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Physics 451 Quantum mechanics I Fall 2012 Sep 10, 2012 Karine Chesnel Quantum mechanics Announcements Homework • Homework 4: T Sep 11 by 7pm Pb 1.9, 1.14, 2.1, 2.2 • Homework 5: Th Sep 13 by 7pm Pb 2.4, 2.5, 2.7, 2.8 Quantum mechanics No student assigned to the following transmitters: 2214B68 17A79020 1E5C6E2C 1E71A9C6 Please register your i-clicker at the class website! Quantum mechanics Ch 1.6 Quiz 4a Uncertainty principle Which statement is accurate for these electronic wave functions? A. Both the position x and the momentum p are fairly well defined B. The position of the particle is fairly well defined but the momentum is poorly defined C. The momentum of the particle is fairly well defined but the position is poorly defined D. Both the position and the momentum are poorly defined. Quantum mechanics Ch 1.6 Uncertainty principle Position x Momentum p i x p 2m E V particle x p p 2 wave 2 h De Broglie formula 1924 Heisenberg’s uncertainty Principle 1927 Quantum mechanics Ch 1.6 Uncertainty principle How to check the uncertainty principle? • Calculate x x2 and x • Calculate p p 2 and p • Estimate the product • Compare to x p 2 Pb 1.9 Quantum mechanics Ch 1 Probability current ( x , t ) ( x, t ) * 2 Density of probability b Probability between two points Pab ( x, t )dx a where dPab J a, t J (b, t ) dt i * J a, t * 2m x x Pb 1.14 Quantum mechanics Ch 2.1 Time-independent Schrödinger equation 2 2 i V 2 t 2m x In general V ( x, t ) Here V ( x) function of x only The potential is independent of time General solution: ( x, t ) ( x) (t ) “Stationary state” Quantum mechanics Ch 2.1 Time-independent Schrödinger equation Plugging the general solution: ( x, t ) ( x) (t ) in the Schrödinger equation 2 1 d 1 d 2 i V 2 dt 2m dx Function of time only Function of space only E Quantum mechanics Ch 2.1 Time-independent Schrödinger equation • Time dependent part: 1 d i E dt d iE dt General solution: (t ) e iE t Quantum mechanics Ch 2.1 Time-independent Schrödinger equation 1 d 2 V E 2 2m dx 2 • Space dependent part: d 2 V ( x) E 2 2m dx 2 Solution (x) depends on the potential function V(x). Global solution: ( x, t ) ( x)eiEt / Stationary state Quantum mechanics Quiz 3b “If the particle is in one stationary state, its expectation value for position is not changing in time.” A. True B. False 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 ) ( x)eiEt / Q Q ( x , ) 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) E 2 2m dx ^ H ^ ^ H H dx E * dx E ^ * ^ H 2 * H 2 dx E 2 * 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 ) n ( x)eiEnt / • Associated expectation value for energy H cn2 En n 1 Quantum mechanics Ch 2.1 Stationary states n ( x, t ) n ( x)eiEnt / Pb 2.1 a) En must be real b) n(x) can always be real c) n(x) is either real or odd, when V(x) is even Pb 2.2 En Vmin Classical analogy: The kinetic energy is always positive! However, in QM, it is possible that En V ( x) at some locations x