1. A n d -w a) F th L b) H c) F th o HINT: P e An ideal, unif ail of neglig istance y (a without frict ind an expre hrough a (ve Lagrangian ap How much tim ind an expre hrough a (ve f the tension Place the zer exists when COMP Co Ea form rope of gible diamete and then rele ion- off the n ession for the rtical) distan pproach and me will it tak ession for the rtical) distan n in the rope ro point of g the two ends REHENSIV Physics University o Classic Aug omplete any ch problem h f constant len er. One end eased), and s nail. Assum e velocity of nce y. Use co d compare th ke for the ro e tension, T, nce y. Check is zero as th ravitational s are even). VE EXAMIN s Departmen of Nevada, R cal Mechanic gust 2014 4 out of 5 pr has the same ngth L is han of the rope i so, starting fr ming the rope f the rope aft onservation he answers. ope to slide c in the rope k your result he rope leave potential en NATION 20 nt Reno cs roblems. e weight. nging, with t is pulled dow rom rest, the e does not lif ter the end o of energy ap completely o after the end t by using it es the nail. ergy at the m y 14 the ends eve wnward a ve e rope begin ft off the nai of the rope h pproach and off of the nai d of the rope to verify tha midpoint of t en, over a sm ery small s to slowly s il, as moved then a il? e has moved at the magni the rope (as mooth slide itude it
Transcript
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thL
b) H
c) F
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HINT: Pe
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ind an exprehrough a (ve
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University o Classic Aug
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2. Suppose we have a particle moving in a central force potential. Consider the following vector, here given in units of 1m ,
rklpA ˆ
,
where p
is the particle’s linear momentum, l
its angular momentum with respect to the origin (force center), and r is the radial unit vector pointing from the origin (force center) to the particle. Note that rp
in units of 1m . For a central potential, rVV , find a
general expression for the first time derivative of A
, that is, find A
. Use your answer to
show that A
is a constant of the motion for the specific central potential rkV .
HINT: The vector formula cbabcacba
may be useful.
3.
a) Show that the following transformation,
2, ApqPpQ (where A is any constant) is a canonical transformation,
i) by evaluating the Poisson bracket PBPQ,
ii) by expressing PdQpdq as an exact differential QqdF ,1 . WARNING: to do this,
you must first use the transformation equations to express Qqpp , and
QqPP , . Hence find the generating function QqFF ,1 that generates this transformation.
b) Write down the Hamiltonian, pqH , , for a particle moving vertically in a uniform
gravitational field. Using the given transformation, find the Hamiltonian PQH ,~
. Show that we can make Q cyclic by choosing an appropriate value for the constant A.
c) With this choice of A, write down and solve Hamilton’s equations for the new canonical
variables, and then use the transformation equations to find equations for tq and tp . Identify all constants produced.
4. Cp
a) F
thb) F
HINT: In
5. Dex
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Classical Mechanics Supplement – August 2014
Newton’s second law
dt
pdF
Lagrangian
VTL Hamiltonian
LqpH ii
i
Hamilton’s equations
q
Hp
p
Hq
Euler- Lagrange equation
0
q
L
dt
d
q
L
Poisson bracket
q
P
p
Q
p
P
q
QPQ PB
,
Comprehensive Examination EM (2014 summer) Department of Physics
Electricity and Magnetism
Answer any four problems. Do not turn in solutions for more than four problems.Each problem has the same weight.
1: Particles of mass M are singly ionized in an ion source Q and accelerated by thevoltage U . They are entering the magnetic field B through a slit S perpendicular tothe plane of paper, as in Fig. 1. (a) What is the velocity of the particles at the slitS? (b) Where do they hit the photoplate? (c) Where do the particles entering themagnetic field at an angle α � 1 with respect to the axis hit the photoplate? (e)How can the mass of the particles be determined by this arrangement?
Fig. 1
2: Let a point charge q be at a distance a in front of an infinitely extending conductingwall. (a) Obtain the electric field normal to the conducting wall. (b) What chargedensity will be induced at the wall? (c) What is the magnitude of the total charge ofthe plane?
Fig. 1
1
Comprehensive Examination EM (2014 summer) Department of Physics
3: Consider a moving charge q in X-direction with velocity v in K-frame (lab frame).There is another observation frame K ′, which is comoving with the charge, where thecharge is at rest. (a) Obtain the electric field (E ′x, E
′y, E
′z) and the magnetic field (B′x,
B′y, B′z) in K ′-frame. (b) Obtain the electric field (Ex, Ey, Ez) and the magnetic field
(Bx, By, Bz) in the K-frame by the Lorentz transformation. (c) Use the obtainedelectric field, illustrate the electric fields from the moving charge when the velocity vis v � c and when it is close to the speed of light.
4: A transmission line consists of two identical thin strips of metal, of width b andseparated by a distance a. Assume that b� a, and neglect edge effects.
a) Is it possible to propagate a TEM mode on this line? Explain why.b) Work out the electric E and magnetic H fields associated with the TEM mode.c) Calculate the net flow of power P .d) Find out the attenuation constant.e) Find out the impedance of the line.f) Find out the series resistance per unit of length.g) Find out the inductance per unit of length.
