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SYRACUSE UNIVERSITY −− PHYSICS DEPARTMENT
Ph.D. Qualifying Examination
Aug. 23, 2008
9:00 a.m. −− 12:00 p.m.
Each of the problems carries equal weight (10 points out of a total of 100
points for the entire two-day exam.)
Show clearly the steps leading to your final answer(s). No books or notes.
Partial credit is given. Pace your work accordingly.
Part I: Classical Mechanics
Canonical Transformations
1. A transformation is canonical if there is a function F such that
pi q̇i − H = Pi Q̇i − K +
dF
dt
(1)
(a) Suppose that F = F2 (q, P, t) − Qi Pi . Expand the total derivative
in Equation 1 to obtain independent equations that, among other
things, give the equations for a canonical transformation.
(b) Suppose that a system is described by the variables qi , pi with
i = 1, 3. Given the generating function
c
F2 (qi , Pi ) = bq12 P1 + q2 P2 + (aq1 + q3 )P3 − (aq1 + q3 )3
3
where a, b, and c are constants. Find the canonical transformation
from the pi , qi to the Pi , Qi .
(c) Would this F2 make a useful Hamilton’s principle function S?
Useful Integrals
� �
�
dx
1
x
= tan−1
2
2
a +x
a
a
�
√
dx
√
=
ln(x
+
a2 + x 2 )
a2 + x 2
1
Wire act
2. A rigid parabolic wire defined by the equation z = ar2 , as shown in the
figure, has its motion constrained such that it can only rotate about
the z-axis. The wire rotates on a frictionless bearing. A bead of mass
m is free to slide along the wire.
(a) Neglecting the mass of the wire, show that the Lagrangian L, for
the system can be written as
�
�
1 ��
L = m 1 + 4a2 r2 ṙ2 + r2 θ̇2 − mgar2
2
(2)
where g is the acceleration due to gravity.
(b) If the motion of the wire is further constrained such that the
angular speed about the z-axis, θ̇ = ω where ω is a constant,
show that the Hamiltonian H can be written as
H=
p2r
1
− mr2 ω 2 + mgar2
2
2
2m(1 + 4a r ) 2
(3)
where pr is the momentum conjugate to r.
(c) Show that in this case (θ̇ = constant), H is not equal to the total
energy of the bead. Why is this? Would the Hamiltonian be equal
to the total energy if θ̇ was unconstrained?
2
Part II: Electromagnetism
Magnetic potential
� the
3. (a) Show that for a constant, uniform magnetic field vector B
vector potential can be chosen to be
� = 1 (B
� ⊗ �r).
A
2
(b) Electric currents I flow in opposite directions along two straight,
parallel conductors of infinite length placed apart a distance d.
�
Determine A.
Dielectrics
4. Consider an infinite dielectric medium with dielectric constant κ = �/�0
with a spherical cavity (vacuum) of radius a. An electric field E0 is
applied parallel to ẑ.
(a) Find the potential and electric field both inside and outside the
cavity.
(b) Interpret the field outside the cavity.
3
Lines of Charge
5. A straight-line charge with constant linear charge density λ is located
perpendicular to the x − y plane in the first quadrant at (x0 , y0 ). The
intersecting planes {x = 0, y ≥ 0}, and {y = 0, x ≥ 0} are conducting
boundary surfaces held at zero potential. Consider the potential, fields,
and surface charges in the first quadrant.
(a) The well-known potential for an isolated line charge at (x0 , y0 ) is
Φ(x, y) = (λ/4π�0 ) ln(R2 /r2 )
where r2 = (x−x0 )2 +(y−y0 )2 and R is a constant. Determine the
expression for the the potential of the line charge in the presence
of the intersecting planes. Verify explicitly that the potential and
the tangential electric fields vanish on the boundary surfaces.
(b) Determine the surface charge density σ on the plane y = 0, x ≥ 0.
(c) Show that the charge per unit length in z on the above plane is
�
2
x0
Qx = − λ tan−1
π
y0
�
(d) What is the total charge in the plane x = 0, y ≥ 0?
Justify your answers.
4
Syracuse Univeristy — Department of Physics
August
25, 2008
-----------------------
PhD Qualifying Examination
August 24, 2008
9:00 AM – Noon
Each of the problems today carries equal weight (10 points out of a total
of 100 points for the entire two-day exam).
