Page 1 of 8

Department of Physics University of California San Diego

Modern Physics (2D) Prof. V. Sharma

Final Exam (Dec 10 2003)

Page 2 of 8

Page 3 of 8

Page 4 of 8

Page 5 of 8

The Hitchhiker’s guide to the galaxy says on its very first page : “ DON’T PANIC !”

Page 6 of 8

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 relative to the enemy ship. (a) What is the speed of the missile relative to you? 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: Mirror Mirror On The Wall [20 pts]

An observer in a rocket moves towards a mirror at speed v relative to the reference frame labeled S in the figure above. The mirror is stationary with respect to S. A light pulse emitted by the rocket travels towards the mirror and is reflected back to the rocket. The front of the rocket is a distance d away from the mirror (as measured by observer in S) at the moment the light pulse leaves the rocket. What is the total travel time of the pulse as measured by the observers in (a) the S frame and (b) the front of the rocket? Problem 3: Designing Photocells: [20 pts] The active material in a photocell has a work function of 2.0 eV. Under reverse bias voltage conditions (needed to generate repulsive stopping voltage) the cutoff wavelength is found to be 350nm. What is the value of the stopping voltage?

Page 7 of 8

Problem 4: Quantum Pool: [20 pts] Gamma rays (high energy photons) of energy 1.02 MeV are scattered from electrons that are initially at rest. If the scattering is symmetric (that is to say that both the electron and photon fly off with the same angle with respect to the incident photon so that φ=θ ) find (a) the scattering angle θ and (b) the energy of the scattered photons. Problem 5: Tiger Hunting in a Quantum Jungle ! [20 pts] Somewhere in the Himalayan mountain range there are rumors of a mysterious Quantum jungle where the value of the Planck's constant 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. Problem 6: Triggering a Transition Between Quantum States [40pts] Consider a particle in an infinite square well described initially by a wave that is superposition of the ground state and the first excited states of the well: [ ]1 2( , 0) ( ) ( )x t C x xψ ψΨ = = + (a) show that the value 1/ 2C = normalizes this wave, assuming 1ψ and 2ψ are themselves normalized. (b) find ( , )x tΨ 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 :

( )

*0 1 2

2 2 2 10 1 2

cos( ) where A= x

1 | | | | and 2

x x A t dx

E Ex x dx x dx

ψ ψ

ψ ψ

< >= + Ω

−= + Ω =

∫

∫ ∫

(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

Page 8 of 8

time it takes for the electron to execute one period of oscillation around the mean position x0 . Problem 7: Rapping About a 2-D Harmonic Oscillator [20 pts] Consider a 2D harmonic oscillator of mass m under a potential 2 2 2 2

1 2 1 21( , , ) ( ) with 2

U x y z m x yω ω ω ω= + <

(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 average potential energy of this state? Problem 8: The Hydrogen Atom [30 pts] Calculate and compare the most probable distances of the electron from the proton in the 2s and 2p states with the radius of the second Bohr orbit in Hydrogen of 4a0.

GOOD LUCK & HAPPY HOLIDAYS !