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FLORIDA INTERNATIONAL UNIVERSITYDEPARTMENT OF PHYSICS
UNIVERSITY PARK, MIAMI, FL33199
Ph.D. Qualifying Exam, Fall 2011 – MODERN PHYSICS
THURSDAY, AUGUST 18, 2011 1PM - 5PM
INSTRUCTIONS:
1. There are nine problems on this exam with
• five in SPECIAL RELATIVITY/MODERN PHYSICS;
• four in QUANTUM MECHANICS.
2. You must solve a total of SIX problems with
• at least two from SPECIAL RELATIVITY/MODERN PHYSICS;
• at least two from QUANTUM MECHANICS.
3. You MUST solve the problems marked REQUIRED.
4. Do each problem on separate sheets of paper and turn in only those problemsyou want to have graded.
5. Write your name and problem number on each page.
1
RELATIVITY / MODERN PHYSICS
1. Xenofo and Ydex are fighting a duel with identical paintball guns. Xenofo is at theorigin of an Earth reference frame and Ydex is at x = L. An earth observer notesthat they fire their guns simultaneously. An observer on a space-craft moving athigh speed in the positive x direction notes that Ydex fired before Xenofo by anamount T . How fast is the space-craft moving relative to the Earth?
2. In the lab reference frame, an electron and a positron emerge from a nuclearreaction with equal speeds of βc. The electron is observed to be moving at anangle θ above the positive x axis, and the positron is moving at an angle θ belowthe positive x axis.
(a) What is the velocity of a moving observer (S’) who sees the two particlesmove outward along the positive and negative y′ axis respectively.
(b) What is the speed of each particle in S’?
3. A ball of mass m and speed βc collides completely inelastically with a ball ofmass 2m moving in the opposite direction with speed v.
(a) If the two balls are at rest after collision, determine v.
(b) Thermal energy, Q, is generated by the collision, and is dissipated to thesurroundings as the balls cool down to the temperature they had before thecollision. Determine Q.
4. REQUIRED In a Stern-Gerlach experiment, silver atoms of mass m, whose spinis due to a single electron, are obtained from an oven at temperature T . Youranswers to all parts should be in terms of given symbols and any other standardphysical constants.
(a) What is their root mean square (RMS) speed?
(b) The atoms pass perpendicularly through a magnetic field of length L thathas a gradient, dB
dz , perpendicular to the beam. For atoms in the two beamshaving the same RMS speed, what force do they experience?
(c) How far apart will they be as they emerge from the field?
2
5. A radioactive nucleus decays via the α-emission reaction
A→ B + α (1)
with a half-life of t1/2.
(a) Calculate the kinetic energy of the α-particle released in the rest frame ofA.
(b) At time t = 0 a pure sample of A is prepared. At time t = T , the activityis measured to be R. What is the original mass of the sample?
3
QUANTUM MECHANICS
6. Consider two particles of mass m interacting with a potential in one dimensionof the form
V (x1, x2) =1
2k1x
21 +
1
2k2x
22 + αx1x2, (2)
where x1 and x2 are the locations of the two particles. Assume that k1, k2, α > 0and k1k2 > α2.
Write down the time independent Schrodinger equation and find all the boundstate energy eigenvalues.
7. Let H be a quantum mechanical Hamiltonian with eigenfunctions |φi〉 and cor-responding eigenvalues Ei with i = 0, 1, 2, · · ·∞. Further assume that
E0 < E1 < E2 < · · · < E∞. (3)
(a) Let |ψ〉 be a normalized wavefunction (not necessarily an eigenfunction ofH). Prove that
〈ψ|H|ψ〉 ≥ E0. (4)
Hint: Expand |ψ〉 in the eigenbasis of H.
(b) Prove that every attractive potential in one dimension, i.e., V (x) < 0 forall x, has at least one bound state. Hint: Consider the normalized wavefunction,
ψα(x) =(
α
π
) 14
e−12αx2
; α > 0 (5)
and calculate,
E(α) = 〈ψα|H|ψα〉; H = − h2
2m
d2
dx2+ V (x). (6)
Show that E(α) can be made negative by a suitable choice of α.
8. Consider a one-dimensional harmonic oscillator in the ground state |0〉 at t =−∞. Let a perturbation
H1(t) = −Xe−t2
τ2 (7)
with X being the position operator be applied between t = −∞ and t = +∞.What is the probability that the oscillator is in the state |n〉 at t = ∞?
