Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 11.1, Problem 11.6P
To determine
The second order in perturbation theory for the general case
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For Problem 8.16, how do I prove the relations and give the correct expressions?
Problem #1
(Problem 5.3 in book). Come up with a function for A (the Helmholtz free energy) and
derive the differential form that reveals A as a potential:
dA < -SdT – pdV [Eqn 5.20]
1. Consider the 2D motion of a particle of mass u in a central force field with potential V(r).
a) Find the r, o polar-coordinate expression of the Lagrangian for this system and write down the
corresponding Euler-Lagrange e.o.m.s.
b) Note that the angular variable o is cyclic. What is the physical interpretation of the correspond-
ing integral of motion? (For the definitions of the italicized terms see this link.)
c) Solve for o in terms of this integral of motion and substitute the result into the Euler-Lagrange
equation for r. Show that the result can be arranged to look like a purely 1D e.o.m. of the form
dVef(r)
(1)
dr
Identify in the process the explicit expression for Vef(r), which will depend among other things on
the integral of motion.
d) Take now
k
V (r) =
with k > 0 to be an attractive electrostatic/gravitational-type potential. Sketch the profile of the
corresponding effective potential function Vef(r). Find the equilibrium solution for the correspond-
ing e.o.m. (1). What…
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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Similar questions
- For Problem 8.35, how do I prove, or perhaps verify, what it is they're asking for?arrow_forwardQ.n.3 A central force is defined to be a force that points radially, and whose magnitude depends on only r. That is, F(r) = F(r) `r. Show that a central force is a conservative force, by explicitly showing that Vx F = 0 Q.n.4 Consider two particles of masses ml and m2. Let m1 be confined to move on a circle of O plane, centered at x = y = 0. Let m2 be confined to move on a circle of radius radius a in the z = b in the z = c plane, centered at x = y = 0. A light (massless) spring of spring constant k is attached between the two particles. a) Find the Lagrangian for the system. Q.n.5 Oral Vivaarrow_forwardProblem 8.14 An infinitely long cylindrical tube, of radius a, moves at constant speed v along its axis. It carries a net charge per unit length λ, uniformly distributed over its surface. Surrounding it, at radius b, is another cylinder, moving with the same velocity but carrying the opposite charge (-λ). Find: (a) The energy per unit length stored in the fields. (b) The momentum per unit length in the fields. (c) The energy per unit time transported by the fields across a plane perpendicular to the cylinders.arrow_forward
- Consider a uniformly charged ring in the xy plane, centered at the origin. The ring has radius a and positive charge q distributed evenly along its circumference. Imagine a small metal ball of mass m and negative charge −q0. The ball is released from rest at the point (0,0,d) and constrained to move along the z axis, with no damping. If 0<d≪a, what will be the ball's subsequent trajectory? repelled from the origin attracted toward the origin and coming to rest oscillating along the z axis between z=d and z=−d circling around the z axis at z=darrow_forwardTwo particles, each of mass m, are connected by a light inflexible string of length l. The string passes through a small smooth hole in the centre of a smooth horizontal table, so that one particle is below the table and the other can move on the surface of the table. Take the origin of the (plane) polar coordinates to be the hole, and describe the height of the lower particle by the coordinate z, measured downwards from the table surface. i. sketch all forces acting on each mass ii. explain how we get the following equation for the total energyarrow_forwardProblem 9.4 For the 2D LHO with K1 = K2 show that and [ê, ²] = 2ihxy, (ê, p}] = -2ihxy Problem 9.5 It follows from the above that [ê., Ĥ] = 0 if K1 = K2 only Work out the equivalent commutator for ê and é, with the Hamiltonian. What do these mean?arrow_forward
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