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UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT

PART I Qualifying Examination

January 22, 2013, 5:00 p.m. to 8:00 p.m.

Instructions: The only material you are allowed in the examination room is a writing instrument and a calculator. You may not store any formulae in your calculator. Paper and question sheets are provided. Each student is assigned a capital English letter; this letter will identify your work on both parts (I and II) of this exam.

In writing out your answers, use only one side of a page, use as many pages as necessary

for each problem, and do not combine work for two different problems on the same page.

Each page should be identified in the upper, right-hand corner according to the following scheme: A 4.3 i.e., student A, problem 4, page 3. Refer all questions to the exam proctor.

In answering the examination questions, the following suggestions should be heeded:

1. Answer the exact question that is asked, not a similar question.

2. Use simple tests of correctness (such as a reasonable value, correct limiting values and dimen-sional analysis) in carrying out any derivation or calculation.

3. If there is any possibility of the grader being confused as to what your mathematical symbols mean, define them.

You may leave when finished.

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1. A spring of negligible mass and force constant k = 400 N/m is hung vertically, and a m = 0.2 kg pan is suspended from its lower end. A butcher drops a M = 2.2 kg steak onto the pan from a height of

h = 0.40 m . The steak makes a totally inelastic collision with the pan and sets the system into vertical simple harmonic motion.

(a) What is the speed of the pan and steak immediately after the collision? (b) What is the period of the subsequent motion? (c) What is the amplitude of that motion? (d) What is the maximum extension of the spring?

How long it takes, from the moment of collision, to reach the maximum extension of the spring?

2. A particle of mass m slides without friction on the surface of a half sphere of radius R in the presence of a uniform gravitational field directed along the negative z axis.

(a) Write the Lagrangian in terms of the generalized coordinates θ and φ . (b) Find the constant of the motion that follows from rotational invariance about the z axis. (c) Find the constant of the motion that follows from translational invari-ance in time.

3. A particle of mass m in one-dimension moves under the action of an attractive potential V (x) that tends to zero at infinity. It is known that the ground state wave function of this particle is

ψ (x) = 1

2 cosh(x / ) ,

where is the characteristic length scale of this bound state and cosh(x) = (ex + e− x ) / 2 is the hyper-bolic cosine function.

(a) What is the ground state energy E0 ?

Hint: at large x ( x ) the potential vanishes and the wave function is proportional to e− x/ , i.e.,

H0 e− x/ = E0 e− x/ , where H0 is the kinetic energy operator.

(b) Find the potential V (x) .

θ

φ

x

y

z

m

R

3

4. (a) Consider a spin ½ particle whose wave function is given by |Ψ〉 = 15

21

⎛⎝⎜

⎞⎠⎟

, with the spin

quantization axis chosen along the z direction. What would be the expectation value of the spin measured along the z direction?

(b) Now a magnetic field is applied along the x axis. This can be described by the Hamiltonian H = −gµBB σ x , where B is the strength of the magnetic field, σ x is a Pauli matrix, and g and µB are constants, the gyromagnetic ratio and the Bohr magneton, respectively. What are the possible eigenenergies of the particle in the presence of the magnetic field?

(c) Suppose at time t =0, the particle was prepared in the spin state |Ψ(0)〉 = 10

⎛⎝⎜

⎞⎠⎟

and is allowed

to evolve under the applied magnetic field in part (b). How does the state |Ψ(t)〉 change with time? (d) From the answer you found in part (c), what is the expectation value of the spin measured along the z direction as a function of time?

Recall: The three Pauli matrices for the spin ½ particle are

σ x =0 11 0

⎡

⎣⎢

⎤

⎦⎥, σ y =

0 −ii 0

⎡

⎣⎢

⎤

⎦⎥, and σ z =

1 00 −1

⎡

⎣⎢

⎤

⎦⎥.

UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT

PART II Qualifying Examination

January 24, 2013, 5:00 p.m. to 8:00 p.m.