4
9. REQUIRED Consider the one dimensional harmonic oscillator.
(a) Prove that[a, a†] = 1. (8)
(b) Prove that
H = hω(a†a +
1
2
)(9)
(c) Prove that
H|n〉 = hω(n +
1
2
)|n〉. (10)
(d) Prove that〈n|T |n〉 = 〈n|V |n〉 (11)
for any eigenstate |n〉.
5
Some useful information
• Some useful information about quantum harmonic oscillators in one dimension:Using standard notation, let ω be the classical frequency of the oscillator andlet m be the mass of the particle. Let |n〉, n = 0, 1, · · · ,∞ label all the boundeigenstates. The coordinate and momentum operators are given by
X =
√h
2mω(a + a†); P = i
√mωh
2(a† − a); (12)
with a† and a being the step up and step down operators obeying
a|n〉 =√
n|n− 1〉; a†|n〉 =√
n + 1|n + 1〉. (13)
• √α
π
∫ ∞
−∞e−αx2
dx = 1. (14)
for all α > 0.
6
FLORIDA INTERNATIONAL UNIVERSITYDEPARTMENT OF PHYSICS
UNIVERSITY PARK, MIAMI, FL33199
Ph.D. Qualifying Exam, Fall 2012 – MODERN PHYSICS
THURSDAY, AUGUST 16, 2012 1PM - 5PM
INSTRUCTIONS:
1. There are nine problems on this exam with
• five in SPECIAL RELATIVITY/MODERN PHYSICS;
• four in QUANTUM MECHANICS.
2. You must solve a total of SIX problems with
• at least two from SPECIAL RELATIVITY/MODERN PHYSICS;
• at least two from QUANTUM MECHANICS.
3. You MUST solve the problems marked REQUIRED.
4. Do each problem on separate sheets of paper and turn in only those problemsyou want to have graded.
5. Write your name and problem number on each page.
1
RELATIVITY / MODERN PHYSICS
1. REQUIRED In the lab system S, a mass m1 = 9 is moving towads a targetmass m2 = 16, which is at rest. The energy of m1 is E1 = 27.75. Assumetime-space dimension is 1 + 1.
(a) Calculate the space momentum p1 of m1. What is its velocity v1?
(b) Find the Lorentz transformation Λ : S → S ′,
Λ =(
cosh ξ − sinh ξ− sinh ξ cosh ξ
), (1)
to the center-of-mass system S ′, i.e. find the numbers for cosh ξ and sinh ξ.What is the boost velocity v?
(c) What are the energies E ′1 and E ′
2 in the center-of-mass system?
(d) Supposes m1 and m2 collide and stick together, forming one combined massM . What is M? Compare M to m1 + m2.
Hints: We are using units such that c = 1 and all masses/energies are given insome unit mu. (For example, choosing mu = 931.5MeV, the target mass couldbe Oxygen 16
8O and the projectile would be Beryllium 94Be).
2. The proper length for one spaceship is twice that of the other. To an observeron earth, both ships have the same length. If the faster spaceship is moving withβ = v
c = 0.95
(a) Find the speed of the slower ship.
(b) If both ships are moving in the same direction, find the speed of the slowership in the rest frame of the faster ship.
Hint: Make sure that the sign of the velocity you find makes sense.
3. An observer in a rocket moves toward a mirror at speed v relative to the mirror.A light emitted by the rocket moves toward the mirror and is reflected back tothe rocket. If the rocket is a distance d away from the mirror when it emits thelight pulse as measured in the mirror’s reference frame, what is the total traveltime of the pulse (from the rocket and back) as measured in
(a) the rest frace of the mirror and
2
(b) the rest frame of the rocket?
4. The activity (decays per second), R, of a radioactive sample decreases to R32 in
time T .
(a) What is the half-life and decay constant?
(b) What fraction of the original number of nuclei will have decayed after T100
and T10 .
5. The radial part of the wavefunction for the Hydrogen atom in the 2p state isgiven by
R2p(r) = Are−r
2a0 , (2)
where A is a constant and a0 is the Bohr radius. Use this to calculate the averagevalues of r, r2 and the standard error in r.