5: Consider a circular loop antenna of radius a located on the z = 0 plane that carriesan AC current given by the real part of I(t) = I0e
it.
a) Calculate the potential vector in the radiation zone.b) Find out the electric and magnetic fields in the radiation zone.c) In the limit λ� a show that the potential and fields in the radiation zone
become those of the magnetic dipole moment of the current distribution.
2
Comprehensive Examination EM (2014 summer) Department of Physics
Supplements of Electricity and Magnetism
Constant parameters
• Electric permitivity of free space ε0 = 8.854× 10−12 (mks) or 1/4π (cgs)
• Magnetic permeability of free space µ0 = 4π × 10−7 (mks) or 4π/c2 (cgs)
• Electron charge e = 1.6× 10−19 [C] or 4.8× 10−10 [esu]
• Electron mass m = 0.91× 10−30 [kg] or 0.91× 10−27 [g]
Maxwell equations
MKS cgs∇ ·D = ρ ∇ ·D = 4πρ (Coulomb′s law)
∇× E +∂B
∂t= 0 ∇× E +
1
c
∂B
∂t= 0 (Faraday′s law)
∇×H− ∂D
∂t= J ∇×H− 1
c
∂D
∂t=
4π
cJ (Ampere−Maxwell′s law)
∇ ·B = 0 ∇ ·B = 0 (Absence of free magnetic poles)
The time averaged poynting vector
S =1
2E× H
The net flow power of the electromagnetic fields
P =∫
S · da
Biot-Savart’s law
Magnetic field δB from a small current δI is
MKS cgs
δB =µ0
4π
δI× (x− x′)
|x− x′|3, δB =
1
c
δI× (x− x′)
|x− x′|3,
here x’ is the location of the current and x is the observation point.
The retarded vector potential
A(x, t) =µ0
4π
∫d3x′
∫dt′
J(x′, t′)
|x− x′|δ(t′ +
|x− x′|c
− t)
3
Comprehensive Examination EM (2014 summer) Department of Physics
A plane wave and the refractive indexA plane wave has the following relation between electric field and magnetic field,
H
E=
√ε
µ,
and the refractive index is given as n =√ε∗µ∗, here ε∗ (µ∗) is the ratio of the electric
permitivity (magnetic permeability).
Lorentz transformation
x′µ = Λµνxν
4-dimensional vectors and tensors
• Lorentz transformation matrix to the frame K ′ moving in X-direction withvelocity β and γ = 1/
√1− β2:
Λµν =
γ −iγβ 0 0iγβ γ 0 00 0 1 00 0 0 1
• Space and time: xµ = (ict, x, y, z)
• Charge and current: jµ = (icρ, jx, jy, jz)
• Potential: Aµ = (icφ, Ax, Ay, Az)
• Velocity wµ = (icγ, γvx, γvy, γvz)
• Lorentz transformation formula of E and B fields (transferred by Λµν).
Complete 4 out of 5 problems. Each problem has the same weight.
1.) Consider a particle bound in a double-well potential. We will approximate the system as a two-level problem, where we let |L> and |R> represent orthogonal state vectors for the particle being in the left and
right well, respectively. Suppose the Hamiltonian for the system is |||| LRRLH where
is a positive real constant.
a.) Find the energy eigenstates and eigenvalues.
b.) If the initial normalized state of the system is RLt ||)0(| , where and are
complex numbers, what is the probability for observing the particle on the right side of the potential at time t?
c.) Suppose instead the Hamiltonian is || LRH . Is this a valid Hamiltonian? Why or why
not?
2.) Consider three identical spin-1/2 particles which are bound in a central potential and interact with each other weakly. Assume the spatial component of the state vector is completely antisymmetric under the exchange of any pair of two particles. The spin component of the state can be expressed in the basis
of eigenkets 321 ,,| mmm of S1z, S2z, and S3z, where m1,m2, and m3 can be -1/2 or 1/2. Here e.g.
.,,|,,| 32113211 mmmmmmmS z
a.) Is it possible to construct a normalized spin state of the system for two of the particles having m = 1/2 and one having m = -1/2? If so, construct one, or else explain why this is not possible.
b.) Construct all possible normalized spin states of the system for all three of the particles having the same m values.
c.) For the states given in (a) and (b), what results are possible for a measurement of the z-component of
the total spin 321 SSSS
?
d.) Write all possible spin states for the system assuming instead that the spatial component of the state vector is symmetric under the exchange of any pair of two particles.
3). Consider a system of three spin-1/2 particles. Assume the initial state of the system is given by
||4|18
1| i . Let the total spin angular momentum be 321 SSSS
.
a.) If zS3 is measured, what results are possible and with what probability?
b.) If instead zS is measured, what results are possible and with what probability?
c.) Suppose instead yS3 is measured, what results are possible and with what probability?
4.) Consider a four-state system with state kets 4|,3|,2|,1| and unperturbed Hamiltonian
E
EH
000
000
0000
0000
0 .
