Physics 115A Midterm #2 Solutions
Sean Litsey
May 29, 2014
Problem 1
Part a
We are given the wavefunction
(x) = A (2 (x) + 3 (x)
(1)
where
nx
2
sin
a
a
n (x) =
(2)
are the eigenfunctions of the innite square well Hamiltonian, with eigenvalues (i.e. energ
Physics 115B Problem Set #1 Solutions
Sean Litsey
April 15, 2014
Problem 1
We would like to show that the energy density u and emissive power E for a cavity are related by
4
E (, T ) .
c
u (, T ) =
(1)
To do this, we need to understand how these two quant
Problem Set 1 (due 4/7)
(Physics 115A, Fall 2014, H. W. Jiang)
1. Show that the relation between the energy density in a cavity, u(, T) and the emissive
power, E(,T) is given by,
u(,T)=4E(,T)/c
[Hint: To do so, look at the figure. The shaded volume elemen
Quantum Mechanics HW 6 (Solutions)
Matthew Poulos
UCLA Department of Physics
(Dated: November 10, 2016)
1.
PROBLEM 1 - 2.10
The ground state of the harmonic oscillator is given by
0 =
1 x 2
1
e 2 ( )
1/4
(1)
q
~
where = m
is the characteristic length s
Physics 115A
Fall 2015
T. Tomboulis
Due 10-27-15
HOMEWORK # 4
Reading Assignment: G Ch. 2, sections 2.1, 2.2, 2.4
Problems:
1. Delta function practice
(a) Evaluate the following integrals:
i.
Z +1
dx exp(|x| + 3) (x + 2)
1
ii.
4 Z
X
h
i
dx cos(x) + 2 (2x
Physics 115A
Fall 2015
T. Tomboulis
Due 10-20-15
HOMEWORK # 3
Reading Assignment: G Ch. 2, sections 2.1, 2.2.
Problems:
1. Show that in any stationary state (x, t) = u(x) exp(iEt/
h) we always have:
< p >= 0 .
2. (a) Calculate the averages and standard de
Physics 115A
Fall 2015
T. Tomboulis
Due 12-01-15
HOMEWORK # 8
Reading Assignment: Note 1, class notes; Griffiths, Ch. 3, sections 3.1 - 3.4, 3.6,
Appendix: Linear Algebra.
Problems:
1. Given the matrix
2
i 1
T = i 2 i ,
1 i 2
(1)
(a) Compute detT and TrT.
Physics 115A
Fall 2015
T. Tomboulis
Due 10-13-15
HOMEWORK # 2
Reading Assignment: G Ch. 1.
Problems:
1. For
1
k + a2
calculate the initial (t = 0) shape of the wave packet
Z
(x, 0) =
dk A(k) ei kx
A(k) =
2
Sketch A(k) and (x, 0) and verify that, for any
Physics 115A
Fall 2015
T. Tomboulis
Due 12-08-15
HOMEWORK # 9
Reading Assignment: Note 1, class notes; Griffiths, Ch. 3, sections 3.1 - 3.4, , 3.5.1,
3.6, Appendix: Linear Algebra.
Problems:
1. One often finds sloppy notation in many QM texts. Consider, f
Physics 115A
Fall 2015
T. Tomboulis
Due 11-24-15
HOMEWORK # 7
Reading Assignment: Griffiths Appendix: Linear Algebra; also, Arfken and Weber,
or Boas or any other Phys. 131 text, is a useful reference for matrices, eigenvectors eigenvalues, orthogonal exp
Physics 115A
Fall 2015
T. Tomboulis
Due 10-06-15
HOMEWORK # 1
Reading Assignment: G, Ch. 1, section 1.1 - 1.4
Problems:
1. Old quantum theory - review (PHY 17)
(a) According to classical electromagnetic theory a charge e moving in a circular
orbit radiate
Physics 115A
Fall 2015
T. Tomboulis
Due 11-17-15
HOMEWORK # 6
Reading Assignment: G. Ch. 2 - omit section 2.3.1, class notes.
Start reading the Appendix: Linear Algebra in G. for refreshing your basic linear algebra.
