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Physics 451 Quantum mechanics I Fall 2012 Sep 12, 2012 Karine Chesnel Quantum mechanics Announcements Homework remaining this week: • Extended Friday Sep 14 by 7pm: HW # 5 Pb 2.4, 2.5, 2.7, 2.8 Note: Penalty on late homework: - 2pts per day Credit for group presentations: Homework 2: 20 points Quiz 5: 5 points Quantum mechanics Announcements No student assigned to the following transmitters: 2214B68 17A79020 1E5C6E2C 1E71A9C6 Please register your i-clicker at the class website! Quantum mechanics Ch 2.1 Time-independent Schrödinger equation • Space dependent part: 2 2 d V ( x) y Ey 2 2m dx Solution y(x) depends on the potential function V(x). ( x, t ) y ( x)e iEt / Stationary state Associated to energy E Quantum mechanics Ch 2.1 Stationary states Properties: • Expectation values are not changing in time (“stationary”): Q Q ( x, )dx i x * with ( x, t ) y ( x)eiEt / Q y Q ( x , )y dx i x * Q is independent of time p m v m d x dt 0 The expectation value for the momentum is always zero In a stationary state! (Side note: does not mean that x and p are zero!) Quantum mechanics Ch 2.1 Stationary states Properties: • Hamiltonian operator - energy 2 d2 V ( x) y Ey 2 2m dx ^ H ^ ^ H y H y dx E y *y dx E ^ * ^ H 2 y * H 2 y dx E 2 y *y dx E 2 H 0 Quantum mechanics Ch 2.1 Stationary states • General solution ( x, t ) cn n ( x, t ) n 1 where n ( x, t ) y n ( x)eiEnt / • Associated expectation value for energy H cn2 En n 1 Quantum mechanics Quiz 6a A particle, is in a combination of stationary states: ( x, t ) cn n ( x, t ) n 1 What will we get if we measure its energy? A. H B. c E n n n C. one of the values D. n En En Quantum mechanics Quiz 6b A particle, is in a combination of stationary states: ( x, t ) cn n ( x, t ) n 1 What is the probability of measuring the energy En? A. 0 B. cn C. cn cn D. cn 2 E. 1 2 cn Quantum mechanics Ch 2.2 Time-independent potential Expectation value for the energy: * H cm m H cn n n1 m 1 ^ ^ H c c m 1 n 1 ^ * m n H y n Eny n ^ H n dx * m ^ H cm* cn En e m 1 n 1 ^ H cn En n 1 2 i ( En Em ) t nm Quantum mechanics Ch 2.2 Infinite square well V(x)=0 for 0<x<a V=∞ 0 a The particle can only exist in this region else x Shape of the wave function? Quantum mechanics Ch 2.2 Infinite square well Solutions to Schrödinger equation: d y E y 2 2m dx 2 2 d 2y 2 k y 2 dx Simple harmonic oscillator differential equation with k 2mE Quantum mechanics Ch 2.2 Infinite square well Solutions to Schrödinger equation: y ( x) A sin kx B cos kx Boundary conditions: At x=0: y (0) 0 At x=a: y (a) 0 y ( x) A sin kx with n kn a Quantum mechanics Ch 2.2 Infinite square well Possible states and energy values: 2 n yn sin a a x n 2 2 2 En 2 2ma Quantization of the energy Each state yn is associated to an energy En ^ H y n Eny n Quantum mechanics Ch 2.2 Infinite square well Properties of the wave functions yn: y 3 , E3 1.They are alternatively even and odd around the center y 2 , E2 2. Each successive state has one more node y 1 , E1 3. They are orthonormal Excited states Ground state 0 a * y my n nm x 4. Each state evolves in time with the factor e iEn t / Quantum mechanics Ch 2.2 Infinite square well Pb 2.4 x Pb 2.5 Particle in one stationary state x 2 p x p p2 2 Particle in a combination of two stationary states x, 0 A(y 1 y 2 ) x, t ( x, t ) oscillates in time 2 x H p evolution in time? expressed in terms of E1 and E2 Quantum mechanics Ch 2.2 Infinite square well Expectation value for the energy: ^ H cn En 2 n 1 The probability that a measurement 2 yields to the value En is cn Normalization c n 1 n 2 1