Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Question
Chapter 11, Problem 11.36P
(a)
To determine
The classical position of the oscillator, assuming it started from rest at the origin
(b)
To determine
Show that the solution to the time-dependent Schrodinger equation for this oscillator can be written as
(c)
To determine
Show that the eigenfunctions and eigenvalues of
(d)
To determine
Show that in the adiabatic approximation the classical position reduces to
(e)
To determine
Show that
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A particle of mass m described by one generalized coordinate q movesunder the influence of a potential V(q) and a damping force −2mγq˙ proportional to its velocity with the Lagrangian L = e2γt(1/2 * mq˙2 − V (q)) which gives the desired equation of motion.
(a) Consider the following generating function: F = eγtqP - QP.Obtain the canonical transformation from (q,p) to (Q,P) and the transformed Hamiltonian K(Q,P,t).
(b) Let V (q) = (1/2)mω2q2 be a harmonic potential with a natural frequency ω and note that the transformed Hamiltonian yields a constant of motion. Obtain the solution Q(t) for the damped oscillator in the under damped case γ < ω by solving Hamilton's equations in the transformed coordinates. Then, write down the solution q(t) using the canonical coordinates obtained in part (a).
An anharmonic oscillator has the potential function
V = 1/2.(k.x^2) + c.x^4
where c can be considered a sort of anharmonicity constant.Determine the energy correction to the ground state of theanharmonic oscillator in terms of c, assuming that H^° is theideal harmonic oscillator Hamiltonian operator.
Consider a particle of spin s = 3/2.
(a) Find the matrices representing the operators S^
x
, S^
y
,S^
z
,
^
Sx
2
and ^
S y
2 within the basis of
^
S
2
and S^
z
(b) Find the energy levels of this particle when its Hamiltonian is given by
^H=
ϵ 0
h
2
( Sx
2−S y
2
)−
ϵ 0
h
( S^
Z )
where ϵ 0
is a constant having the dimensions of energy. Are these levels degenerate?
(c) If the system was initially in an eigenstate Ψ0=(
1
0
0
0)
, find the state of the system at time
Chapter 11 Solutions
Introduction To Quantum Mechanics
Ch. 11.1 - Prob. 11.1PCh. 11.1 - Prob. 11.2PCh. 11.1 - Prob. 11.3PCh. 11.1 - Prob. 11.4PCh. 11.1 - Prob. 11.5PCh. 11.1 - Prob. 11.6PCh. 11.1 - Prob. 11.7PCh. 11.1 - Prob. 11.8PCh. 11.1 - Prob. 11.9PCh. 11.3 - Prob. 11.10P
Ch. 11.3 - Prob. 11.11PCh. 11.3 - Prob. 11.12PCh. 11.3 - Prob. 11.13PCh. 11.3 - Prob. 11.14PCh. 11.3 - Prob. 11.15PCh. 11.3 - Prob. 11.16PCh. 11.4 - Prob. 11.17PCh. 11.5 - Prob. 11.18PCh. 11.5 - Prob. 11.19PCh. 11.5 - Prob. 11.20PCh. 11.5 - Prob. 11.21PCh. 11.5 - Prob. 11.22PCh. 11 - Prob. 11.23PCh. 11 - Prob. 11.24PCh. 11 - Prob. 11.25PCh. 11 - Prob. 11.26PCh. 11 - Prob. 11.27PCh. 11 - Prob. 11.28PCh. 11 - Prob. 11.29PCh. 11 - Prob. 11.30PCh. 11 - Prob. 11.31PCh. 11 - Prob. 11.33PCh. 11 - Prob. 11.35PCh. 11 - Prob. 11.36PCh. 11 - Prob. 11.37P
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