QUANTUM MECHANICS
6. REQUIRED Eight electrons are placed in a three-dimensional infinite cubicwell with sides of length L.
(a) Determine the ground state energy.
(b) Determine the energy of the first excited state.
7. This problem involves the case of a harmonic oscillator in the ground state (n =1). In some cases you may use physics arguments instead of direct calculation tofind the answers.
(a) Find 〈x〉 and 〈x2〉.(b) Find 〈p〉 and 〈p2〉.(c) Use your results from the previous two parts to determine ∆x∆p.
(d) What is special about this case and why?
3
8. Consider a quantum mechanical particle of mass m constrained to move in aone dimension. Let x label the space variable and let x ∈ [0, L]. Also assumeperiodic boundary conditions: x = 0 is identified with x = L. As such, you canalso visualize space as a circle with perimeter L. Let this particle interact witha potential V (x) which satifies the property
V (x + a) = V (x) (3)
for all x. In addition a satisfies the relation Na = L with N being an inte-ger. Assuming no degeneracies in the spectrum of bound states, prove that alleigenfunctions satisfy the condition
ψ(x + a) = Cψ(x) (4)
with C being one of the N ’th root of unity, i.e; CN = 1.
9. Consider Hermitian matrices γi, i = 1, 2, 3, 4 that obey
γiγj + γjγi = 2δijI; i, j, = 1, 2, 3, 4. (5)
The Kronecker delta is defined as
δij ={
1 if i = j0 if i "= j
. (6)
(a) Show that the eigenvalues of γi are ±1 for all i. Hint: Go to the eigenbasisof γi, and use the equation for i = j.
(b) By considering the relation (5) for i "= j and noting that trace is cyclic,namely; Tr(ACB) = Tr(CBA), show that γi is traceless for all i.
(c) Show that γi cannot be odd dimensional matrices.
4
Some useful information
• The ground state eigenfunction of a harmonic oscillator is
ψ(x) =(
mω
πh
) 14
e−mωx2
2h . (7)
• √α
π
∫ ∞
−∞e−αx2
dx = 1. (8)
for all α > 0.
5
FLORIDA INTERNATIONAL UNIVERSITY
DEPARTMENT OF PHYSICS
UNIVERSITY PARK, MIAMI, FL 33199
INSTRUCTIONS
There are nine problems on this exam with
• four in SPECIAL RELATIVITY/MODERN PHYSICS;
• five in QUANTUM MECHANICS.
You must solve a total of SIX problems with
• at least two from SPECIAL RELATIVITY/MODERN PHYSICS;
• at least two from QUANTUM MECHANICS.
You MUST solve the problems marked as REQUIRED.
Do each problem on separate sheets of paper and turn in only those problems you want
to have graded.
Write your assigned number (DO NOT WRITE YOUR NAME) and problem number on
each page.
Ph.D Qualifying Exam Modern Physics Thursday, August 22, 2013
1. (Special Relativity/Modern Physics)
Derive the relativistic explanation of superluminal motion.
Superluminal motion is the apparent faster-than-light motion seen in some radiogalaxies
and other sources. Consider an active galactic nucleus (A) that emits a relativistic jet
from its center at time t1. This jet has a speed v in a direction that makes an angle ! with
respect to the line joining the observer on earth (O) and the origin of the relativistic jet at
time t1 (A). Light emitted from this jet is seen by this observer as a streak in the sky.
(a) Obtain an expression for the apparent transverse speed of this streak as a
function of v, ! and the speed of light c in the limit where the distance OA is very
very large.
(b) Find the maximum value of the apparent transverse speed for a fixed v.
(c) Show that this maximum value can be larger than c even though v has to be less
than or equal to c.
2. (Special Relativity/Modern Physics)
An Imperial Star Destroyer passes a rebel observation post at a relativistic speed. The
observation post"s sensors measure the length of the Star Destroyer to be 800m. (You
can assume that this is an instantaneous measurement of the length).
(a) Assuming that the length of the Star Destroyer is 1,600 m at rest, how fast is the
ship traveling relative to the observation post? Leave your answer in terms of c,
the speed of light.
(b) In response to the approaching Star Destroyer, the observation post sends an
alert signal to the rebels on a nearby planet. If the signal travels at the speed of
light, how fast does this signal travel according to the observer on the Star
Destroyer? Explain your answer.
