The de Broglie Duality of Matter
E= mc2 (Einstein); E =h (Planck)
Photon momentum P =mv =mc
Photoelectric effect = particle
Diffraction = wave
h
c
p=
; = ,
c
h
=
mv
h
p = = mv
Electron is both a wave because it has wavelength,
and also a particle because
Problem: If the wavefunction of an electron is proportional
to e-r/b, where b is a constant and r is the distant from the
nucleus, calculate the relative probabilities of finding the
electron inside a region of volume 1.0 pm3 located at a)
the nucleus, b)
Chem 131A Homework
Week 4
1. [9.3(b)] Calculate the energy separations in joules, electronvolts, and as a wavenumber
between the levels (a) = 3 and = 1; (b) = 7 and = 6 of an electron in a one-dimensional
nanoparticle modeled by a box of length 1.5 nm.
(a
ECSE-6968 Quantum mechanics applied to semiconductor devices
Instructor information:
Prof. E. F. Schubert, Rensselaer Polytechnic Institute, Department of Electrical, Computer, and
Systems Engineering, 110 Eighth Street, Troy NY 12180, Phone: 518-276-8775
Max Planck (18581947)
Established equation E = h
Louis De Broglie (1892-1987)
Attributed wave-like properties to particles
2
The postulates of quantum mechanics
2.1 The five postulates of quantum mechanics
The formulation of quantum mechanics, also calle
Chemistry 131A Credit by Examination
Sample questions
1. The kinetic energy of a proton is 1 eV. Find the magnitude of its velocity.
2. Two electrons are initially at rest and separated by a distance of 10-6 m. They
accelerate due to their mutual
Chem 131A Discussion
Week 3
1. Use Eulers formula to show that
1
cos() = ( + )
2
Write an equivalent expression in terms of exponential functions for sin().
sin() =
1
( )
2
2. Determine which of the following functions are eigenfunctions of the inversion
Leo Esaki
(1925)
18
Tunneling structures
18.1 Tunneling in ohmic contact structures
Attempt rate of carriers in a semiconductor
Calculation of the success rate using the WKB approximation
Calculation of the current-vs.-voltage characteristic
Contact resis
16
High doping effects
16.1 Screening of impurity potentials
Variations of the electrostatic potential are reduced in magnitude by the spatial redistribution of
free carriers. Variations in the band-edge potential can occur due to local doping concentrati
15
Impurities in semiconductors
15.1 Bohrs hydrogen atom model
Shallow impurities are of great technological importance in semiconductors since they determine
the conductivity and the carrier type of the semiconductor. Shallow impurities are defined as
im
14
Carrier concentrations
The activation energy of impurities will be frequently used in this chapter. It is useful to recall
the interdependence of free energy, internal energy, enthalpy, entropy, and activation energy. To
do so, consider the electronic
James Maxwell (18311879)
Established velocity
distribution of gases
Ludwig Boltzmann (18441906)
Established classical statistics
Enrico Fermi (19011954)
Established quantum statistics
13
Classical and quantum statistics
Classical MaxwellBoltzmann statisti
Enrico Fermi
(1901-1954)
11
Time-dependent perturbation theory
11.1 Time-dependent perturbation theory
In the preceding sections, we have considered time-independent quantum mechanical systems, in
which wave functions and state energies do not depend on t
10
Perturbation theory
Quantum mechanical systems may be exposed to perturbations including external electric fields,
magnetic fields, or electromagnetic radiation. Due to such perturbations, the quantum system
considered here is stimulated and, as a cons
9
Approximate solutions of the Schrdinger equation
9.1 The WKB method
The Schrdinger equation has analytic solutions only for very few selected potential energies
U(x). For example, the infinite square well has an analytic solution. If the one-dimensional
8
Applications of the Schrdinger equation in periodic semiconductor
structures
8.1 Free electrons
Before considering electrons in the periodic potential of a semiconductor crystal, we first
consider electrons in free space that is in an environment in whi
7
Applications of the Schrdinger equation in nonperiodic semiconductor
structures
7.1 The infinite square-shaped quantum well
The infinite square-shaped well potential is the simplest of all possible potential wells. The
infinite square well potential is
Chem 131A Homework
Week 6
1. [14.1(b)] The rotation of a molecule can be represented by the motion of appoint mass moving
on the surface of a sphere with angular momemtum quantum number = 2. Calculate the
magnitude of its angular momentum and the possible
Chem 131A Discussion
Week 4
1. Convert each of the following photon wavenumbers into a wavelength, and energies in J and eV:
a) 10 cm1; b) 1500 cm1; c) 15000 cm1; d) 25000 cm1; e) 40000 cm1. Identify the region of
the electromagnetic spectrum for each as
Chem 131A: Quantum Principles
Week 7
1. Show that the moment of inertia for rigid rotor can be written as I = r2 , where r = r1 + r2
(the fixed separation of the two masses) and is the reduced mass.
2. The J = 0 to J = 1 transition for carbon monoxide (12
Chem 131A Homework
Week 10
1. [E23.2(b)] Suppose that a molecular orbital has the (unnormalized) form 0.727 + 0.144.
Find a linear combination of the orbitals and that is orthogonal to this linear combination
and determine the normalization constants
Chem 131A Homework
Week 9
1. [19.1(a)] The classical picture of an electron is that of a sphere of radius r = 2.82 fm. On the
basis of this model, how fast is a point on the equator of the electron moving? Is this answer
plausible?
The speed of rotation i
Chem 131A Discussion
Week 10
1. Describe and account for the variation of first ionization energies along Period 2 of the
perdiodic table. Would you expect the same variation in Period 3?
2. A series of lines in the spectrum of neutral Li atoms rise
Chem 131A Homework
Week 9
1. [19.1(a)] The classical picture of an electron is that of a sphere of radius r = 2.82 fm. On the
basis of this model, how fast is a point on the equator of the electron moving? Is this answer
plausible?
2. [20.1(b)] What are t
Chem 131A Week 0 Homework
1.4b) Draw Lewis (electron dot) structures of (a) O3, (b) CIF3+, (c) N3
1.6b) Use VSEPR theory to predict the structures of (a) H2O2, (b) FSO3, (c) KrF2, (d) PCl4+
2.3b) Consider a harmonic oscillator with B=0; relate the total e
Chem 131A Homework
Week 5
1. [12.1(b)] Calculate the zero-point energy of a harmonic oscillator consisting of a rigid CO
molecule adsorbed to a metal surface by a bond of force constant 285 N m1.
8.31021 J
2. [12.5(b)] Locate the nodes of the harmonic osc
Chem 131A Discussion
Week 5
1. Discuss the physical origin of a) quantization of energy for a particle confined to motion inside a
one-dimensional box, and b) quantum mechanical tunneling.
a) In quantum mechanics, particles are said to have wave chara