Quantum Mechanics 102

Tunneling and its Applications

Interference of Waves and the Double Slit Experiment

Waves spreading out from two points, such as waves passing through two slits, will interfere

d

Wave crestWave troughSpot of constructive interference

Spot of destructive interference

Interpretation 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) = |(x,t)|2 dx = |(x)|2 dx |(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(x) = B sin (nx/a)

(x) |(x)|2n=2

n=3

Only certain wavelengths = 2a/n are allowedOnly certain momenta p = h/ = hn/2a are allowedOnly certain energies E = p2/2m = h2n2/8ma2 are

allowed - energy is QUANTIZEDAllowed energies depend on well width

What about the real world? 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 Can pick energies for electron by adjusting a

|(x)|2

n=1n=2

x

Putting Several Wells Together

How does the number of energy bands compare with the number of energy levels in a single well?

As atom spacing decreases, what happens to energy bands?

What happens when impurities are added?

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

Escaping quantum wells Thanks to quantum mechanics, an electron has a non-zero

probability of appearing outside of the well This happens more often than thermal escape

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 = [82m(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 increasesthe 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-sideso 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

Randomness 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.,

TT22

1RR

22

1RT

2

1TR

2

1 ii

Adding amplitudes

Lower detector:

Upper detector:

TT22

1RR

22

1RT

2

1TR

2

1 ii

02

1

2

12

2

122

22

22

1

22

122

2

iii

What wave function would give 50% at each detector?

Must have |a| = |b| = |c| = |d| = 1/4

Need |a + b|2 = |c+d|2 = 1/2

TTRRRTTR dcba

TT22

RR22

RT22

1TR

22

1 ii