(c) As the Star Destroyer passes and moves away from the post, the post"s sensors
monitor Darth Vader in his private gymnasium on the Star Destroyer practicing
his light saber skills. If the sensors measure infrared photons from the light saber
with # = 2500 nm, determine the wavelength of light that Darth Vader observes.
Ph.D Qualifying Exam Modern Physics Thursday, August 22, 2013
3. (Special Relativity/Modern Physics) REQUIRED
When a muon decays at rest,
why can the electron only have a maximum energy of 53 MeV? The mass of the muon
is 106 MeV/c2 and you can assume that the neutrinos are massless. You can also
ignore the mass of the electron.
4. (Special Relativity/Modern Physics)
A large number of hydrogen atoms are in the n = 3 state.
(a) Considering all possible transitions that the atoms can make before reaching
their ground state, what wavelengths would you observe in the emission
spectrum?
(b) If one such atom, initially at rest, undergoes a transition directly from the n = 3
state to the ground state,
i. what is the momentum of the emitted photon, and
ii. with what kinetic energy will the atom recoil?
Ph.D Qualifying Exam Modern Physics Thursday, August 22, 2013
5. (Quantum Mechanics)
Consider three spin 1/2 particles, each bound in the 1-dimensional harmonic oscillator
potential. The single particle state vectors are given by |n,m>1, |n,m>2 and |n,m>3, where
n is the energy state number (n=0,1,2,3,...) and m is the z-component of the spin (m =
±1/2).
(a) Assume that the particles have equal mass, but are distinguishable.
i. Determine the ground state energy of this system of distinguishable particles.
ii. Determine the ground state vector(s) as a product of the single particle state
vectors. What is the degeneracy of the ground state?
(b) Now assume that the particles are identical fermions.
i. Determine the ground state energy of this system of identical particles.
ii. Determine the ground state vector(s) as a product of the single particle state
vectors. What is the degeneracy of the ground state?
6. (Quantum Mechanics)
A hydrogen atom is in the ground state. A pulsed electric field E(t) = E0!(t) is applied to
the atom. Calculate the transition probability to the excited state n with the energy En
and the eigenfunction !nlm (n,l, and m, are quantum numbers) and calculate the
probability of the atom remaining in the ground state.
7. (Quantum Mechanics) REQUIRED
A particle with mass m is in the n!th bound state (with the energy En and the
wavefunction "n(x)) in a 1-D square potential well ( the potential depth is V0 and the
width is w). Find
(a) the probability of the particle outside the potential well and
(b) the expectation values of V(x) and V2(x) under the condition that V0 >> En.
8. (Quantum Mechanics)
An operator A, representing observable A, has two normalized eigenstates #1 and #2
with eigenvalues a1 and a2, respectively. Operator B, representing observable B, has
Ph.D Qualifying Exam Modern Physics Thursday, August 22, 2013
two normalized eigenstates !1 and !2 with eigenvalues b1 and b2, respectively. The
eigenstates are related by "1 = (#3 !1 - !2)/2 and "2 = (!1 + #3 !2)/2.
(a) Observable A is measured, and the value a2 is obtained. What is the state of the
system (immediately) after this measurement?
(b) If B is now measured, what are the possible results and what are their
probabilities?
(c) Right after the measurement of B, A is measured again. What is the probability of
getting a2? [ Note that the answer would be quite different if I had told you the
outcome of the B measurement.]
9. (Quantum Mechanics)
Find the differential cross section for elastic scattering of a particle initially traveling
along the z-axis from a non-spherical, double delta potential
where is a unit vector along the z-axis.
(a) Since the potential is not spherically symmetric, obtain the scattering amplitude
from the general equation
where is the momentum transfer.
(b) Since the incident particle is initially traveling along the z-axis, and it scatters
elastically from the potential, the magnitudes of the momenta are the same
before and after the collision. Let $ be the angle of the outgoing momentum with
respect to the z-axis. Find the differential cross section d%/d& = | f($) |2 in terms
of constants and the magnitude of the momentum, k.
