Unformatted text preview: MIDTERM EXAMPLE 2 SIMKS
8 Speed of light in free space c = 2.99792458 × 10 m s Planck’s constant h = 6.5821188926 × 10
h = 1.054571596 × 10 Electron charge e = 1.602176462 × 10 – 16 – 34 Neutron mass
Proton mass m 0 = 9.10938188 × 10 C – 31 m n = 1.67492716 × 10 – 27
– 27 – 23 eV s Js – 19 m p = 1.67262158 × 10 Electron mass –1 kg
kg
kg Boltzmann k B = 1.3806503 × 10 constant k B = 8.617342 × 10 Permittivity of free space ε 0 = 8.8541878 × 10 Permeability of free space µ 0 = 4π × 10 Speed of light in free space
Avagadro’s number N A = 6.02214199 × 10 Bohr radius a B = 0.52917721 ×10 m –1 c = 1 ⁄ ε0 µ0 –7 –5 Hm eV K – 12 2 Inverse finestructure constant α –1 = 137.0359976
4πε 0 hc
α –1 = e2 Applied quantum mechanics –1 Fm –1 –1 23 – 10 4πε 0 h
a B = m0e 2 JK mol –1 PROBLEM 1
The first four energy eigenvalues and eigenfunctions for an electron with effective mass
*
m e = 0.07 × m 0 confined to the asymmetric potenital well sketched in the following
figure and bounded by barriers of infinite energy for x < 0 nm and x > 50 nm are:
E1 = 0.0396 eV, E2 = 0.1508 eV, E3 = 0.2034 eV, E4 = 0.2134 eV
(a) Sketch the corresponding wavefunctions and explain the qualitative differences
between each wavefunction. (50%)
(b) What can you say about the dipole matrix elements 〈1x 3〉 and 〈1x 4〉 for optical transitions? (50%)
Potential energy, V (eV) V→∞ V→∞ V0 = 0.2 eV 0.2 0.0
0 10 20
30
Position, x (nm) 40 50 PROBLEM 2
In classical mechanics, the Hamiltonian for a onedimensional harmonic oscillator with
motion in the xdirection at frequency ω is
2 2
m0 ω 2
p
H =  +  x
2
2m 0
where m0 is the mass of the particle and p is the particle momentum. m0 ω 1 ⁄ 2
ˆ
ip
ˆ
ˆ
(a) Introduce the operator b =  x +  and show that the quantum 2h m 0 ω
mechanical Hamiltonian for a onedimensional harmonic oscillator may be symmehω ˆ ˆ † ˆ † ˆ
ˆ
trized such that H =  ( b b + b b ) . (15%)
2
ˆ ˆ† ˆ† ˆ
ˆ ˆ
ˆ† ˆ†
ˆ ˆ†
(b) Using the fact that [ b, b ] = b b – b b = 1 and [ b, b ] = [ b , b ] = 0 , show
ˆ† ˆ
ˆ
that H = hω ( b b + 1 ⁄ 2 ) . (15%)
ˆ
(c) Show that the groundstate wave function ψ n = 0 = 0〉 is defined by b 0〉 = 0 .
(30%)
(d) Find the normalized ground state wavefunction. (20%)
(e) At time t = 0 the harmonic oscillator is prepared in a superposition state such
1
that the total state function is ψ ( x, t = 0 ) =  ( 1〉 + 3〉 ) . What is the time depen2
dence of the state probability and what is the expectation value of the position operator,
ˆ
x ? (20%)
∞ In answering this question, you may wish to use the standard integral ∫e
–∞ – ax 2 dx = π
 .
a PROBLEM 3
(a) Find the onedimensional density of states for electrons of mass m. (20%)
(b) Show that at low temperatures onedimensional electron conductance can be
quantized. (60%)
(c) What is the maximum frequency of operation of a onedimensional conductance
device with capacitance 1 fF (1015 F)? (20%) PROBLEM 4
(a) Classically (i.e, from Maxwell’s equations) change in charge density, ρ is related to
divergence of current density, J . Use this fact and the timedependent Schrödinger
wave equation describing the motion of particles of mass m and charge e in a real poteieh *
*
nial to derive the current operator J = –  ( ψ ∇ψ – ψ ∇ψ ) . (40%)
2m
(b)
If a wave function in freespace can be expressed as
i ( kx – ωt ) i ( – k x – ωt ) 2 2 show that particle flux is proportional to A – B .
+ Be
ψ ( x, t ) = Ae
(30%)
(c) Calculate the maximum current flowing through a wire of area 100 nm2 due to a
density n = 1018 cm3 of electrons with average energy E = 28 meV and mass m = m0,
where m0 is the bare electron mass. (30%) Applied quantum mechanics ...
View
Full
Document
This note was uploaded on 02/01/2009 for the course EE 539 taught by Professor Levi during the Fall '06 term at USC.
 Fall '06
 Levi

Click to edit the document details