Physics 2D Quiz Final Exam
Department of Physics, UCSD
Summer Session II - 2009
Prof. Pathria
5 September 2009
Instructions
1. There are EIGHT questions on the exam — you may attempt ANY SIX.
2. All questions are of EQUAL value.
3. Please write your answers in the blue book and make sure that your secret
code number is written on all pages with indelible ink.
1
Some Useful Numbers, Equations, and Identities
Speed of light: c = 2.998×108 m/s
Planck’s constant: h = 6.626×10−34 J·s
h̄ =
h
2π ;
1 eV = 1.602 × 10−19 J
Coulomb’s constant: k = 8.99×109 Nm2 /C2
Electron Charge: e = 1.602×10−19 C
Electron Mass: me = 9.11×10−31 kg = 0.511MeV/c2
Rydberg Constant: R = 1.097×107 m−1
Atomic Mass Unit: u = 1.6606×10−27 kg = 931.5 MeV/c2
Proton Mass: mp = 1.673×10−27 kg = 938.3 MeV/c2 = 1.0073 u
Neutron Mass: mn = 1.675×10−27 kg = 939.6 MeV/c2 = 1.0087 u
Compton wavelength for an electron:
h
me c
Compton scattering formula: λ0 − λ0 =
= 0.00243 nm
h
me c
(1 − cosθ)
Photo-electric equation: eVs = hf − φ = h (f − f0 )
Momentum for a relativistic particle: p = γm0 u,
γ=
q 1
2
1− vc2
Energy for a particle: E = K + mc2 = γmc2
Energy-momentum relation (particle): p =
1
c
p
E 2 − m20 c4 =
1
c
√
2m0 c2 K + K 2
Energy-momentum relation (photon): E = pc
Relative velocity: u0 =
u−v
1− uv
c2
Doppler Effect (light source approaching observer): fobs = f0
De Broglie wavelength: λ = h/p
2
q
1+ vc
1− vc
2
2
h̄ d ψ
Schrödinger Equation: − 2m
dx2 + U (x)ψ = Eψ
1-Dimensional Normalization Condition:
R∞
−∞
ψ ∗ ψdx =
R∞
−∞
|ψ|2 dx = 1
Harmonic Oscillator Potential: U = 12 mω 2 x2
For a Hydrogen-like atom:
2
2
ke Z
- Energy: En = − 2a
2 ,
0 n
n = 1, 2, 3, 4, . . .
h2
4π 2 me ke2
- Bohr radius: a0 =
= 0.529×10−10 m
Volume element in spherical coordinates: dV = r2 sinθdrdθdφ
Ground state Wavefunction for Hydrogen: Ψ (r, θ, φ) =
p
Root-Mean-Square deviation: ∆r = r¯2 − r̄2
Expectation Value for an operator Q: Q̄ =
sin2 θ =
1
2
[1 − cos (2θ)]
cos2 θ =
1
2
[1 + cos (2θ)]
R∞
2
0
R∞
0
Rb
a
1
2
pπ
2
x2 e−αx dx =
α,
1
4
p
all space
1
a0
dV Ψ∗ [Q]Ψ
α>0
π
α3 ,
α>0
xn e−x dx = − xn + nxn−1 + n(n − 1)xn−2 + . . . + n! e−x |ba
R∞
0
e−αx dx =
R
√1
π
xn e−x dx = n!
3
or
3/2
4πr2 dr
e−r/a0
1. Two space stations A and B are at rest relative to one another. A rocket
ship is sent from A towards B at at speed of 0.6c. The captain of the rocket
ship measures the time of flight from A to B as 200 seconds and sends a radio
signal immediately upon passing B.
(a) What is the distance between A and B as in the space stations’ frame
of reference and how much time (by the space stations’ clocks) elapsed
between the departure of the rocket from A and the arrival of the radio
signal at A?
(b) Immediately upon receiving the signal, station A sends out a second
rocket, at a speed of 0.8c, to chase and finally overtake the first rocket
(which is still maintaining its original speed of 0.6c relative to the space
station). What is the speed of the second rocket relative to the first?
(c) How much time (according to the captain of the first rocket) elapsed between his own departure from A and the meeting up of the two rockets?
2. (a) A rocket ship leaves Earth at a speed of 0.28c. It reports back to Earth by
radio every hour according to its clocks. What will be the time interval
between successive reports received back on Earth?
(b) If the rocket ship returns a the same speed directly toward Earth and
continues to report as before, what will be the new time interval between
successive reports?
(c) If the whole trip (outbound and inbound) lasted 48 hours according to
the rocket ship’s clocks, how long did it last according to the Earth-bound
clocks?
3. A neutral pion π 0 , of rest energy 135 MeV, travelling at a speed of 0.6c with
respect to the laboratory frame of reference decays into two photons:
π 0 −→ γ1 + γ2
(a) If the decay process causes the photons to be emitted in opposite directions
along the pion’s original track, what will the energies of the photon be (in
the laboratory frame of reference)?
(b) If, on the other hand, the photons are emitted at equal forward angles
with respect to the direction of motion of the pion, what will these angles
be and what will the photon energies be?
4
4. X-ray photons with an energy of 300 keV (1 keV = 103 eV) undergo Compton
scattering from a target. If a scattered photon is detected at 30 degrees relative
to the incident direction, find:
(a) the Compton shift at this angle,
(b) the energy of the scattered photon,
(c) the speed of the recoiling electron, and
(d) the angle at which the electron recoils.
5. Consider a photon and an electron that have a common wavelength λ. The
energy Eγ of the photon is twice the kinetic energy Ke of the electron.
(a) What is the value of λ in nm?
(b) What are the values of Eγ and Ke in MeV?
(c) What is the speed of the electron in terms of c?
6. The nuclear potential that binds protons and neutrons inside the nucleus of
an atom is often approximated by an infinite square well, i.e. U = ∞ outside
the well. Imagine a proton confined in such a well of width 10−5 nm — a typical
nuclear dimension.
(a) If the proton undergoes a transition from the state n = 2 to the state
n = 1 and in doing so emits a photon, what will the energy of the photon
be, what will its wavelength be, and in what region of the spectrum will
it belong?
(b) In the above transition, how much momentum will the photon carry and
with what velocity will the proton recoil?
5
7. (a) Write down the Schrödinger equation for a simple harmonic oscillator in
one dimension. Show by substitution that the function
2
ψ(x) = Cxe−αx
is a solution of this equation, provided that the constant α is chosen
appropriately. What is the appropriate value of α in terms of the mass m
and the angular frequency ω of the oscillator? What is the corresponding
value of the energy E of the system?
(b) If the oscillator, with the same energy E as found above, were treated classically, what would be the amplitude of its motion be? Call this amplitude
A.
(c) Go back to the quantum oscillator and determine the values of x for which
the probability density P (x) of the state in question is at a maximum.
Compare these values of x with the classical amplitude A.
8. For the Hydrogen atom in the ground state, calculate
(a) the average distance r̄, of the electron from the nucleus,
(b) the root-mean-square deviation in the value of r, (call it ∆r), and
(c) the probability that the electron is found to be in the region lying between
r̄ − ∆r and r̄ + ∆r.
6