Exercises on Wave Mechanics on One Dimension
1. Show that classical traveling wave (x, t) = sin (kx t) is not a valid
solution to the free particle (V(x) = 0), time dependent SE.
2. Show that (x, t) = exp(ikx + it) is an invalid solution to the free parti
Jim Wiss
November 14
1. Exam is open book , open notes. Bring a calculator. Exam covers HW 5
-HW 7, but only covers non-degenerate perturbation theory. Degenerate
theory is on the nal.
2. Exam emphasizes hydrogen selection rules ( = 1 , m = 0, 1) and
pari
Exercises on Hydrogen and the Central Force Problem
1. By inserting a trial wavefunction of u(r) = r exp (r) into
2 d2 u
h
+
2 dr 2
for
h
2
( + 1)
Ze2
2r 2
4 o r
u=E u
(C2)
= 0, obtain a value for and the ground state energy E.
2. According to the virial
Exercises on Quantum Measurements
1. Demonstrate the Hermitian property of the momentum operator by proving:
+
+
dx (x) i (x)
h
x
h
i (x)
x
dx
=
(x)
I suggest you use integration by parts (which is equivalent to integrating
both sides of the equation:
Exercises on Quantum Tunneling
1. Consider an electron of mass m in a quantum state which is a mixture of
of a ground state and rst excited state
(x, t) 1 (x) ei1 t + a ei 2 (x)ei2 t
where a and are modulus and phase of the amplitude of the 2 piece
relati
Exercises on Quantum Mechanics in Multidimensions
1. Show that Lz =
h
i
h
i
x y y x =
by assembling the pieces of Eq. (24)
in the Dimensions lecture notes.
2. Recall the expression for the quantum current
J=
h
Im
m
(a) Write wave function in polar for
Exercises on Bound States in One Dimension
1. We argued that both the wave function and its derivative is continuous
across a boundary between two nite potential steps. Is the double derivative continuous as well? Give an argument to defend your answer.
2
PHYSICS 485
Atomic Physics and Quantum Theory
Fall Semester 2013
Study Problems for the Final - Wednesday December 11th
Problems
1. E&R Problems 11.12, 11.13, 11.17, 11.20, 11.26, 11.27, 11.29, 11.30
2. E&R Problem
13.4, 13.7, 13.9, 13.10, 13.14, 13.16, 1
ECE 455
Exam #1
E t
c
0
gD() =
M 0 2 c 2
M
exp
2kT
2kT 02
1
2
C = kC NC
i1
j
SE =
c3
8n 3 h 3
2
8n
2
POUT = Pe
T
T
1
TOPT = ( 0 ) 2 L
A ij
B21 = A21
Gain / Pass
1I SAT
I() =
Loss / Pass
, B12 = B21
g2
g1
I(t) ~ E 02
g
0() = SE () N 2 2 N1
Physics 485, Fall 2016
Homework 3
Due Monday September 19th
Turn in the homework in the Physics 485 mailbox on Loomis 2nd floor by 8:00 PM.
Problem 1: Energy eigenstate decomposition
Townsend Problem 3.4
Problem 2: Particle in a box
Townsend Problem 3.6
P
Physics 485, Fall 2016
Homework 2
Due Monday September 12th
Turn in the homework in the Physics 485 mailbox on Loomis 2nd floor by 8:00 PM.
Problem 1: Crystal Diffraction
Townsend Problem 2.14
Problem 2: Phase and group velocities
Townsend Problem 2.24
Pr
UIUC Physics 485: Solutions for HW 1
September 6, 2016
Problem 1:
(a):
KEmax = h W
hc
W
=
= 2 eV
(b):
Let Vs = Stopping potential. Then we have,
eVs = KEmax = 2eV.
This implies,
Vs = 2V.
(c):
Cutoff wavelength occurs when
hc
cutof f
= W.
i.e.,
cutof f =
Exercises on Perturbation Theory
1. Consider the ground state of a harmonic oscillator:
0 =
1/2
exp
x2
2
where = m/
h
We apply a perturbation of the form V = (1/2)k x2 .
(a) Use rst order perturbation theory to compute E Eo + E. This
is easy to do from G
Exercises on Assorted Aspects of Atomic Physics
1. Consider the case of symmetric state where two spin 0 electrons are in the
same quantum state : (x1 , x2 ) = N m (x1 ) m (x2 ) N |mm . Find the
normalization for this case.
2. Consider two identical spin
Exercises on Variational Methods
1. Imagine you are on a desert island that has a good set of integral tables,
but no quantum mechanics books. For some reason escape from the island
depends on your ability to get the rst excited state energy of the harmon
Final exam date will be announced when known.
Physics 485 Practice Final Exam
1. Two identical, non-interacting electrons are put in a centered, one dimensional box of width a. This problem deals with the two correlation product
x1 x2 (or the correlation
Fall 2012
Physics 485 Practice Midterm
I think this is longer but of comparable diculty as your exam and should
provide good practice. The exam instructions and useful information are at the
end.
V = Vo
E = Vo
2
V=0
-0.1 nm
x
b
0
1. An electron with mass
Extra Physics 485 Practice Final Problems
1. In this problem you will use the variational technique to estimate the energy of the
first excited state energy for a particle of mass m in a potential of the form U = g x 4 .
Assume the particle travels only i
Energy Bands in Crystals
This chapter will apply quantum mechanics to a one dimensional, periodic
lattice of potential wells which serves as an analogy to electrons interacting with
the atoms of a crystal. We will show that as the number of wells becomes
Variational Methods
The variational technique represents a completely dierent way of getting
approximate energies and wave functions for quantum mechanical systems. It
is most frequently used to compute the ground state, but can be extended to
compute the
Quantum Tunneling
In this chapter, we discuss the phenomena which allows an electron to quantum tunnel over a classically forbidden barrier.
9 eV
10 eV
Rolls back
99% of time
Rolls over
1% of the time
10 eV
10 eV
This is a strikingly non-intuitive process
Assorted Aspects of Atomic Physics
As preparation for a discussion of the multielectron atom, we discuss some
very fundamental quantum mechanics concerning quantum states constructed
from indistinguishable particles. A consequence of the quantum mechanics
Quantum Measurements
This chapter will discuss some important aspects of the measurement of physical observables in quantum systems. As we will see there are many non-intuitive
, and surprising aspects to the nature of quantum measurement. We will also
di
Perturbation Theory
Although quantum mechanics is beautiful stu, it suers from the fact that
there are relatively few, analytically solveable examples. The classical solvable
examples are basically piecewise constant potentials, the harmonic oscillator an
Bound States in One Dimension
In this chapter we will concern ourselves with obtaining stationary state solution of the time independent Schrdinger Equation:
o
h
2 2
(x) + V (x)(x) = E(x)
2m x2
(1)
for particles (such as electrons) bound in one dimension
Quantum Mechanics in Multidimensions
In this chapter we discuss bound state solutions of the Schrdinger equation
o
in more than one dimension. In this chapter we will discuss some particularly
straightforward examples such as the particle in two and three