Show clearly the steps leading to your final answer(s). No books or notes are
to be consulted. Partial credit is given. Please pace your work according to
the available time.
1
Part III: Statistical Physics
6. A condensed matter experimentalist working on graphite has found
that the specific heat of graphite is proportional to the square of the
temperature, at low temperatures. You, a theorist, are curious about
this, as many solids have a specific heat proportional to the cube of
temperature, at low temperature. In discussions with the experimentalist you learn that (1) graphite has a layered structure where the
coupling between the carbon atoms is much weaker from layer to layer
than within a layer and (2) graphite is an insulator. You wish to see if
Debye theory can be adapted to provide an explanation of the specific
heat, in the approximation that the graphite layers can be treated as
a two-dimensional solid.
(a) What is the name for the excitations that dominate the energy at
low temperatures?
(b) If the c� is the longitudinal wave speed and ct is the transverse wave
speed of sound in the graphite layers, what is the number of normal
modes in the frequency range (ω, ω + dω) for graphite. If it makes your
work easier, you can assume that graphite is a square lattice of atoms.
[Hints: first find how many modes are in a given wave vector range,
assuming N atoms? how is the energy related to wave-vector or mode
number?]?
(c) What is the average internal energy of the system? (Do not evaluate the integral, as it can only be done numerically.)
(d) What is the specific heat of the system?
(e) In the limits of (i) low temperatures and (ii) high temperatures,
how does the specific heat depend on temperature?
(f) Does this two-dimensional Debye model account for the experimentalist’s findings?
2
7. You are an experimentalist studying a biopolymer interacting with
small molecules. The biopolymer has N binding sites: a small molecule
can attach to the long biopolymer at any of these sites. The free energy
is lower when two small molecules bind to adjacent sites than when they
bind to non-adjacent sites. The model you construct has an energy, the
Hamiltonian H, where (i) each small molecule has a chemical potential
µ = −kT ln(W ) to be attached to the biopolymer (µ is the free energy
difference between a site with a bound molecule and an empty site)
and (ii) at each point where two small molecules are adjacent, there is
a gain in energy of J = −kT ln(K) (see the figure below on this page).
(a) A configuration of this system can be described by a sequence of
numbers �i , where i = 1, . . . , N and �i = 0 when no small molecule is
bound to the polymer at site i and �i = 1 when a small molecule is
bound to the polymer at site i. Write down the effective energy function H for a configuration of bound molecules given by �i .
(b) When J = 0 (i.e., K = 1), what fraction of the binding sites are
occupied by a small molecule?
(c) For general W and K, let ZN be the partition function for the given
H for a biopolymer with N sites.
number of
� Show that the expected
∂ ln ZN
bound small molecules �n� = � i �i � is given by W ∂W .
(d) Write the constrained partition functions ZN (�N = 1) and ZN (�N =
0), ZN = ZN (�N = 0) + ZN (�N = 1), in terms of ZN −1 (�N −1 = 1) and
ZN −1 (�N −1 = 0).
(e) Briefly describe, in a sentence or two, how you would continue to
find how ZN depends on N . You do not need to actually compute �n�
for this exam for general W and K.
3
Part IV: Quantum Mechanics
8. Consider a physical system that is well described as a frictionless pendulum with a mass m at the end of an arm of length � in a gravitational
field g. Treat this pendulum as a purely quantum mechanical system
(no temperature or decoherence effects). Estimate how long the pendulum can stay inverted, that is, balanced with the mass above the
point of support. Construct your answer by following these steps:
(a) Write a Hamiltonian for the pendulum.
(b) Show that angular momentum and angular position of the pendulum cannot be simultaneously determined.
(c) Write an uncertainty principle relationship between angular momentum and angular position.
(d) Show that, classically, a pendulum that is displaced by an angle φ0
from an inverted vertical position will “fall” in a time proportional to
ln(φ0 ).
(e) How does the time to rotate by an angle π depend on the angular
momentum L?
(f) Estimate an upper bound on the time for the pendulum to remain
vertical, by adjusting the two times in (d) and (e) to be the same.
(If you have sufficient time on this exam, you can compute the maximum time in seconds for the case m = 10−3 kg, � = 10−2 m, and
g = 10 m/s2 .)
4
9. Suppose that one has an Ω− particle, which has spin 32 , orbiting another particle with angular momentum quantum number � =1. That
� 2 = s(s + 1)�2 = 3 · 5 · �2 and L
� 2 = �(� + 1)�2 = 2�2 .
is S
2 2
(a) What are the possible values for the total angular momentum quantum number j?
