Post on 19-Dec-2015
Exam 2
Mean was 67 with an added curve of 5 points (total=72) to match the mean of exam 1.
Max after the curve = 99Std Dev = 15
Grades and solutions are on D2L
Correspondence principleProposed by Bohr: Quantum physics results should match classical physics results in the appropriate regions (large quantum number n).
As n increases, the quantum probability averages out to flat across the well. This is exactly what is predicted by classical physics.
Only really tiny wires have noticeable quantum effects at thermal energies
Quantum probability
Classical probability
0 a
a1a2
a1a2
Quantum probability
Classical probability
0 a
n=1 n=20
wire
Ene
rgy
x0 a
V(x)
Region I Region II Region IIIxDex )(III)sin()cos()(II kxBkxAx xEex )(I
)()(
II2
2II
2
xkdxxd )(
)(III
22
III2
xdx
xd )()(
I2
2I
2
xdxxd
exponent must be dimensionless.
Ene
rgy
x0 a
V(x)
Particles in classically forbidden regions
Outside well: E < VInside well: E > VOutside well: E < V
Eparticle
EVm
2
What are the units of ?
A. J B. J-1 C. m2 D. m-1 E. J-1/2
xDex )(III
Small means slow drop off
Ene
rgy
x0 a
V(x)
Particles in classically forbidden regions
Eparticle
How far does the particle extend into the forbidden region?
A measure of the penetration depth is
EVm
2
xDex )(III
)(a
xea )(
Large means fast drop off
/1
For an electron with (V-E) ~ 3 eV this is only 10-10 m (~ size of an atom). Not very far!
Distance at which (x) is reduced by a factor of 1/e.
V(x)
0 a
V(x)
0 a
EEE
Which of the four possible scenarios (A,B,C,D) would give the shortest penetration depth /1
EVm
2
xDex )(III
Eparticle
AC
BD
AD
BC
Small implies large .
Large comes from large V and/or small E
Answer C gives largest value for (V-E).
wire
Consider two very long wires separated by a small gap:
wire
This is an example of a potential barrier.
Quantum tunneling through potential barrier
Quantum tunneling occurs when a particle which does not have enough energy to go over the potential barrier somehow gets to the other side of the barrier.
Eelectron
This is due to the particle being able to penetrate into the classically forbidden region. If it can penetrate far enough (the barrier is
thin enough) it can come out the other side.
Quantum Tunneling Sim
Real( )Electron can penetrate into the barrier, but is reflected eventually.
“Transmitted” means continues off to the right forever, i.e. the wave function does not go down to zero.
Electron encounters potential step
Real( )
Copper wire #1 Copper wire #2CuO
Quantum tunneling
If the potential increase has a finite width, it is a potential barrier and the electron can tunnel out of Region IL
L
Tunneling probability
The probability to tunnel across a barrier of length L is approximately:
A) P ~ L
B) P ~ -L
C) P ~ e-L
D) P ~ e-2L
decays as e-L
decays as e-2L
Quantum tunneling probabilityThe probability of tunneling depends on two parameters:
EVm
21. The parameter measures how quickly the exponential decays and =1/ is the penetration depth (how far the wave function penetrates).
L
L
2. The width of the barrier L measures how far the particles has to travel to get to the other side.
The quantum tunneling probability is
LeP 2As increases (penetration depth decreases), probability decreases.
As L increases (barrier width increases), probability decreases.