Lecture12 - Conserved quantities Quantities that do not...

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1 Physical Chemistry Lecture 12 Mathematics of Quantum Mechanics Waves Classical waves are periodically varying functions of time and space Described by an amplitude, A , a wavelength, , and a period, T . Alternative to is the frequency, . A wave’s speed depends on the wavelength and frequency. Speed of sound Speed of light v 2 sin ) , ( T t x A t x Wave motion The amplitude at a point varies as a function of time The amplitudes at a time vary in space Convenient to define the wave vector, the angular frequency, and the phase Standing waves, which are the result of pinning with boundary conditions, can be written in a simple form By taking the derivatives, one can show that a wave obeys the wave equation :   t kx A t x k sin , 2 2 ) ( ) ( ) , ( t f x A t x 2 2 2 2 2 2 1 1 t x k Conserved quantities Quantities that do not change in time are said to be conserved Examples Also called constants of motion Conserved quantities define a system’s state They are unique characteristics of the system when in the state We seek conserved quantities as a way to describe a state conserved not is quantity conserved a is quantity conserved a is z z x x p dt dp E dt dE p dt dp 0 0 0 Eigenvalue equations Special relationship between a function and an operation Operation on the function yields the same function multiplied by a constant For any operator, only a certain group of functions can satisfy such an equation Establishes the relationship ALL functions having this relationship to a particular operator form a complete set Usually distinguished by a number (often appended as a subscript to the function Each member of the set is a distinct, special case Eigenvalue equations give a means to find states corresponding to constants of motion  n f kx A k kx A dx d Example f K f O sin sin ˆ 2 2 2 Schroedinger’s equation Means to solve for wave functions of a system corresponding to constant energy states
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This note was uploaded on 02/02/2012 for the course CHEM 419 taught by Professor Staff during the Fall '10 term at University of Delaware.

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Lecture12 - Conserved quantities Quantities that do not...

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