(b) If the quantum number j for the total angular momentum for the
Ω− particle is 52 and the z-component of the total angular momentum
is jz = 12 , measured in units of �, what is the probability that the Ω−
particle has a sz (the z-component of the spin) equal to 12 (measured in
units of �)? [Hint: apply the operator that lowers the z-component of
the total angular momentum to the |s = 32 ; � = 1� state often enough
to construct the jz = 12 state.]
5
10. Suppose that the two lowest energy levels of an atom are nearly degenerate. That is, the difference in the two energy eigenvalues, Ea and
Eb are much closer to each other than to any other energy levels and
are the two lowest eigenvalues. Suppose that an external (classical)
time-dependent electric field E(t) modifies the Hamiltonian by adding
a term γE(t)(|a��b| − |b��a|).
(a) What are the units of γ? Why must γ be purely imaginary? Since
this is the case, one can write γ = ig, with g real.
(b) Suppose that E(t) is weak and that E(t) = E0 e−|t|/τ , where τ has
units of time. Initially, i.e., in the limit t → −∞, the atom is in state
|a�. To the lowest order in E0 that gives a non-zero result, using timedependent perturbation theory, what is the probability that the atom
is in state |b� at t = +∞?
(c) For very large τ , how does your answer behave? Explain how your
answer agrees or disagrees with the adiabatic theorem.
6
Syracuse Univeristy — Department of Physics
August 29, 2009
PhD Qualifying Examination
August 29, 2009
9:00 AM – Noon
Each of the problems today carries equal weight (10 points out of a total
of 100 points for the entire two-day exam).
Show clearly the steps leading to your final answer(s) - answers without
explanation will not be given credit. No books or notes are to be consulted.
Partial credit is given. Please pace your work according to the available time.
1
Part I: Statistical Physics
1. In the Landau theory of phase transitions, the state of a system is
described by an “order parameter” ψ. The free energy F (ψ, T ) of a
system is a non-singular smooth (say, analytic) function both in the
temperature T of the system and the order parameter ψ. In the simplest case, the order parameter ψ is a single scalar, i.e., a real number.
The order parameter ψ is zero when the system is in a disordered (high
temperature) phase, T > Tc . For this problem, assume that Landau
theory is the proper description of phase transitions.
(a) Write down a general expansion of the free energy F (ψ, T ), in powers of ψ, assuming that the free energy is independent of the sign of ψ
[i.e., F (−ψ, T ) = F (ψ, T )].
(b) Which coefficient of this expansion reverses sign at the phase transition, usually?
(c) What is the simplest form for the dependence of this coefficient on
the temperature T and the phase transition temperature Tc ?
(c) Given the results of (a), (b), and (c), how does ψ depend on T , for
T < Tc (that is, what is the dominant functional dependence of ψ on
T , for T just below Tc )?
2
2. Suppose that a satellite orbiting the Earth has a container of helium.
Suppose that this container is hit by a small meteoroid: this impact
opens a small hole in the side of the container. The hole is small
enough that gas leaks slowly: the temperature of the gas in the container is maintained at temperature T and non-equilibrium effects can
be neglected. The temperature T is high enough that the helium is a
classical ideal gas. The helium atoms have mass m.
(a) What is the average kinetic energy of the atoms inside of the container?
(b) What is the probability distribution P (v) for the speeds v = |�v | (�v
is a velocity vector, so that the speed v is the magnitude of �v ) of the
atoms?
(c) Consider the atoms escaping (that is, effusing) from the container.
How does the probability of an atom escaping during a small time interval depend on the speed of the atom?
(b) What is the probability distribution Pe (v) for the speed of escaping
atoms? You do not need to give a normalized distribution: just give
the dependence of the distribution on v that is correct up to a constant
factor that might depend on m and T . Note that Pe (v) is not the
same distribution as the distribution P (v) of atom speeds within the
container.
(c) What is the average kinetic energy of the atoms escaping from the
container?
3
Part II: Quantum Mechanics
3. In this problem, assume that you that you have a single uncharged
point particle whose quantum mechanical behavior is described by the
standard time-dependent Schrodinger equation in 3 dimensions.
