Chem 455 Fall 2012
–
Homework 5
–
due Mon Oct 29 5:00pm (in box 65 on 3
rd
floor)
1.
Tunneling.
An electron with energy 0.200 eV runs into a potential step of height 3.50 eV. What is the
penetration depth of the electron?
The penetration depth is 1/
K
, with
√
. For an electron with
m
=
m
e
,
E
= 0.200 eV and
V
0
= 3.50 eV, this gives
K
= 9.31 nm
1
and
1/
K
= 0.107 nm.
2.
Tunneling.
Consider an electron with 0.800 eV kinetic energy approaching a 1D potential barrier of
height 1.00 eV and width 1.10 nm. What is the probability the particle tunnels across the barrier?
Using the expressions
√
√
we get
K
= 2.29 nm
1
,
q
= 1.25, and therefore
T
= 0.0165, so above one percent.
3.
Tunneling.
The maximum safe current in a copper wire with a diameter of 3.0 mm is about 20 A. For
higher currents, the wire melts. In an STM (scanning tunneling microscope), a current of 0.50 nA
passes from the tip to the surface in a filament of diameter 1.2 nm. Compare the current density
(current per crosssection area, in A mm
2
) in the copper wire with that in the STM.
Copper:
STM tip:
The current density in the STM tip is over 150 times larger than in the copper wire.
4.
Operators.
Do the operators for kinetic energy
̂
and for linear momentum
̂
commute? To find out, evaluate the commutator
̂
̂
by applying it to an arbitrary function.
Let’s write out the commutator of the two operators:
[
̂
̂]
̂
̂ ̂
̂
Next, we evaluate each of the two terms separately:
̂
̂
̂
̂
̂
̂
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '06
 Reinhardt
 Electron, Uncertainty Principle, expectation value, ground state

Click to edit the document details