Download Final Exam - University of California San Diego

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Department of Physics
University of California
San Diego
Modern Physics (2D)
Prof. V. Sharma
Final Exam
(March 17, 2005)
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Problem 1: Top Gun ! [30 pts]
An enemy spaceship is moving towards your starfighter with a speed, as
measured in your frame, of 0.40c. The enemy ship fires a missile towards
you at a speed of 0.70c according to the bad guy in the enemy ship. (a) What
is the speed of the missile in your frame? Express your answer in terms of
the speed of light c. (b) If you measure that the enemy ship is 8.0×106 km
away from you when the missile is fired, how much time, measured in your
frame, will it take the missile to reach you?
Problem 2: Sustainable Fusion Will Solve Energy Crisis ! [20 pts]
UCSD physicists and engineers are working on the design of a sustainable
fusion reactor. When commercially viable, it can solve the world’s everincreasing demand for clean energy. In a fusion reactor, two Deuterium
nuclei fuse to form one Helium nucleus. The mass of the Deuterium nucleus
is 2.0136u and that of Helium nucleus is 4.0015u. (a) How much energy is
released when 1.0kg of Deuterium undergoes fusion? (b) The annual energy
consumption in the US is of the order of 1.0×1019 J. How much Dueterium
must react to produce this amount of energy?
Problem 3: Estimating The Planck Constant :
[20 pts]
In a photoelectric experiment where a Sodium photocathode is used, one
finds a stopping potential of 1.85V for a wavelength of 300nm and a
stopping potential of 0.82V for a wavelength of 405nm. From these data
alone find (a) a value for the Planck constant (b) the work function for
Sodium (c) the cutoff wavelength for Sodium.
Problem 4: Tiger Hunting in a Quantum Jungle ! :
[30 pts]
Somewhere in the Himalayan mountain range there are rumors of a
mysterious Quantum jungle where the value of the Planck's constant
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h is much larger than our usual world. Imagine that you are in this quantum
jungle where h=50 J.s !! Sher Khan, the tiger, runs past you in the bushes a
few meters away. The tiger, weighing 100kg, is known to be in a region
about 4m long. (a) What is the minimum uncertainty in his speed? (b)
Assuming this uncertainty in his speed to prevail for 10 seconds, determine
the uncertainty in his position after this time.
A Fuzzy Sher Khan
Problem 5: Triggering a Transition Between Quantum States: [40pts]
Consider an electron in an infinite 1-D square well (located at x=0, x=L)
described initially by a wave function that is superposition of the ground
state and the first excited states of the well: Ψ ( x, t = 0) = C [ψ 1 ( x) +ψ 2 ( x)] (a)
show that the value C = 1/ 2 normalizes this wave, assuming ψ 1 and ψ 2 are
themselves normalized. (b) find Ψ ( x, t ) at any later time t. (c) show that the
superposition state is not a stationary state, but that the average energy of
this state is the arithmetic mean (E1+E2)/2 of the ground state energy E1 and
the first excited state energy E2. (d) show that the average particle position
<x> oscillates with time as :
< x >= x0 + A cos(Ωt ) where A= ∫ x ψ 1*ψ 2 dx
x0 =
1
2
(
2
2
|
ψ
|
+
|
ψ
|
x
dx
x
dx
1
2
∫
∫
)
and Ω =
E2 − E1
=
(e) Evaluate your results for the mean position x0 and the amplitude of
oscillation A for an electron in a well of length 1.0 nm. (f) Now calculate the
time it takes for the electron to execute one period of oscillation around the
mean position x0 .
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Problem 6: Rapping About a 2-D Harmonic Oscillator : [30 pts]
Consider a 2D harmonic oscillator of mass m under a potential
U ( x, y , z ) =
1
m(ω12 x 2 + ω22 y 2 ) with ω1 < ω2
2
(a) Write the appropriate time-independent Schrodinger equation for this
oscillator. (b) Write the wavefunction for the first excited state including the
normalization constant (let’s call it A) (c) Normalize the wavefunction and
calculate the value of the normalization constant A (d) What is the energy
of this state? (e) Is this state degenerate? Why (not)? (f) What is the average
potential energy of this state?
Problem 7: An Excited Hydrogen Atom:
[30 pts]
Calculate and compare the most probable distances of the electron from the
proton in the (a) 2s and (b) 2p states with the radius of the second Bohr orbit
in Hydrogen of 4a0.
GOOD LUCK!