Review: The finite square well
Today’s class:
Outside well
(E<V):
(Region I)
Quantum tunneling
Outside well (E<V):
(Region III)
Inside well (E>V):
(Region II)
d 2ψ III ( x)
= α 2ψ III ( x)
dx 2
d 2ψ II ( x)
= −k 2ψ II ( x)
dx 2
ψ II ( x) = C sin(kx) +D cos(kx)
ψ III ( x) = Be −αx
Energy
4.7 eV
Eelectron
V=0 eV
0
L
x
1) ψ(x) has to be continuous:ψ II ( L) = ψ III ( L)
2) ψ(x) has to be smooth: ψ ' II ( L ) = ψ ' III ( L )
3) ψ(|x|
∞)
0 (required for normalization)
Q1:
Review: ‘Penetration depth’
ψ ( x) = Be −αx
E<V: Classically forbidden region
0
Eelectron
0
L
ψ ( x) = Be −α ( x − L )
Penetration depth: Distance 1/α
over which the wave function
decays to 1/e of its initial value at
the potential boundary (here ψ(L)):
2m
(V − E )
h2
, with α =
ψ
ψ (L)
ψ ( L) / e
1/α
x
2m
(V − E )
h2
with: α =
L
What changes would increase how far the wave
penetrates into the classically forbidden region?
A. Decrease potential depth (= work-function of metal)
B. Increase potential depth
C. Decrease wire length L
D. Increase wire length L
E. More than one of the above
For V-E = 4 eV, 1/α ~ 0.1 nm (very small ~ an atom!!!)
Wide vs. narrow finite potential well
V
V-E
E1 ≈
L
π 2h 2
2mL2eff
α=
2m
(V − E )
h2
Smaller V-E smaller α larger penetration depth
6
Q2:
ψ(x) V(x)
Eelectron
Eelectron
0
a little later…
L
Wire#1
Wire#2
If very very long Wire#2 gets closer and closer to our short wire#1,
what will happen?
a. electron is “shared” between wires, with fraction in each
constant over time
b. there is a probability the electron will flow away through wire 2
c. electron will jump back and forth between wire 1 and wire 2
d. electron stays in wire 1.
e. something else happens.
Q3:
Ψ(x,t): standing wave
even latter...
Ψ(x,t): traveling wave
Electron flows away
http://phet.colorado.edu/en/simulation/quantum-tunneling
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. stop.
B. be reflected back.
C. exit the wire and keep moving to the right.
D. either be reflected or transmitted with some probability.
E. dance around and sing, “I love quantum mechanics!”
Answers to clicker questions
Q1: E (A&C work)
Q2: B
Q3: B