Ph 12b Solution Set 6
1. a) The Schrodinger equation for this system can be written:
2
2 2 2
+
+
2m x2
y 2
z 2
1
+ m 2 (x2 + y 2 + z 2 ) = E
2
We try a solution by separation of variables:
(x, y, z ) = X (x)Y (y )Z (z )
Plugging this in and dividing by ,
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Physics 12b
One-Dimensional Scattering
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All the relevant mathematics is in Libo , sect
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Physics 12b
3-Dimensional Quantum Physics
3 6 98
These plots were produced using the IQ program from the book Quantum Mechanics on the Macintosh" by Siegmund Brandt and Hans Dieter Dahmen Springer Verlag, 1991.
All the relevant mathematics is in Libo , se
Lecture notes, Caltech Physics 12b (Quantum Physics), Winter 2013, Gil Refael
I.
1
CLUMSY BEGINNINGS
If you ignore for a bit what you hear from people inbvolved in the LHC, or characters in the big-bang theory, what
is the most important particle for mode
Physics 12b
Midterm Exam
2/6/14
This exam should be completed in four hours, working by yourself. Breaks are permitted.
In your studies, do not use problem/solution books, or look at problems or exams from
previous years of Phys 2 or 12.
Please use a b
Ph 12 Homework 7
1
Question 1
H=
1.1
h
2 2
VL (x + a) VR (x a)
2m x2
Part a
In this section, we write down the solution for two bound modes assuming in
turn that V1 or V2 is 0.
eL (x+a)
x a
L L (x+a)
|L =
,
L e
x > a
|R =
eR (xa)
R R (xa)
R e
xa
.
x
Ph 12b Solution Set 6
1. a) We look for a solution of the form:
eikx + reikx
tei x
( x) =
where E =
22
k
2m
and E + V0 =
22
2m
x<0
x>0
. Requiring continuity of and
d
dx
at x = 0 gives:
1+r =t
ik (1 r) = i t
Which can be solved to give:
r=
t=
or, in term
Ph 12b Solution Set 4
1.
(a) We have:
(x, 0) = Ax(L x)
The normalization condition is:
L
L
dx| (x, 0)|2 =
1=
0
dxA2 x2 (L x)2
0
A2 L5
30
where weve used Mathematica to evaluate this integral. This means we should take:
=
30
L5
See the gure below for a pl
Ph 12b Solution Set 2
1. (a) The time derivative of the total probability of nding the particle anywhere is:
d
dp
=
dt
dt
=
dx
dx| |2
+
t
t
Now if we use the Schrodinger equation:
i
2
2
=
+ V (x)
t
2m x2
along with its complex conjugate, we can plug i
Physics 12b Problem Set 1: Solutions
January 17, 2013
Problem 1: Wiens Law ala Planck
This problem is a question of denitions. Wiens law T max = cw , with cw = 0.0029K.m, is a
relation between the wavelength corresponding to the maximum intensity of emiss
Ph 12b Midterm Solutions
Problem 1
Our system is composed by a particle conned on a ring, and it is described by the hamiltonian
H | = E |
2
(1)
2
where H = 2m x2 is the selfadjoint operator that represents it.
a. We want to nd eigenstates of this hamilto
1
Mid-term
Instructions:
This exam must be taken in one sitting over the period of 5 hours from start to end. Please note on the top of
your solution the starting and nishing time.
you may use any personal notes, course hand-outs, and one book of your c
Physics 12b
Assignment VI
2/20/14
Reading:
This week: Griths sections 2.52.6; Next week: Griths chapter 4.
Problems:
Due in Ph12 IN box across from 264 Lauritsen or in class, 10:30 AM Thursday, 2/27/14.
Note that some of these problems require solving lon
Physics 12b
Final Exam
3/13/14
This exam should be completed in ve hours, working by yourself. Breaks are permitted (and not counted against the time limit), but you are under the honor system!
Please use a blue book, and put your name on it.
The exam
Physics 12b
3-Dimensional Quantum Physics
3 6 98
These plots were produced using the IQ program from the book Quantum Mechanics on the Macintosh" by Siegmund Brandt and Hans Dieter Dahmen Springer Verlag, 1991.
All the relevant mathematics is in Libo , se