(a) If the particle is a free particle, that is, there is no external potential,
explain why momentum p� and position �x can not be simultaneously
defined: give a paragraph or two of physical explanation, using both
words and an equation defining the momentum of a particle.
(b) Now consider the general Schrodinger equation, where there is a
non-zero potential V (�x). Use the Schrodinger equation to show that
the probability density P (�x, t) = ψ ∗ (�x, t)ψ(�x, t) obeys a conservation
equation, that is, the rate of change of P is the divergence of a current
�j(�x, t). In your derivation, clearly state how �j(�x, t) is defined in terms
of ψ and its gradients(s) and conjugate.
(c) Compute the probability
current for a free
of mass m with
�
� particle
2
wave function ψ(�x, t) = Aei�p·�x/� + Be−i�p·�x/� e−ip t/2m�.
4
4. If a charged particle with charge q is subject to a uniform external
� then, the only effect on a Schrodinger equation demagnetic field B,
scription of the electron is to replace the momentum operator by the
�
� is the classical
canonical momentum, that is, p� → p� + qcA , where A
vector potential (a real vector field, not an operator) from electromagnetism.
(a) Use this replacement to write the Schrodinger equation for a charged
spinless particle in an external magnetic field, for general V (�x) and
� t).
A(x,
(b) In some electronic devices used in experiments, mobile electrons
are confined to move in a very narrow layer of material, so that the
electrons are effectively confined to move in two dimensions, the x
and y directions. For simplicity, take the electron to have spin zero.
� =
Take the charge of the electron to be q. Make the gauge choice A
�
(−By/2, Bx/2, 0) for a static uniform magnetic field B = (0, 0, B).
Suppose that any external potential is zero. Write down the Schrodinger
equation for the electron subject to this magnetic field confined to two
dimensional motion.
� is
(c) Assume that the wave function for the confined electron in B
ikx
−iEt/�
an eigenstate of the energy of the form ψ(x, y, t) = e φ(y)e
.
Show that φ(y) satisfies a time-independent Schrodinger equation for
a harmonic oscillator and deduce the ground state energy value for the
electron.
5
� = (0, 0, B).
5. Consider a spin- 12 particle subject to a magnetic field B
For this problem, ignore the spatial degrees of freedom and consider
only the spin degree of freedom.
(a) What are the ground state(s) and excited states(s) and corresponding energies for this particle?
(b) At time t = 0, the particle is in its ground state. Suppose that one
briefly turns on a magnet that adds a time-dependent magnetic field
� oriented in the x-direction and with magnitude Bx (t) during the
∆B,
time interval 0 ≤ t ≤ T . During, before, and after the application of
� is held constant at B. Please assume
this field, the z-component of B
that Bx is weak and that T is brief. To lowest order in perturbation
theory, what is the probability that the particle will be in the ground
state just after t = T ?
(c) Now consider the case where the particle is in its ground state (for
Bx = 0) at t = 0. Let Bx be constant and non-zero during the interval
0 ≤ t ≤ T and Bx = 0 for t < 0 or t > T . Compute the probability
that the particle is in its ground state after time T , where Bx need not
be weak and where T could be long.
6
SYRACUSE UNIVERSITY −− PHYSICS DEPARTMENT
Ph.D. Qualifying Examination
Aug. 30, 2009
9:00 a.m. −− 12:00 p.m.
Each of the problems carries equal weight (10 points out of a total of 100
points for the entire two-day exam.)
Show clearly the steps leading to your final answer(s). No books or notes.
Partial credit is given. Pace your work accordingly.
Part I: Classical Mechanics
Rigid Rotor
1. A rigid body is composed of four particles each of mass M located at
the following Cartesian coordinates:
√
√
√
√
√
√
{0, 0, 2}, {0, − 2, 0}, {1, −1/ 2, −1/ 2}, {−1, −1/ 2 − 1/ 2}
(a) What is the inertial tensor in the given coordinate system?
(b) Find the principle moments of inertia and the principle axes.
(c) Suppose that the body is rotating about the z axis with angular
frequency !ω . What is L?
1
r
g
θ
m
Bob
2. The figure shows a pendulum bob of mass m suspended from a light coil
spring of force constant k and equilibrium length r0 . The pendulum
oscillates in the plane, with the angle between the bob and the vertical
given by θ, as shown. The spring is constrained to contract only in the
direction parallel to "r .
(a) Show that the Hamiltonian for this system is given by
!