Ph.D Qualifying Exam Modern Physics Thursday, August 22, 2013
Modern Physics Ph.D. Qualifying Exam Fall 2014 Florida International University Department of Physics Instructions: There are nine problems on this exam. Five on quantum mechanics (Section A), and four on general modern physics (Section B). You must solve a total of six problems with at least two from each section. You must also do the problems marked Required. Do each problem on its own sheet (or sheets) of paper. Turn in only those problems you want graded. Write your student ID number on each page but not your name. You may use a calculator and the math handbook as needed. Section A 1. A particle is moving inside a 1-‐D potential 𝑉 𝑥 = ∞ for x<0; 𝑉 𝑥 = !!!!!
! for
x>0. Find the particle energy. 2. A particle of mass m moves between two infinite potential walls separated by a distance d.
a) Find the ground state energy. b) One of the potential walls is suddenly and instantaneously moved to a
distance 2d. What is the probability of finding the particle in the new ground state?
3. An electron with magnetic moment µ0 and spin up at t=0 enters a region of a static magnetic field 𝐻 = −𝐻!𝑧. There is also a AC magnetic field given by 𝐻! = 𝐻! cos 𝜔𝑡 𝑥 + 𝐻! cos 𝜔𝑡 𝑦. What is the probability of finding the electron in the spin up state at time t? 4. A particle with positive energy (E > 0) encounters a Dirac delta-‐function potential barrier
𝑉 𝑥 = 𝛼𝛿 𝑥 where α is positive real constant with the units of energy×length.
a) Solve the Schrödinger equation for the regions x < 0 and x > 0. If you make any assumptions, you must verify your answer to receive full credit.
b) State and apply the appropriate boundary conditions for the potential interface at x = 0. Obtain expressions relating the different coefficients.
c) Using the results from part b), determine the transmission coefficient T for a particle incident and transmitted through this potential barrier. Discuss the limiting cases for T as the energy E approaches zero (𝐸 → 0) and E approaches infinity (𝐸 → ∞).
5. Required Consider an unperturbed harmonic oscillator with Hamiltonian 𝐻! =
!!
!!+ !
!𝑚𝜔!𝑥! ,
with eigenstates of the form |𝑛 , and energy eigenvalues 𝐸!! = 𝑛 + !
!ℏ𝜔.
A perturbation of the form 𝐻! = !
!𝛽𝑚𝜔!𝑥!
is added to the system, with β being a small parameter. a) Using the definition for the harmonic oscillator raising and lowering
operators, show that the expectation value of x2 can be written as
𝑛 𝑥! 𝑛′ =ℏ
2𝑚𝜔𝑛! + 1 𝑛! + 2 𝛿!,!!!! + 𝑛′ 𝑛! − 1 𝛿!,!!!! + 2𝑛! + 1 𝛿!,!!
𝑛!
b) Using non-‐degenerate perturbation theory, obtain the first-‐order correction
to the energy. c) Using non-‐degenerate perturbation theory, obtain the second-‐order
correction to the energy. You may only consider the n = 0 and n = 1 states when doing the expansion.
Hint: Recall the following relationships 𝑎± =
!!ℏ!"
∓𝑖𝑝 +𝑚𝜔𝑥 𝑎!|𝑛 = 𝑛 + 1|𝑛 + 1
𝑥 = ℏ!!"
𝑎! + 𝑎! 𝑎!|𝑛 = 𝑛|𝑛 − 1
Section B 6. Light of wavelength 300 nm strikes a metal plate, producing photoelectrons that move with a speed of 0.002c.
a) Determine the work function of the metal. b) What is the critical wavelength for this metal, so that photoelectrons are
produced? c) What is the physical significance of the critical wavelength?
7. Provide a brief qualitative description for each physical phenomenon:
a) Compton Effect. b) Difference between bosons and fermions. c) Michelson Morley Experiment. d) Rutherford scattering.
8. Required A pion, 𝜋!, has a rest mass of 139.57 MeV/c and is moving with a speed 𝛽 = 0.5𝑐. It then decays into a muon/neutrino pair, 𝜇!𝜈! , with the muon traveling off at an angle of 100 relative to the original direction of the pion. What is the energy and momentum of the muon and what is the angle, energy, and momentum of the neutrino? The muon has a rest mass of 105.65 MeV/c and you may assume the neutrino is massless 9. Police radar is used to measure the speed of oncoming cars. It works by broadcasting microwaves of a precisely known frequency, f. The moving car reflects these waves that are then picked up by the radar gun. The gun computes the speed of the car by measuring the beat frequency (which is small compared to the broadcast frequency) between the broadcast frequency and the detected frequency.
a) Find the frequency of the reflected wave in terms of f, v, and c. b) Show that the speed of the car is given by 𝑣 =≅ !
