Test 1 Solution for Introduction to Quantum Mechanics II | PHY 4605, Exams of Physics

Download Test 1 Solution for Introduction to Quantum Mechanics II | PHY 4605 and more Exams Physics in PDF only on Docsity! PHY4605–Introduction to Quantum Mechanics II Spring 2005 Test 1 Solutions February 9, 2005 1. Short Answer. Must attempt (only) 3 of 4. Circle answers to be graded. (a) Sketch an experiment to measure the Aharonov-Bohm effect, and comment on the significance of the electromagnetic potentials A and φ in quantum mechanics as opposed to classical physics. interference pattern B=0 2 1 A=0 B=0 Changing B in solenoid changes relative phases of electrons travelling along “paths” 1+2 shifts interference pattern at screen. Classically e− would need to move through region of B 6= 0 to feel an effect. Not true in Quantum Mechanics. (b) Comment on the significance of unitary transformations in quantum me- chanics. Indicate how the Hamiltonian must transform under such a uni- tary transformation in order to preserve the Schrödinger equation. Unitary transformations play a special role in quantum mechanics because they preserve matrix elements of observables. The Hamiltonian, for exam- ple, must transform as H → U−1HU at the same time as states transform as ψ → U−1ψ. This means 〈ψ|H|χ〉 → 〈ψ′|H ′|χ′〉 = (〈ψ|U) (U−1HU) (U−1|χ〉) = 〈ψ|H|χ〉 (c) a) Explain why, in a Stern-Gerlach (SG) apparatus, a beam consisting of neutral particles in different spin states are split into different beams. b) If a single spin-1 particle in state |` = 1,m = 1〉 (spin quantization axis ẑ) is sent into a SG apparatus aligned along ŷ, state how many possible paths the particle can take. (c) a) In a magnetic field a magnetic moment has (classically) an energy Edipole = −~µ ·B, and if B is inhomogeneous the particle experiences a force given by −∇Edipole = ∇(~µ ·B) = ∇( e m S ·B) = e m ∇(SzBz), where in last step I arbitrarily assumed Bz was in z-direction. Thus the force on a particle will be different depending on the value of Sz, split- ting the particles with a given initial velocity into different beams. The quantum mechanical version of this argument, involving calculating the expectation value of p, is given in Griffiths, and reaches the same conclu- sion. (c)b) Since each eigenvalue of the spin experiences a different force, an unpolarized spin-1 particle beam is split into 3 different beams in general, corresponding to the 2S +1=3 independent components with different Sz. (d) A broad beam of x-rays is incident on an ammonia molecule in its ground state. i. Explain why interference fringes are observed on a photographic plate. ii. Design a gedanken-experiment to determine if the N atom in the molecule is above or below the plane defined by the three H atoms. The ammonia atom in its ground state is in a linear superposition of the quantum states with the H-atom below and above the plane, as shown. A wide beam of electrons, photons, etc. of short wavelength will therefore scatter as shown, producing spherical wavelets from each N atom which would–in principle–interfere at an appropriately placed screen. If one could on the other hand produce a very narrow beam of particles oriented such that the N atom would scatter only if it were in the top or bottom state. Scattering will then constitute measurement of the top or bottom state, and collapse the wave function.