"
1
1
1
p2r + 2 p2θ − mgr cos θ + k (r − r0 )2
H=
2m
r
2
(1)
where pr and pθ are the momenta conjugate to the r and θ generlized coordinates respectively.
(b) Write down Hamilton’s equations for this system.
(c) What dynamical variable (if any) is constant for this system? Is
the Hamiltonian equal to the total energy?
2
Part II: Electromagnetism
A sphere
3. A point charge q is a distance d > a from the center of a grounded
conducting sphere of radius a and center at O.
(a) Find the induced charge on the sphere. Hint: use the method of
images.
(b) If instead the conducting sphere is isolated and neutral, find the
induced dipole moment.
(c) For case (b), find the force on the charge q in the dipole approximation with the dipole located at O. Compare this dipole force
with the large d limit of the exact force on charge q.
Legendre Polynomials
4. The potential outside a sphere of radius R that has azimuthal symmetry
is given by
! " R #(l+1)
ψ(r, θ) =
Vl Pl (cos θ)
r
l
(a) Using the properties of the Legendre polynomials, compute Vl for
the case where the potential on the sphere is given by V (θ).
(b) If all the charges are on the surface of the sphere, show that the
charge density is given by
σ(θ) =
1 !
(2l + 1)Vl Pl (cos θ)
4πR l
(c) Use the above result to compute the dipole moment for a sphere
with potential +V for θ < 60◦ and −V for θ > 120◦ .
3
Magnetic Monopoles
5. (a) Write down Maxwell’s equations. Also write down the relationship
! and the vector potential A.
!
between the magnetic field B
(b) Show that
!
! · d!l
A
around any loop gives the magnetic flux through that loop.
(c) How would Maxwell’s equations change if magnetic monopoles
existed?
! is no longer generally useful if mag(d) Why would we expect that A
netic monopoles are present?
! in spherical coordinates (with orthonor(e) Nevertheless show that A
mal basis vectors) given by
Ar = Aθ = 0; Aφ =
g(1 − cos θ)
4πr sin θ
reproduces nearly everywhere the Coulomb-like field of a magnetic
point charge g located at r = 0. Is this valid for all values of
(r, θ, φ)? (The idea is a bit like using a scalar magnetic potential.)
(f) Calculate the upward flux of the magnetic field through a circular
loops of radius a with its center on the z-axis a distance ±d from
the origin. Is the result different from what you would expect from
a point charge by symmetry?
Useful formula for orthogonal coordinates:
∇ψ =
$
"
∂ψ 1 ∂ψ
1 ∂ψ
,
,
∂r r ∂θ r sin θ ∂φ
%
$
#
%
$
%
1
∂Fθ
1 ∂(rFθ ) ∂Fr
∂
1 ∂Fr 1 ∂(rFφ )
sin θFφ −
−
−
∇×F! =
r̂+
θ̂+
φ̂
r sin θ ∂θ
∂φ
r sin θ ∂φ
r ∂r
r
∂r
∂θ
4
SYRACUSE UNIVERSITY −− PHYSICS DEPARTMENT
Ph.D. Qualifying Examination
Aug. 21, 2009
------2010
9:00 a.m. −− 12:00 p.m.
Each of the problems carries equal weight (10 points out of a total of 100
points for the entire two-day exam.)
Show clearly the steps leading to your final answer(s). No books or notes.
Partial credit is given. Pace your work accordingly.
Useful integrals:
!
!
√
√
a2
−
x2 dx
"
# $%
1 √ 2
2
−1 x
2
=
x a − x + a sin
2
a
'
()
√
1& √ 2
x a + x2 + a2 ln x + a2 + x2
2
# $
!
dx
−1 x
√
=
sin
a
a2 − x2
!
'
(
√
dx
2 + x2
√
=
ln
x
+
a
a2 + x2
a2 + x2 dx =
1
Part I: Classical Mechanics
Global Warming
1. Consider the linear triatomic molecule below, which is a simple model
for carbon dioxide. The central atom is more massive than the two
(identical) exterior atoms, i.e. M > m. The molecule has three degrees of freedom. The position of the atoms can be described by the
coordinates x1 , x2 and x3 .
m
m
Define x1 , x2 and x3 such that the system is in equilibrium when they
are all zero. Assuming that each atom oscillates in a simple manner
with the same frequency ω, i.e.
