* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Final Exam - University of California San Diego
Survey
Document related concepts
Future Circular Collider wikipedia , lookup
Quantum vacuum thruster wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Uncertainty principle wikipedia , lookup
Photoelectric effect wikipedia , lookup
Quantum tunnelling wikipedia , lookup
Renormalization wikipedia , lookup
Renormalization group wikipedia , lookup
Atomic nucleus wikipedia , lookup
Electron scattering wikipedia , lookup
Old quantum theory wikipedia , lookup
Nuclear structure wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Transcript
Page 1 of 8 Department of Physics University of California San Diego Modern Physics (2D) Prof. V. Sharma Final Exam (March 17, 2005) Page 2 of 8 Page 3 of 8 Page 4 of 8 Page 5 of 8 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 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 Page 7 of 8 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 . Page 8 of 8 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!