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Physics 451 Quantum mechanics I Fall 2012 Karine Chesnel Phys 451 Announcements Monday Sep 3: NO CLASS (Holiday) Homework • Homework 1: Today Aug 31st by 7pm Pb 1.1, 1.2, 1.3 • Homework 2: Group presentations Sep 5th • Homework 3: F Sep 7th by 7pm Pb 1.4, 1.5, 1.7, 1.8 Help sessions: T Th 3-6pm Phys 451 Announcements Next class Sep 5th Group presentations Will count as homework 2, 20 points plus 5 quiz points for presenting Famous scientist who contributed to the foundation of Quantum Mechanics Einstein Schrödinger Heisenberg Dirac Pauli Born Bohr Planck De Broglie Quantum mechanics Probabilities Discrete variables Examples of discrete distributions: Distribution of scores in a class Age pyramid for a certain population (Utah, 2000) Quantum mechanics Probabilities Discrete variables Distribution of the system Probability for a given j: N ( j) N ( j) P( j ) N Average value of j: j jP( j ) j 0 Average value of a function of j Average value f ( j ) f ( j ) P( j ) j 0 “Expectation” value Example: number of siblings for each student in the class Quantum mechanics Quiz 2a What is the definition of the variance? 2 A. j B. j2 C. j2 j2 D. j j E. j 2 2 2 j2 Quantum mechanics Probabilities Discrete variables j j j The deviation: j 0 Variance j j j 2 The standard deviation 2 2 j 2 j 2 2 Quantum mechanics Probabilities Discrete variables The variance defines how wide/narrow a distribution is intensity brightness Distribution of scores in a class Wide: large Spectral analysis of a photograph Narrow: small Quantum mechanics Probabilities Continuous variables The probability of finding the particle in the segment dx The density of probability: P ( x)dx ( x) b Probability to find the particle between positions a and b: ( x).dx a Normalization: ( x).dx 1 Quantum mechanics Probabilities Continuous variables x Average values: x ( x)dx f ( x) f ( x). ( x)dx Variance: x x 2 2 2 f ( x) f ( x) 2 2 2 Quantum mechanics Connection to Wave function Density of probability (now function of space and time): ( x, t ) ( x, t ) 2 Normalization: ( x, t ) dx 1 2 Solutions ( x, t ) have to be normalizable: - needs to be square-integrable Quantum mechanics Quiz 2b Is this wave function square integrable or not? ( x, t ) Aeit kx A. YES B. NO Quantum mechanics Normalization of Wave function Normalization: ( x, t ) dx 1 2 Can stay normalized in time? If satisfies the Schrödinger equation and is normalizable, then indeed d 2 ( x, t ) dx 0 dt