Instructions: The only material you are allowed in the examination room is a writing instrument and a calculator. You may not store any formulae in your calculator. Paper and question sheets are provided. Each student is assigned a capital English letter; this letter will identify your work on both parts (I and II) of this exam.

In writing out your answers, use only one side of a page; use as many pages as necessary

for each problem, and do not combine work for two different problems on the same page.

Each page should be identified in the upper, right-hand corner according to the following scheme: A 4.3 i.e., student A, problem 4, page 3. Refer all questions to the exam proctor.

In answering the examination questions, the following suggestions should be heeded:

1. Answer the exact question that is asked, not a similar question.

2. Use simple tests of correctness (such as a reasonable value, correct limiting values and dimen-sional analysis) in carrying out any derivation or calculation.

3. If there is any possibility of the grader being confused as to what your mathematical symbols mean, define them.

You may leave when finished.

1. Suppose that the circuit pictured below has been connected for a long time when suddenly, at time t = 0 , the switch S is thrown from A to B, bypassing the battery εo.

a) What is the current at any subsequent time t?

b) What is the total energy delivered to the resistor R?

c) Show that this energy is equal to the energy originally stored in the inductor L.

2. (a) Write down Maxwell's equations.

(b) Show that the electric field E and magnetic induction

B can be expressed in terms of a scalar po-

tential Φ and vector potential A as

B = ∇×

A , and

E = −∇Φ− ∂

A∂t

.

(c) Derive the continuity equation (i.e., the conservation of electric charge) ∂ρ∂t

+∇⋅J = 0 from

Maxwell's equations.

3. One mol of monoatomic ideal gas undergoes the cycle 1→ 2→ 3→ 4→1 shown in the figure.

(a) In terms of p0 , V0 and the ideal gas constant R , find the pressure, volume

and temperature in each of the four states 1,…,4 . (b) Find the total work W per cycle. (c) Find the total heat 1Q absorbed per cycle. (d) Determine the efficiency η of a thermal engine based on this cycle.

Compare η with the efficiency ηC of a Carnot engine working between T1 and

T3 .

4. Conduction electrons in doped graphene can be regarded as a quasi-two-dimensional degenerate Fermi gas. The (kinetic) energy ε of an electron in doped graphene is a linear function of its momentum

p =| p | , i.e., ε = ε( p) = v p , where v is a constant velocity. The density of states of the conduction electrons in graphene is linear in energy, i.e., a(ε ) ≡ dN (ε ) / dε = Aε , where A is a constant. Let there be electrons in the graphene sheet. We are interested in the “low temperature” properties of the system, i.e., when T ε F / kB , where ε F is the Fermi energy (i.e., the chemical potential µ(0) at

T = 0 ) and kB is the Boltzmann constant.

N

2

3

V

p

3p0

p0

V0 2V0

1 4

(a) Find the Fermi energy ε F ?

Hint: ε F = µ(0) can be determined from the condition N = a

0

εF∫ (ε )dε .

(b) Show that for T ε F / kB the chemical potential has the form: µ(T ) ≈ ε F −α ⋅(kBT / ε F )2 , and determine the expression of the coefficient α .

Hint: Start from the condition N = a(ε )dε

exp[β(ε − µ)]+10

∞

∫ , where β = 1/ kBT , and apply the Sommer-

feld expansion:

f (ε )dεexp[β(ε − µ)]+10

∞

∫ ≈ f0

µ

∫ (ε )dε + π 2

6(kBT )2 f ′(µ) ≈ f

0

εF∫ (ε )dε + f (ε F ) ⋅(µ − ε F )+ π 2

6(kBT )2 f ′(µ)

where f (ε ) is an arbitrary function of ε ≥ 0 , and f ′(µ) = df (ε )

dε ε=µ

(c) Show that the total energy of the system U (T ) ≈U (0)+ λ ⋅(kBT / ε F )2 , and determine the expres-sions of U (0) and λ . (d) Find the temperature dependence of the specific heat C(T ) = ∂U (T ) / ∂T .