Download Problem Set 11

Problem Set 11
Prof. J. Gerton
Due Wednesday November 30, 2011 at the beginning of class
Problem 1
(20 pts.) Scanning tunneling microscope
In a Scanning Tunneling Microscope (STM), a narrow gap between a specimen and
the tip of a probe acts as a potential barrier to electrons bound to the specimen. A
small bias between the specimen and the probe acts as a potential barrier of height
V0 , and electrons can tunnel in this barrier to be detected at the probe as a small
current. The tunnel current is very sensitive to the gap width a between the probe
and the specimen.
(a) What is the momentum of the electrons inside the barrier? Take the mass of the
electrons to be me and their energies to be E < V0 .
(b) Argue that the wave function inside the barrier is of the form
ψ(x) = e−κx/~ .
(1)
To answer this question, first recall the expression for the coefficient κ in terms
of E, V0 , and me .
(c) What is the penetration length δ for the wave function in (b)? (Hint: the penetration length is the length at which the wave function is equal to 1/e.)
(d) Compute the penetration length δ, using E = 1 eV, V0 = 1.1 eV, a = 0.1 nm,
and me = 511 keV.
(e) The transmission coefficient in this region is T = e−2κ a/~ (with no exponentially
growing solution appearing), while the electron current density depends on T as
e V0
T ≡ j0 T.
(2)
4π 2 aδ ~
What is the measured current density (in units of C/m2 s), using the numerical
values for the parameters given in (d)?
j=
(f) By how much does the current changes if the width a changes by ∆a = 1 pm?
Use the expression
∂j ∆j = ∆a.
(3)
∂a
1
2
Problem 2
(10 pts.) Three-dimensional box
A particle is confined in a potential well given by V (x, y, z) = 0 for −L/2 < x < L/2,
0 < y < 2L and 0 < z < 3L and V = +∞ outside these ranges.
(a) Write down the generic expression for the energy.
(b) Write down the wave function for the ground-state. Pay attention to the ranges
of the potential.
(c) Assume that the particle in the box is an electron, and the size of the box is
L = 1 nm. How much energy (in eV) is needed to make the electron jump to the
first excited state? To the second excited state?
Problem 3
(10 pts.) The QHO in 3-dimensions - Part 1
Consider a Quantum Harmonic Oscillator (QHO), embedded in the three spacial
dimensions.
(a) Write down the energy level Enx ,ny ,nz for a QHO in 3D with quantum numbers
nx , ny , nz ≥ 0 and frequency ω0 . To do this, recall what was done for the 2D
QHO in the Homework Set 10.
(b) Write down the normalized wave function for the ground state psi3D
0 (x, y, z).
Again, recall the 2-dimensional case from HW10.
(c) What is the Schroedinger equation for the generic 3D wave function ψnx ,ny ,nz (x, y, z)?
Use Cartesian coordinates (x, y, z).
Problem 4
(10 pts.) The QHO in 3-dimensions - Part 2
Consider the QHO in 3D from Problem 2.
(a) We now switch to spherical coordinates (r, θ, φ). Write down the ground state
wave function in these coordinates, ψ03D (r, θ, φ). For this, use the result for the
ground-state wave-function in Cartesian coordinates ψ0,0,0 (x, y, z), and apply the
appropriate transformation of coordinates.
(b) Write down the Schroedinger equation for the 3D QHO in these coordinates, and
show that ψ03D (r, θ, φ) solves the Schroedinger equation with energy E = E0,0,0 .
course name
PS #
3
(c) In spherical coordinates, new quantum numbers nr , l and ml are used, instead of
nx , ny and nz . One of the first excited level has nr = 0, l = 1 and ml = 0, with
(normalized) wave function
2
ψ010 (r, θ, φ) = A1 r e−ν r cos θ,
and ν = mω0 /2~. Using the normalization in spherical coordinates
Z 2π
Z π
Z +∞
2
dφ |ψ(r, θ φ)|2 = 1,
dθ sin θ
dr r
0
0
prove that A1 =
q
√
8 2
π 3/2
(4)
(5)
0
ν 5/4 .
(d) What is the probability that a measurement of ψ010 (r, θ, φ) gives ψnx =1,ny =0,nz =0 (x, y, z)?
course name
PS #