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 = 99 Std Dev = 15 Grades and solutions are on D2L

Correspondence principle Proposed 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 Quantum probability Classical probability 0 a n=1 n=20

wire Energy x 0 a V(x) Region IRegion IIRegion III

exponent must be dimensionless. Energy x 0 a V(x) Particles in classically forbidden regions Outside well: E < V Inside well: E > V Outside well: E < V E particle What are the units of  ? A. J B. J -1 C. m 2 D. m -1 E. J -1/2

Small  means slow drop off Energy x 0 a V(x) Particles in classically forbidden regions E particle How far does the particle extend into the forbidden region? A measure of the penetration depth is Large  means fast drop off For an electron with (V-E) ~ 3 eV this is only m (~ size of an atom). Not very far! Distance at which  (x) is reduced by a factor of 1/e.

V(x) 0 a 0 a E E E Which of the four possible scenarios (A,B,C,D) would give the shortest penetration depth E particle 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. E electron 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 I L 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 -2  L  decays as e -  L   decays as e -2  L

Quantum tunneling probability The probability of tunneling depends on two parameters: 1. 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 As  increases (penetration depth decreases), probability decreases. As L increases (barrier width increases), probability decreases.