(0)
xi = xi eiωt ,
(1)
use Lagrange’s equations to obtain the normal modes of oscillation of
the molecule.
For each solution you obtain for ω, derive the equations of motion for
the system. Describe (using words and/or a diagram) the motion of the
atoms in the molecule for each solution. Are the solutions you obtain
consistent with conservation of momentum? Explain your answer with
reference to your solutions.
2
The Simple Harmonic Oscillator Again
2. A canonical transformation between the variables p and q and the variables P and Q can be generated by any function F2 (q, P ). The equations for the transformation are
p=
∂F2
∂F2
; Q=
∂q
∂P
Now suppose that we want to use a canonical transformation to solve
the harmonic oscillator
1
H = (p2 + q 2 )
2
where we want the new variable P (p, q) = H(p, q) so that Hamilton’s
equations are easy to solve.
(a) Use the equation P = 12 (p2 + q 2 ) to obtain an integral for F2 .
(b) From this F2 , obtain Q(q, P ).
(c) From the last two steps, obtain explicitly the transformation equations and the inverse:
q(Q, P ); p(Q, P ); and Q(q, p); P (q, p)
(d) Solve Hamilton’s equations using the variables Q and P . Note:
this is trivial.
(e) By using the transformation equations, convert your answer to the
variables q and p.
3
Part II: Electromagnetism
A Ring
3. A current I passes through a ring of radius a that lies in the x-y-plane
centered an the origin.
(a) Compute Bz on the z-axis by elementary means.
(b) Expand Bz in a power series in z. Keep two non-vanishing terms.
(c) The magnetic potential is defined so that B = ∇φM in currentfree regions of space. Determine a power series for φM valid on
the z-axis near the origin.
(d) Write an expression for φM valid near the origin but off the axis.
(e) Compute B near the origin in terms of Legendre polynomials.
Fun with Dipoles
4. A conducting sphere of radius b is put at a potential V , and then
disconnecetd from the battery.
(a) A fixed dipole p0 is a distance R " b from the center of the sphere.
What is the energy U of the dipole? Also, compute the force on
the dipole by using the formula F = −∂U/∂r.
(b) A small object of polarizability η (p = ηE) is also placed a distance R from the center of the sphere, where the polarization also
happens to be po . Compute the force on it, also by using the
formula F = −∂U/∂r. However, note that in this case
1
U = p · E.
2
(c) Explain the similarities and differences between the two cases.
4
Magnetic Potential
5. Determine the vector potential A for two infinitely long, parallel wires
each carrying a current I, but in opposite directions. The wires are a
distance d apart. Check your result by answering the following questions:
(a) Calculate A on the line (z-axis) exactly mid-way between the two
wires. Is this what you would expect?
(b) Check that ∇ · A = 0.
(c) Show that near one of the wires the result is as expected.
5
Syracuse University — Department of Physics
August 25, 2010
------------------------
PhD Qualifying Examination
August 22, 2010
9:00 AM – Noon
Each of the problems today carries equal weight (10 points out of a total
of 100 points for the entire two-day exam).
Clearly show the steps leading to your final answer(s). No books
or notes are to be consulted. Partial credit is given. Please pace your work
according to the available time.
1
Part III: Statistical Physics
6. An elastic string is held fixed at two ends: the fixed ends of the string
are separated by a distance L. The tension in the string is τ . This string
is in thermal equilibrium with its environment, which is at temperature
T , and is described by classical (not quantum) mechanics. Gravity and
other external fields are not important. Let x be the position along the
string, 0 ≤ x ≤ L. For this problem, please compute the expected mean
square displacement of the middle of the string from its rest position in
the y-direction, which is perpendicular to the x-axis. That is, compute
�y 2 (L/2)�, where the notation �. . .� indicates a thermal expectation
value. Consider only small (linear) vibrations of the string.
Here is some information, some of which might or might not be useful
for solving this problem:
• If the string has a displacement in one direction yn (x) = An sin(nπx/L),
with amplitude An , the energy En of that displacement is
En =
τ n2 π 2 A2n
4L
• The speed of waves in such an elastic string is c =
�
τ
,
ρ
where the
mass of the string is m = ρL, so that ρ is the linear mass density.
• The infinite sum of the inverse of the�
squares of all positive integers
2
1
is π6 = ζ(2) = 112 + 212 + 312 + . . . = ∞
k=1 k2 .
