P237S05HW5sol

P237S05HW5sol - Physics 237 Spring 2005 Solution to...

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Physics 237 Spring 2005 Solution to Homework Assignment #6 6-2. The time dependent Schrodinger equation is: i h  t   h 2 2 m 2 x 2 V . We are given: x , t Ae i kx t . Hence, we get: (a) i h  t h & (b) h 2 2 m 2 x 2 V   h 2 k 2 2 m V . In both (a) & (b), the prefactor on the right hand side is the total energy. Hence, the wave function obeys the time-dependent Schrodinger equation. The classical wave equation is: 1 c 2 2 t 2 2 x 2 . Using the given wave function, we get: 1 c 2 2 t 2   2 c 2 & 2 x 2   k 2 . Since ck , the wave function obeys the classical wave equation. 6-10. The ground state wave function is: 1 x   2 L sin x L . The probability density is hence: 1 x   2 2 L sin 2 x L . If we want to determine the probability of finding the particle in some small interval Δ x centered around a position x , we can simply multiply this by the probability density evaluated at x . (a) For instance, for x = L /2 and Δ x = 0.002 L , the probability is equal to: 1 L 2 2 x 2 L sin 2 L L 2 0.002
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This note was uploaded on 10/15/2008 for the course PHYS 237 taught by Professor Stephonalexander during the Spring '08 term at Penn State.

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P237S05HW5sol - Physics 237 Spring 2005 Solution to...

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