Physics 451 Quantum mechanics I Fall 2012 Sep 10, 2012 Karine Chesnel

Announcements Quantum mechanics 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 Homework

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Uncertainty principle Quiz 4a Which statement is accurate for these electronic wave functions? A.Both the position x and the momentum 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 mechanicsCh 1.6

Uncertainty principle Position x Heisenberg’s uncertainty Principle 1927 particle De Broglie formula 1924 wave Momentum Quantum mechanicsCh 1.6

Uncertainty principle Quantum mechanicsCh 1.6 Pb 1.9 How to check the uncertainty principle? Calculate and Estimate the product Compare to

Quantum mechanicsCh 1 Probability current Pb 1.14 Density of probability Probability between two points where

Quantum mechanicsCh 2.1 Time-independent Schrödinger equation In general Here function of x only The potential is independent of time General solution: “Stationary state”

Quantum mechanicsCh 2.1 Time-independent Schrödinger equation Plugging the general solution: in the Schrödinger equation Function of time only Function of space only

Quantum mechanicsCh 2.1 Time-independent Schrödinger equation Time dependent part: General solution:

Quantum mechanicsCh 2.1 Time-independent Schrödinger equation Solution  (x) depends on the potential function V(x). Space dependent part: Global solution: Stationary state

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

Quantum mechanicsCh 2.1 Stationary states Properties: Expectation values are not changing in time (“stationary”): with is independent of time The expectation value for the momentum is always zero In a stationary state! (Side note: does not mean that and are zero!)

Quantum mechanicsCh 2.1 Stationary states Properties: Hamiltonian operator - energy

Quantum mechanicsCh 2.1 Stationary states General solution where Associated expectation value for energy

Quantum mechanicsCh 2.1 Stationary states Pb 2.1 a) E n 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 Classical analogy: The kinetic energy is always positive! However, in QM, it is possible that at some locations x