Notes on Eigenvalues
Math 107 - Summer 2012
Eigenvalues and Eigenvectors
Definition: Given a square matrix A we define its eigenvalues and eigenvectors through
the following equation,
Av = v
The number is called the eigenvalue of the matrix A and the vect
Key ideas and formulas for statistical physics
(overlaps with Ch6 of Philip Nelsons Biological Physics)
Goal: to characterize macroscopic systems consisting of many microscopic
parts such that the macroscopic systems are in equilibrium (thermodynamic
equi
Diagonalization, Transformation, Generators and Commutators
Phy 464 - Fall 2013
In this note we focus on topics specifically pertinent to quantum mechanics. Please consult
the note Eigenvalues and Eigenvectors for a general introduction to eigenvalues and
Jim Lambers
MAT 415/515
Fall Semester 2013-14
Lecture 3 Notes
These notes correspond to Section 5.2 in the text.
Gram-Schmidt Orthogonalization
We have seen that it can be very convenient to have an orthonormal basis for a given vector
space, in order to
Postulates of Quantum Mechanics and Discrete Systems
Phy 464 - Fall 2014
In this note we discuss the basic postulates and properties of quantum mechanics and show
their application to some discrete systems, in particular two state systems and the one
dime
a) The wire is equipotential.
F
because it carries a current and has a finite conductivity and thus has a nonzero electric field inside.
b) If one doubles the current I through the wires, the electric field in the wire:
Doubles
~ and the geometry of the w
As always, a conductor in electrostatic equilibrium:
has no electric field inside or tangent to the surface;
is equipotential, as one can move a small test charge around through zero field doing no work.
Hence:
b) Equal to the potential at B.
The easies
y
dQ
d
R
dE x
dE
x
dEy
A quarter of a ring of total charge Q and radius R is oriented as shown in the figure above. Find the electric
field at the origin (magnitude and direction) from direct integration. Show all work.
There are several ways to coordinat
1
2
I1
I3
1
I2
3
6V
2
4V
Suppose we (more or less randomly) assign the directions above to the unknown currents. In that case the
junction rule is:
I1 = I2 + I3
We can evaluate the loop rule for the two (equally arbitrary) loop directions 1 and 2 above:
6
negative charge closer than positive charge
+
+
+
E=0
+
+
+
+
+q
F
Eext
Even though there is no net charge on the conducting sphere, there is inexhaustible negative and positive free
charge that rearranges to cancel the external field of the point charge
e+
F
v
B
e
~ (for a positive charge) we see that a force will deflect the positive positron
Using the right-hand rule for ~
vB
~
to the left if B is:
a) into
the page.
Fm
q
E down, B in
v
Fe
The electric and magnetic forces must balance in order for the c
F1
+q
F1
F2
a
q
+q x
a
The magnitude of the force from the horizontal and vertical corner charges is just:
ke q 2
a2
The directions are vertically up (from the lower right hand corner) and to the right (from the upper left hand
corner). The force from the
b
a
+
V
Assume that the voltage puts a charge Q on the inner conducting shell and Q on the outer shell. From Gausss
Law:
ke Q
Er = 2
r
and the potential difference between the two shells is:
V = V =
ke Q ke Q
a
b
(follows either from integration of the fi
y
Q
R
x
A quarter of a ring of total charge Q and radius R is oriented as shown in the figure above. Find the electric
field at the origin (magnitude and direction) from direct integration. Show all work.
y
L
x
x
A rod with uniform charge per unit length
R
C
+Q0
L
At time t = 0 the capacitor in the LRC circuit above has a charge Q0 and the current in the wire is I0 = 0
(there is no current in the wire).
a) Find (or remember) Q(t). Dont forget to define , the shifted frequency of this system.
b) Draw a qua
magnetic field ends
hand
Bin
R
A conducting loop of wire with a small but finite resistance R sits in and perpendicular to a powerful magnetic
field that ends at the left edge of the wire loop as shown. You grab the loop and try to pull it to the left, ou
B
A
+Q
A charge Q sits close to the inner surface of a hollow conductor in electrostatic equilibrium as shown above. Is the
potential at A:
a) Greater than the potential at B.
b) Equal to the potential at B.
c) Less than the potential at B.
d) Zero.
e) Ne
z
a
I
A circular loop of wire of radius a is carrying a current I counterclockwise (viewed from above) around the
z-axis. It is located in the x-y plane, centered on the origin as drawn.
a) Using the Biot-Savart law, find the magnetic field at an arbitrar
Problem 1.
B(uniform)
I
I
30
a
I
b
c
In the figure above, three wire (segments) are shown, each of length , each carrying a current I, in a uniform
magnetic field. Under or on each figure indicate:
a) The magnitude of the magnetic force acting on the wire
Find the capacitance of this arrangement any way you like.
Note well, this is equivalent to two capacitors in parallel with area A/2 and plate separation d. Thus the easiest
way is to use the two rules:
C = r C0
for a capacitor with a dielectric and
Ctot
A/2
A/2
+Q
d
1
2
Q
In the figure above, a parallel plate capacitor with cross-sectional area A and plate separation d is drawn. The space
between the plates is filled with two dielectrics with relative permittivities r = 1 and r = 2 of thickness d and
are
I(t)
a
b
d
r
R
In the figure above, I(t) = I0 sin(t) in the long, straight wire to the left. A rectangular conducting loop of
resistance R, width a and length b sits a distance d from and in the plane of the wire. Find:
a) The magnetic field at an arbitra
R
I
a
b
In the picture above, a circular capacitor is being charged by a current I. Using Amperes Law and the Maxwell
Displacement Current, derive a formula for the magnitude of the magnetic field at the two points shown (one
at radius a < R from the axis
b
a
+
V
A spherical capacitor has inner radius a and outer radius b
Find the capacitance of this arrangement. Show All Work!
Show that when b = a + with a the capacitance has the limiting form C = 0 A/ (parallel plate result)
where A is the area of the
2q
+a
q
+a
q
2q
Find the total potential energy of the arrangement of four charges above.
2q
+a
q
+a
q
2q
What is the potential at the center of the square of four charges shown (at the origin)? Note that the square has
sides of length 2a.
a) V = ke q/a
b
2q
+a
q
+a
2q
q
Find the total potential energy of the arrangement of four charges above.
ke q 2
2ke q 2
4ke q 2
2ke q 2
2ke q 2
2ke q 2
Etot = +
2a
2a
2a
2a
2 2a 2 2a
or
Etot =
5 2ke q 2
4ke q 2
4a
a
2q
+a
q
+a
q
2q
What is the potential at the center o
A list of named bands of the electromagnetic spectrum are given out of order below:
a) Visible light
b) X-Rays
c) Radio Waves
d) Ultraviolet
e) Gamma Rays
f) Microwaves
g) Infrared
Fill the letters into the boxes below in order of decreasing wavelength (i
+q
+q
a
q
a
+q x
In the figure above, four charges are located at at the corners of a square of side length a. Find the force on the
upper right hand charge.
z
+q
x
+q
+q
+q
y
Four positive charges of magnitude +q are located on at positions (0, a, 0), (0
A conducting wire of resistivity , length and cross sectional area A is carrying current I. Answer the following
short questions about this situation. Briefly explain your answers.
a) The wire is equipotential.
A) T
B) F
b) If one doubles the current I th
+q
A charge q is sitting a short distance away from an uncharged conducting sphere. Is there a force acting on this
charge? If so, in what direction (towards the sphere or away from the sphere) does the force point? Indicate why
you answer what you answer