Physics 451 Quantum mechanics I Fall 2012 Sep 12, 2012 Karine Chesnel

Homework remaining this week: Extended Friday Sep 14 by 7pm: HW # 5 Pb 2.4, 2.5, 2.7, 2.8 Announcements Quantum mechanics Note: Penalty on late homework: - 2pts per day Credit for group presentations: Homework 2: 20 points Quiz 5: 5 points

No student assigned to the following transmitters: Announcements Quantum mechanics Please register your i-clicker at the class website! 2214B68 17A E5C6E2C 1E71A9C6

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

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

Quiz 6a A. B. C. one of the values D. Quantum mechanics A particle, is in a combination of stationary states: What will we get if we measure its energy?

Quiz 6b A.0 B. C. D. E. Quantum mechanics A particle, is in a combination of stationary states: What is the probability of measuring the energy E n ?

Quantum mechanicsCh 2.2 Time-independent potential Expectation value for the energy:

Quantum mechanicsCh 2.2 Infinite square well x 0a The particle can only exist in this region V(x)=0 for 0<x<a V=∞ else Shape of the wave function?

Quantum mechanicsCh 2.2 Infinite square well Solutions to Schrödinger equation: Simple harmonic oscillator differential equation with

Quantum mechanicsCh 2.2 Infinite square well Solutions to Schrödinger equation: Boundary conditions: At x=0: At x=a: with

Quantum mechanicsCh 2.2 Infinite square well Possible states and energy values: Quantization of the energy Each state  n is associated to an energy E n

Quantum mechanicsCh 2.2 Infinite square well Properties of the wave functions  n : x 0a 1.They are alternatively even and odd around the center 2. Each successive state has one more node Ground state Excited states 3. They are orthonormal 4. Each state evolves in time with the factor

Quantum mechanicsCh 2.2 Infinite square well Pb 2.4 Particle in one stationary state Pb 2.5 Particle in a combination of two stationary states evolution in time? oscillates in time expressed in terms of E 1 and E 2

Quantum mechanicsCh 2.2 Infinite square well Expectation value for the energy: The probability that a measurement yields to the value E n is Normalization