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Unformatted text preview: is odd p2 —2 = a
2— p p H— ê aL2 H— ê aL2 Therefore
D px = Printed by Mathematica for Students 2a `
The momentum uncertainty does not depend on time because px commutes with the Hamiltonian, so momentum is a constant of `
¶
X px \ = Ÿ ¶ p y H p, tL
4 `2
¶
X px \ = Ÿ ¶ p2
Ch6ProblemSetKey.nb 2 2 yH p, tL =
= ¶ a
p — p 2
Ÿ¶ „ p p ‰ ¶ a
—  Ÿ¶ „ p p ‰ ap
—2  =0 ¬༼ integrand is odd a2 p2
—2 = a
2— p H— ê aL2 H— ê aL2 p Therefore
D px = —
2a `
The momentum uncertainty does not depend on time because px commutes with the Hamiltonian, so momentum is a constant of
the motion. We see that a DxD p = B 1+J 2 ht
m a2 2 N FB —
2a F= —
2 1+J ht
m a2 2 N ¥ —
2 for all times t so the uncertainty principle is always satisfied by this state. We also see that the minimum uncertainty state occurs at time t = 0,
when the state is a Gaussian wavepacket. Problem 6.15
Given a particle of mass m in the onedimensional potential energy well
0 0<x<L
¶ elsewhere V HxL = in the state I yHxL = 1+Â
M
2 2
L sinI px
M
L + 1
2 2
L sinI 0 2px
M
L 0< x < L
elsewhere at time t = 0. ü Part a
The energy eigenstates yn \ of the system are those that satisfy the timeindependent Schrödinger equat...
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This note was uploaded on 02/01/2014 for the course PHYC 491/496 taught by Professor Akimasamiyake during the Fall '13 term at New Mexico.
 Fall '13
 AkimasaMiyake

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