Quantum Mechanics — Daily Schedule Term 4

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Week 1 — Introduction to Quantum-Mechanical Interference — Introduction to Complex Variables

Monday, Jan. 12 — Reading: Churchill, Brown & Verhey (hereafter, just “CBV”), Sections 1 and 2 of Chapter 1; Feynman, Volume III Chapter 1 — Problem Set 1: All the evens, and optionally the odds, in the CBV reading — Seminar leader for Feynman Chapter 1: Keevan
Thursday, Jan. 15 — Reading: CBV, Sections 3 and 4 of Chapter 1; Feynman, Chapter III-2 — Problem Set 2: All the evens, and optionally the odds, in the CBV reading; Feynman 27.12, 27.14 and 27.21 — Seminar leader for Feynman Chapter 2: Lucinda

Week 2 — Mathematics of Classical and Quantum-Mechanical Wave Interference and Diffraction

Monday, Jan. 19 — Reading: CBV, Sections 5 and 6 of Chapter 1; Start Feynman, Chapter I-29 and part of I-30 — Problem Set 3: All the evens, and optionally the odds, in the CBV reading — Seminar leader: Grisha
Thursday, Jan. 22 — Feynman, Conclude Chapter I-29 and part of I-30 — Problem Set 4: Finish Feynman’s exercise 21.1 (to do both the even and odd case you are going to need another identity for the sum of the odd powers of z like the one you derived for all the powers of z in CBV problem 14); Derive the actual function of angle (not just the location of nodes and anti-nodes) for all the situations Feynman detailed in Chapter I-29; In Feynman’s exercise book, do Problems 21.4, 21.5, and 21.6 — Seminar leader: Keevan

Week 3 —

Monday, Jan. 26 — Reading: Continue Feynman through Chapter 3 Section 2 (through means up to and including) — Problem Set 4: Wolfson, Chapter 14, Problems #24 (see commentary on Problem 24 below) and #46, Feynman Problems 27.6, 27.7, and 27.20, and my Gaussian problem below
Commentary on Problem 24: Make it a four-part problem: (a) Put y₁(x,t)=f₁(x-vt) into the equation. Take the partial derivatives. What does that demand about v? This is a right-moving wave (assuming the direction of increasing x is to the right). Is there any restriction on the function f₁ other than it be twice differentiable? (b) Put y₂(x,t)=f₂(x+vt) into the equation. Does it work with the same demand on v? This is a left-moving wave. There is no restriction on f₂ other than it be twice differentiable. Popular choices for f₁ and f₂ are the sines and cosines you so often see. (c) Given that y₃(x,t) and y₄(x,t) are solutions, show that y₃(x,t)+y₄(x,t) is a solution. This is the principle of superposition. The sum of any two solutions is a solution! Of course the difference is a solution too. In fact any linear combination is c₃×y₃(x,t)+c₄×y₄(x,t). (d) If the tank is 1.5 inches deep, and the acceleration of gravity is 32 ft / second-squared, what is the speed of the water waves in feet per second? Plugging in the numbers is elementary, but a simple problem like this is a good opportunity to check that you can track units carefully, including the conversion factor from feet to inches.
Gaussian Problem: In quantum mechanics, a Gaussian probability amplitude, e^(-x²/2a). when squared, becomes a Gaussian probability distribution, and it is something which is e^(-x²/a). (a) Integrate this function from minus infinity to infinity. What would the amplitude have to be multiplied by so that after squaring and integrating you got 1? This is called normalization. (b) Take your normalized probability distribution, multiply it by x², and integrate that from minus infinity to infinity. You can do this integral without looking it up if you change 1/a to α and realize that the extra x² in the integrand can be obtained by taking minus the derivative respect to α of the integral you previously did. Sneaky, eh?
Thursday, Jan. 29 — Conclude Chapter III-3

Week 4 — Exam 1 —

Monday, Feb. 2 — Exam 1 on CBV, Chapter 1, Wolfson, Chapter 14, and Feynman, Chapters I-29, I-30, III-1, III-2, and III-3 (glossing Feynman’s emphasis on crystal diffraction patterns, even though they are experimentally and scientifically extremely important for discovering the lattice structure of crystals)