• The infinite sum of the inverse of the squares
�∞ of1odd positive inπ2
1
1
1
1
tegers is 8 = 12 + 32 + 52 + 72 + . . . = k=0 (2k+1)2 .
2
7. Consider a nondegenerate monatomic ideal gas, where the atoms of
mass m can move in 3 dimensions.
(a) What is the equation of state for this ideal gas?
(b) What is the average energy of this ideal gas with N particles at
temperature T in volume V and at pressure P ?
(c) Initially you have N particles of this gas in volume V and at temperature T . You then compress the gas at constant temperature
to volume V /2 while keeping the temperature constant:
i. What is the change in the entropy of the gas? Justify your
answer.
ii. What is the change in the energy of the gas? Justify your
answer.
(d) Initially you have N particles of this gas in volume V and at
temperature T . You then reversibly compress the gas to volume
V /2 while keeping it thermally isolated (no heat flow from or to
the environment):
i. What is the change in the entropy of the gas? Justify your
answer.
ii. What is the change in the energy of the gas? Justify your
answer.
3
Part IV: Quantum Mechanics
8. A particle of mass m is confined by a one-dimensional harmonic potential, U (x) = 12 mω 2 x2 . The particle is described by the Schrödinger
equation. A ground state wave function is
ψ0 (x) =
� mω � 14
π�
e−
mωx2
2�
(a) What is the ground state energy of this particle (you can simply
state it from memory)?
(b) Suppose that you only knew that the ground state wave function
2
was a Gaussian function. That is, assume that ψ0 (x) ∝ e−bx .
Show that the above wave function has the minimum value of �E�
out of all possible Gaussian wave functions.
(c) Suppose that the potential is perturbed, with Uα (x) = U (x) +
αδ(x), where α can be taken to be “small”
i. What are the units of α?
ii. When we state that α is small, what quantity must we compare α with? That is, what dimensionless ratio tells us whether
α is small?
iii. Compute the change in the ground state energy due to this
perturbation, to first order in α, using time-independent perturbation theory.
Possibly
information:
� ∞ useful
�
−ax2
dx
e
= πa
−∞
�∞
�
2
dx x2 e−ax = (2a)−1 πa
−∞
�∞
�
�π
−bx2 d2 −bx2
2 ∞
2 −2bx2
dx
e
e
=
−4b
dx
x
e
=
−b
2
dx
2b
�−∞
� π −∞
∞
1
−bx2 2 −bx2
dx
e
x
e
=
4b
2b
�−∞
�
∞
π
3
−2bx2 4
dx
e
x
=
16b2
2b
−∞
4
9. Consider a spin- 12 particle in a external time independent uniform mag� The particle is fixed (it has no linear momentum). The
netic field B.
Hamiltonian for this particle is then
� ·B
�.
H = −γ S
� what are the possible energy eigenvalues for this particle?
(a) Given B,
� and the state of the particle at time t = 0 is |ψ(0)�, what
(b) Given B
� and B?
�
is the state |ψ(t)� at time t, written in terms of t, S,
� = B0 ẑ and |ψ(0)� = | ↑�, where | ↑� is the spin-up (in the z(c) If B
direction) eigenstate, what is the probability of the particle being
in the spin-down eigenstate, | ↓� at time t?
� = B0 x̂ and |ψ(0)� = | ↑�, where | ↑� is the spin-up (in the z(d) If B
direction) eigenstate, what is the probability of the particle being
in the spin-down eigenstate, | ↓� at time t?
5
10. Suppose that particle physicists discover a stable particle which has a
mass m, spin 52 , and zero charge. Astrophysicists wish to estimate the
size of stars that might be made up entirely out of these new particles. Assume that the particles do not interact with each other, that
the temperature of the star is very low, and that the particles are
non-relativistic (they move much more slowly than the speed of light).
Assume the star to have uniform density.
(a) Does the gravitational energy decrease or increase as R decreases?
(Assume constant total mass M ).
(b) What type of pressure can these particles exert to keep the star
from collapsing: give a qualitative (heuristic) explanation. Your
answer should be made up of two or three sentences.
(c) Why would the star be unstable if the particles instead had spin
2?
(d) Now compute the size of the star in more detail. Compute the
total energy of N = M/m particles confined to a sphere of radius
R and find the value of R that minimizes the total energy. Explain
your computation. If you don’t finish the computation, outline the
steps you would carry out.
6