Problems:
1. A particle in a harmonic
Midterm 1 Solutions
Matthew Poulos
UCLA Department of Physics
(Dated: October 26, 2016)
1.
PROBLEM 1
Conservation of momentum requires
hi
= pe cos ,
c
(1)
hf
= pe sin .
c
(2)
Furthermore, conservation of energy requires
hi + mc2 = hf +
p
m2 c4 + p2e c2 .
Quantum Mechanics 115A - Midterm Exam # 1
(Date: April 25th, 2016)
Problem 1
1) Ans: b
The expectation value is the average value of repeated measurements on an ensemble of identically
prepared systems.
2) Ans: b
The Fourier transformation of a delta func
Midterm II Practice
1.
A particle in an infinte square well has its initial wave funtion an even mixture of the
first two stationary states:
(x, 0) = A [1 (x) + 2 (x)] .
(a) Normalize (x, 0).
(b) Compute < x > .
(c) If you measure the energy of this part
HW #9 (due Friday, 12/2)
(Physics 115A, Fall 2016)
1. Derive the following inverse Fourier transform relation using the Dirac bra-ket
notation.
( p)
1
2
dx( x)e
ipx /
2. Consider a particle in a one-dimensional infinite potential box of length a. Its w
Quantum Mechanics HW 4 (Solutions)
Matthew Poulos
UCLA Department of Physics
(Dated: October 26, 2016)
1.
PROBLEM 1
We first note, that the function (x) can be written in terms of its Fourier transform as
Z
dk
eikx (k).
(x) =
2
(1)
Similarly, x(x) can
Midterm II Practice (Solutions)
Matthew Poulos
UCLA Department of Physics
(Dated: November 13, 2016)
1.
PROBLEM 1
A particle in an infinte square well has its initial wave funtion an even mixture of the first two stationary states:
(x, 0) = A[1 (x) + 2 (x
Physics 115A
Fall 2015
T. Tomboulis
Due 11-03-15
HOMEWORK # 5
Reminder: Our Midterm is scheduled on Thursday, November 5.
Reading Assignment: G Ch. 2, sections 2.1, 2.2, 2.4, 2.5, 2.6, class notes.
Problems:
1. Set up the solutions for the Schrodinger equ
Homework #1 Solutions
1. (a) The radius in the Bohr model is
a=
n2 ~2
mke e2
where ke = 1/40 , and the velocity is
v=
ke e2
n~
Direct substitution gives
f=
v
mke2 e4
=
2r
2~3 n3
(b) We are considering n 1, so we can use n1 as a small expansion
parameter.
Sample exam questions
1) Circle all of the following sentences that are correct:
a. fMRI has is noninvasive but optogenetics is invasive.
b. fMRI has better spatial resolution than tetrodes.
c. fMRI has worse temporal resolution than MEG
d. None of the ab
PHYS 115A Spring 2017
Assignment 3
Wave function and bound solutions of Shrdinger equation in one dimension
Due in lecture Monday May 1
2
1. A particle of mass m is in the state
real constants
(x; t) = Ce
E mx2
h
i Et
h
, where E and C are positive
(a) Fi
PHYS 115A Spring 2017
Assignment 7
QM formalism, Dirac Notation
Due in lecture Wednesday May 31
1. Eigenstates of a Hermitian operator must be orthogonal unless they share the same
eigenvalue. However, these degenerate eigenstates can be combined to form
John Miller
PHYS 115A, Week 2 Discussion
1. Noethers Theorem, Discrete Case. Find the integral of motion if an infinitesimal
transformation of coordinates and time
qi qi + i (~q, t) ,
t t + T (~q, t) ,
0
leaves the action functional
Z
t2
S=
t1
d
~q, t dt
PHYS 115A Spring 2017
Assignment 6
Harmonic oscillator in one dimension, time dependent solutions of Shrdinger equation
Due in lecture Monday May 22
1. Gri ths, Introduction to QM, Problem 2.13
2. Gri ths, Introduction to QM, Problem 2.17
3. Gri ths, Intr