect path with bomb?
No
interference,
even if bomb
does not
detonate!
Interpretation
• Wave theory does not explain why bomb detonates half the
time
• Particle probability theory does not explain why changing
position of mirrors affects detection
• Neither explains why presence of bomb destroys
interference
• Quantum theory explains both!
– Amplitudes, not probabilities add - interference
– Measurement yields probability, not amplitude - bomb detonates
half the time
– Once path determined, wavefunction reflects only that possibility presence of bomb destroys interference
Quantum Theory meets Bomb
• Four possible paths: RR and TT hit upper detector,
TR and RT hit lower detector (R=reflected,
T=transmitted)
• Classically, 4 equally-likely paths, so prob of each
is 1/4, so prob at each detector is 1/4 + 1/4 = 1/2
• Quantum mechanically, square of amplitudes must
each be 1/4 (prob for particular path), but
amplitudes can be imaginary or complex!
– e.g.,
1
1
1 i
1 i
Y  TR 
RT 
RR 
TT
2
2
2 2
2 2
Adding amplitudes
1
1
1 i
1 i
Y  TR 
RT 
RR 
TT
2
2
2 2
2 2
1 1
Y  
0
2 2
2
• Lower detector:
2
1 i 1 i
2  2i
• Upper detector: Y 


1
2 2 2 2
2 2
2
2
2
What wave function would give
50% at each detector?
Y  a TR  b RT  c RR  d TT
• Must have |a| = |b| = |c| = |d| = 1/4
• Need |a + b|2 = |c+d|2 = 1/2
Y
1
2 2
TR 
1
2 2
RT 
i
2 2
RR 
i
2 2
TT
Use to determine if Bombs are
any good
• Roger Penrose, The Large, the Small and
the Human Mind, CUP, pp.66-70
• R. Penrose, Shadows of the mind, Oxford
University Press, Oxford.
Before the Next Class
• Do Activity 24 Evaluation before class
Thursday (why not right now!)
• Do Homework 25
• Reading Quiz 25 (Chapts. 9 and 10).
• Exam on Monday, April 22.
• Projects due Monday, April 29.