!!!"#$!𝑐 for a car moving
much less than the speed of light.
Modern Physics Ph.D. Qualifying Exam Fall 2015 Florida International University Department of Physics Instructions: There are nine problems on this exam. Six on quantum mechanics (Section A), and three on general modern physics (Section B). You must solve a total of six problems with at least two from each section. You must also do the problems marked Required. Do each problem on its own sheet (or sheets) of paper. Turn in only those problems you want graded. Write your student ID number on each page but not your name. You may use a calculator and the math handbook as needed. Section A 1. An electron (spin ½) at rest is in a uniform magnetic field 𝐵 = 𝐵!𝚥 + 𝐵!𝑘
(By>>Bz). The Hamiltonian is zz
yy S
meBS
meB
HHH +=+= '0 (m is the electron
mass and e is the electron charge). Find the first-‐order perturbation corrections of the electron spin wave functions.
2. (Required) A particle of mass m is confined to move freely in a ring of radius R. (a) Find the energies and the eigenfunctions of the particle. (b) A perturbation term
𝐻! =𝑉!,−𝛼 < 𝜑 < 0𝑉!, 0 < 𝜑 < 𝛼 0, elsewhere
is added to the particle, where φ is the azimuthal angle and α is an arbitrary fixed value. Find the first order corrections of the energies for the three lowest energy states. 3. An electron in a hydrogen atom jumps from n = 4 to n = 1 state. (a) Is energy absorbed or emitted in this process? If energy is emitted, which spectral line series is this transition? Explain. (b) Determine the energy transfer in eV and photon wavelength in nm for this process. (c) Explain why classical physics fails to correctly describe the emission spectrum for the hydrogen atom. (d) Discuss Bohr’s model of the hydrogen atom. In your discussion, be sure to include his three revolutionary postulates that led to this model.
4. A 1-‐D particle in an infinite square well (0 ≤ x ≤ a)has the initial wavefunctionΨ 𝑥, 0 = 𝐴𝑥(𝑎 − 𝑥) inside the well and Ψ 𝑥, 0 = 0 outside the well. (a) Find the normalization constant A. (b) Find the expectation value for the position x. (c) Find the expectation value for x2. (d) Use the results of (a) and (b), find the standard deviation σx.
(e) If σ p = 10a , is the Heisenberg Uncertainty Principle satisfied?
5. An operator 𝐴, representing observable A, has two normalized eigenstates Ψ!and Ψ! with eigenvalues a1 and a2, respectively. Operator 𝐵, representing observable B, has two normalized eigenstates Φ!and Φ! with eigenvalues b1 and b2 , respectively.
The eigenstates are related by Ψ1 =3Φ1 −Φ2
2 , and Ψ2 =
Φ1 + 3Φ2
2.
(a) Observable A is measured, and the value of a2 is obtained. What is the state of the system immediately after this measurement? (b) If B is now measured, what are the possible results and what are their probabilities? (c) Right after the measurement of B, A is measured again. What is the probability of getting a2 ? 6. There are three particles and four distinct single-‐particle states Φ1 ,Φ2 ,Φ3 , and Φ4 . How many different three-‐particle states can be formed, (a) if the three particles are distinguishable, (b) if the three particles are identical fermions, and (d) the three particles are identical bosons? Section B 7. In a nuclear physics experiment, a photon beam with the energy of Eγ is incident on a fixed proton target (liquid hydrogen). Suppose you are interested in production of the strange particle Λ, via the reaction of γ+p K+ +Λ, what is the minimum Eγ you need to produce the Λ? The masses of the proton, K+, and Λ are 0.938, 0.494 and 1.116 GeV, respectively. 8. (Required) Consider the process of Compton scattering. A photon of wavelength λ is scattered off a free electron initially at rest. Let λ’ be the wavelength of the photon scattered in a direction θ relative to the photon incident direction. (a) Find λ’ in terms of λ and θ and universal constants. (b) Find the kinetic energy of the recoiled electron. 9. (a) A laser emits a pulse of light which travels at a speed of c (in vacuum) relative to the laser. Does this mean that the speed of the laser relative to the light pulse is also c? Give your reasoning for your answer.