Consider adding a time-independent perturbation to this system of the form
0000
00
00
000
W . You may assume |||||,| E .
a.) What is the degeneracy of each unperturbed energy level?
b.) Use perturbation theory to find the energy shifts due to this perturbation up to the 2nd order.
c.) Find the “correct” zero-order kets to which the perturbed kets reduce to in the limit that .0,
5.) Consider a one-dimensional simple harmonic oscillator with potential .ˆ2
1)( 22 xmxV
a.) Let |n> denote the energy eigenstates. What are the energy eigenvalues?
b.) Consider an initial state at t=0 given by 0|]ˆexp[]2/||exp[| 2 a where is a complex
number. What is the state of the system at later times t?
c.) Now add a time dependent perturbation
0,0
0,ˆ)(
t
txAtW , where A is a small positive constant,
so that matrix elements of W can be treated as small when compared with matrix elements of V. Assume the initial state of the system for t < 0 is the ground state of the harmonic oscillator.
Using 1st order time-dependent perturbation theory, what is the probability for the oscillator to remain in the ground state as a function of time t >0 ?
Quantum Theory Supplement - August 2014
Schrodinger’s Equation:
ψψ Ht
i =∂∂
Hamiltonian:
Vm
H +∇−= 22
2
Raising and lowering operators:
Angular Momentum:
>±±−+=>± 1,|)1()1(,| mjmmjjmjJ
>++=>
>−=>
−=
+=
+
+
1|1|ˆ
1||ˆ
ˆˆ2
ˆ
ˆˆ2
ˆ
nnna
nnna
pm
ixma
pm
ixma
ωω
ωω
1
Comprehensive Examination Physics Department University of Nevada, Reno Statistical Mechanics August 2014 Complete any 4 out of 5 problems. Each problem has the same weight. 1. Particles in a magnetic field.
When a particle with spin 2
1 is placed in a magnetic field H, its energy level is split into
H and H . Suppose a system consisting of N such particles is in a magnetic field H and is kept at temperature T.
a) Find the partition function and the Helmholtz free energy F with the help of the canonical distribution.
b) Find the entropy S and internal energy U using the results from (a). What is the entropy in two limiting cases of T0 and T? Explain your answer.
c) Find the total magnetic momentum M of this system with the help of the canonical distribution. What is a relation between U and M? Explain your answer.
Hint: .H
FM
d) Find the heat capacity CH and sketch it as a function of (kBT/H). Comment on the graph.
Hint: .)( HH T
UC
2. Formal thermodynamic manipulations.
From the fundamental thermodynamic relation show that
)1()()( ,2
2
, NPNTP
T
VT
P
C
where ,,, PNV and PC are volume, number of particles, pressure, and heat capacity at the constant pressure.
Hint: The fundamental thermodynamic relation is dNPdVTdSdU , so the
independent variables are SNV ,, .To prove the desired expression you need to define the thermodynamic potential that has independent variables of interest in (1) and to use a Maxwell relation approach
2
3. Ideal gas. An ideal gas consisting of N particles of mass m (classical statistics being obeyed) is enclosed in an infinitely tall cylindrical container of a cross section placed in a uniform gravitational field, and is in the thermal equilibrium.
a) Find a classical partition function of this system. b) Calculate the Helmholtz free energy and mean energy of the system and compare it with
ideal gas results. c) Calculate the heat capacity at constant volume of this system and compare it with the
result for an ideal gas. Hint: the translational Hamiltonian per particle is given by
.2
222
mgzm
pppH zyx
trans
4. Chain. There is a one-dimensional chain consisting of N elements (N>>1), as is seen in the figure. Let the length of each element be a and the distance between the end points x. a) Find the entropy of this chain as a function of x. b) Obtain the relation between the temperature T of the chain and the force (tension) X
which is necessary to maintain the distance x, assuming the joints to turn freely. c) For x << Na, what would be the tension X?
Hint: in order to specify a possible configuration of the chain, you may consider indicating successively, starting from the left end, whether each consecutive element is directed to the right (+) or to the left (-). For example, in the case shown in the figure, we have (+ + - + + + - - - + + - + + +).
3
5. Quantum gas. a) Prove that in the non-relativistic case of 3-D Fermi gas, the Fermi energy is
)2(2
)6
(2
3/22
mg
nF
where n is a particle density and g is a weight factor arising from the „internal structure“ (for example, g=2 for electrons). Hint: you may consider using the following expression of the total number of particles N
through the density of states F
Ndaa
0
.)(:)(
b) How will the expression for the Fermi energy shown in (2) change for the case of 2-D Fermi gas?
c) Obtain the numerical estimates of the Fermi energy (in eV) and the Fermi temperature (in
K) for the electron gas in the interior of white dwarf stars with n=1030 cm-3. Are the electron energies in the relativistic regime?
d) Obtain the numerical estimates of the Fermi energy (in eV) and the Fermi temperature (in K) for the conduction electrons in silver with the concentration of atoms 5.76x1022 cm-3. Compare to the results for the electron gas in the interior of white dwarf stars (c).
4
Statistical Mechanics Supplement – August 2014
Thermodynamic potentials
-TS
+PV
Stirling’s approximation for large N .)ln()!ln( NNNN
Multiplicity, entropy, partition functions, and chemical potential
