Modern Physics For Scientists And Engineers
2nd Edition
ISBN: 9781938787751
Author: Taylor, John R. (john Robert), Zafiratos, Chris D., Dubson, Michael Andrew
Publisher: University Science Books,
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Chapter 7, Problem 7.4P
To determine
String’s transverse velocity.
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Show that a gaussian psi (x) = e ^(-ax^2) can be an eigenfunction of H(hat) for harmonic oscillator
1. Compute T(hat)*psi
2. Compute Vhat* psi - assume V operator is 1/2w^2x^2
3. Write out Hbar*psi and identify terms so Hber*psi=E*psi is true
4. From cancellation find a
5. insert back a to Schrodinger eq above and find E
Sketch f(y) versus y. Show that y is increasing as a function of t for y < 1 and also for y > 1. The phase line has upward-pointing arrows both below and above y = 1. Thus solutions below the equilibrium solution approach it, and those above it grow farther away. Therefore, ϕ(t) = 1 is semistable.
The system has a particle (its mass is m) in the vacuum, no potential energy (V(x) = 0). The Hamiltonian of the system
ħ² 8²
is given by Ĥ = -2m dx²
The initial wave functions at time t = 0 is
where a is a real constant.
Question #1: Find the time depended wave functions V(x, t) = ?
Question #2: Also, for the wave function (x, t) (that has been found above), find the probability current density
defined as
Note:
+∞
1000
+∞
[too
where ã and ß are complex constants;
v (z,t = 0) = (=) *exp(- 02²)
1
j = 2
2m
8
(¥* (x)pV(x) + V(x) (pV(x))*)
r+∞
88
da exp(-ã+²) = √
dx
dr exp (-ãx² - ikx) =
= √= exp(-1/²)
dk exp
(-²+ ika) = √³ exp(-82²)
4
ax²+bx+c = a (x +
= a (x + 2)² + c - 50
2a
4a
c
Chapter 7 Solutions
Modern Physics For Scientists And Engineers
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