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University of California, Santa Barbara Department of Physics Physics 115A - Midterm - Winter 2006 A.N. Cleland
Closed notes, closed book; you are allowed one sheet of formulas and a calculator.
Some identities that you might nd useful:
eu d
Physics 115A, Problem Set 9
Due Tuesday, November 24 @ 5pm in the box outside Broida 1019 (PSR)
Suggested reading: Griffiths Section 3.4-3.6.
1
Proof that 0 = 1?
=
Consider a Hermitian operator A with some eigenvector |ai with real eigenvalue a, A|ai
a|a
Physics 115A, Solution Set 2
1
Macroscopic Wavelengths
The idea that matter has a wave-like nature pertains not just to particles, but to macroscopic objects as well. Its just typically the case that the relevant wavelength is far too
small to detect; its
Physics 115A, Solution Set 3
1
E and V for normalizable solutions
Griffiths Problem 2.2:
Show that E must exceed the minimum value of V (x), for every normalizable solution to
the time-independent Schr
odinger equation. What is the classical analog to thi
Midterm exam Part II. Please work all 3 problems. Dont worry if you cant finish, but *do*
move on to!the next part if you get stuck.!
!
1. (28 pts) A particle of mass m is confined in an infinite square well of width L:
V = 0 for 0 x L; and V = , otherwis
Practice Midterm problems
Please work using the posted cover sheet, which contains the information you will be provided
for the actual midterm. (If needed, additional integrals will be provided with the problem, as in
problem 2 below).
For these practice
Physics 115A
Practice Midterm
Seth Koren
Problem 1: Half-Harmonic Oscillator
(a) While it is possible to solve this problem by setting up the Schrodinger equation for this
potential and imposing these boundary conditions, this is difficult and time-consum
Physics 115A, Solution Set 4
Due Tuesday, October 20 @ 5pm in the box outside Broida 1019 (PSR)
Suggested reading: Griffiths 2.3
1
Constructing States
Given the explicit solution for the ground state of the harmonic oscillator,
m! 1/4 m! 2
e 2~ x
0 (x) =
Physics 115A, Solution Set 7
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From the Square Well to the Delta Well
We can think of the Dirac delta function as the limiting case of a rectangle of unit area as
the height goes to infinity and the width goes to zero, so we should be able to obtain the
p
Physics 115A, Problem Set 7
Due Tuesday, November 10 @ 5pm in the box outside Broida 1019 (PSR)
Suggested reading: Griffiths Section 2.6, 3.1. If you have not studied linear algebra
recently, please read the Appendix in Griffiths carefully.
1
From the Squ
Physics 115A, Problem Set 5
Due Tuesday, October 27 @ 5pm in the box outside Broida 1019 (PSR)
Suggested reading: Griffiths 2.4. Study for the midterm!
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The 3D Harmonic Oscillator
Next quarter, when we encounter quantum systems in three spatial dimension
PHYS 115A Midterm Solutions Winter 2008
Problem 1
Recall that the normalized energy eigenstates and energies for an innite well of width a extending from x = a/2 to x = +a/2 are: E n = n2 n (x) = n (x) = 2 2 2ma2 2 nx cos a a 2 nx sin a a for odd n =
Physics 115A, Problem Set 1
Due Tuesday, September 29 @ 5pm in the box outside Broida 1019 (PSR)
Suggested reading: Griffiths 1.1-1.5
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Discrete statistics at the Tour de France
Related to Griffiths section 1.3.1
Suppose were lucky enough to gather all 61
Physics 115A, Solution Set 1
NC
1
Discrete statistics at the Tour de France
Related to Griffiths section 1.3.1
Suppose were lucky enough to gather all 61 winners of the Tour de France in one
room, alive and dead (greetings, zombie Jacques Anquetil!). One
Physics 115A, Problem Set 8
Due Tuesday, November 17 @ 5pm in the box outside Broida 1019 (PSR)
Suggested reading: Griffiths Section 3.1-3.4.
1
Hilbert Space
1. Show that the set of all square-integrable functions (that is, the set of functions f (x)
Rb
f
Physics 115A, Problem Set 2
Due Tuesday, October 6 @ 5pm in the box outside Broida 1019 (PSR)
Suggested reading: Griffiths 1.5-1.6, 2.1
1
Macroscopic Wavelengths
The idea that matter has a wave-like nature pertains not just to particles, but to macroscopi
Physics 115A, Problem Set 4
Due Tuesday, October 20 @ 5pm in the box outside Broida 1019 (PSR)
Suggested reading: Griffiths 2.3
1
Constructing States
Given the explicit solution for the ground state of the harmonic oscillator,
0 (x) =
m 1/4
~
m
e 2~ x
2
Physics 115A, Problem Set 3
Due Tuesday, October 13 @ 5pm in the box outside Broida 1019 (PSR)
Suggested reading: Griffiths 2.1-2.3
1
E and V for normalizable solutions
Griffiths Problem 2.2:
Show that E must exceed the minimum value of V (x), for every n
Physics 115A, Problem Set 6
Due Tuesday, November 3 @ 5pm in the box outside Broida 1019 (PSR)
Suggested reading: Griffiths Section 2.5, 2.6
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SHO in Momentum Space
= p2 + 1 m 2 x
The harmonic oscillator Hamiltonian in any representation is H
2 . So lets
Phys 219
Midterm Exam
02/12/15
Instructions: Show all work. Write clearly. Number all pages turned in and number
responses by question and part. Make sure your exam has five sides printed (six
questions on two sides and three sides of information). Good l
Jimmy Shen
PHYS 115A
Homework #4
October 29, 2014
2.38
(a)
In the new well of width 2a the allowed energies and wave functions are:
q
2 2 2
2
En = n p h 2 ;
yn ( x ) = 2a sin np x
2a
2m(2a)
The wavefunction at t = 0 is:
(q
2
p
for x < a
a sin a x
Y (x) =
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"* a .5
HIM? = ﬁuzgéé7 {2+z/.
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(, PLO/0., flu/e) Lat meg/G- +y¢22
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TL: ﬂth QXCJMX
Problem set 5 Physics 115A Due Friday Oct 28
1. To find the momentum space probability distribution for a particle in the ground stationary
state of the h
HW 3 Physics 115A Due Friday October 14 by 6 PM.
You should be making the switch to Dirac notation for this homework. For the harmonic
oscillator problems, use the ladder operators at every opportunity!
Physics 115A Problem set 2 Due Friday Oct 7 by 6 PM
1. Griffiths 1.9
2. Griffiths 1.14
3. Griffiths 2.1
4. Griffiths 2.2
5. Griffiths 2.7
6. Consider any normalizable probability distribution, r(x), that is defined over the range
- < x <+ .
Physics 115A
Homework 1
Problem 1.1
(a) The mean of the squares of the ages is
j 2 =
1(14)2 + 1(15)2 + 3(16)2 + 2(22)2 + 2(24)2 + 5(25)2
3217
=
460
14
7
Griths finds the mean age to be j = 21, so j2 = 212 = 441
j
14
15
(b) 16
22
24
25
j
21
21
21
21
21
21