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 8, Problem 8.23P
(a)
To determine
The matrix elements of H’ and h value.
(b)
To determine
The new ground state energy in second-order perturbation theory.
(c)
To determine
The variational principle bound on the ground state energy.
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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]
8.8** Two masses m₁ and m₂ move in a plane and interact by a potential energy U (r) = kr². Write
down their Lagrangian in terms of the CM and relative positions R and r, and find the equations of
motion for the coordinates X, Y and x, y. Describe the motion and find the frequency of the relative
motion.
7.10 A particle of charge e moves in a central potential V(r) superimposed onto a uniform
magnetic field B whose vector potential is A = B × r/2. (a) Show that if B is a weak
field, so that effects proportional to B² can be neglected, then
where
H(r, p):
p²
2m
+ V(r) − µ · B,
e
2mc
(b) Write down Hamilton's equations in this approximation.
μ =
-rxp.
Chapter 8 Solutions
Introduction To Quantum Mechanics
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- Part barrow_forwardnumber 6 a b c pleasearrow_forward8.8 ** Two masses m, and m, move in a plane and interact by a potential energy U(r) = Įkr?, Write down their Lagrangian in terms of the CM and relative positions R and r, and find the equations of motion for the coordinates X, Y and x, y. Describe the motion and find the frequency of the relative motion.arrow_forward
- For Problem 8.35, how do I prove, or perhaps verify, what it is they're asking for?arrow_forwardDerive Eq.(7.13) by using set of equations (7.2 a,b,c and d).arrow_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
- 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…arrow_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_forwardAssume MKS units... Let Q be an open subset of R³. Let B: :Q - R³ be a continuous vector .field, representing a magnetic field in 3-D space. 7 Let P be a particle with charge q E R and mass m > 0. If p is at position (x. y, z) in Q and R³ is the velocity of p, at time t, then p feels a force 7(7,7) given by - 7(7,J) := q V × B (7) . Suppose that p moves along a curve C as time t varies from a to b, and that p has position vector (t) and instantaneous velocity (t) at time t. ř (1) Explain why the two vectors 7'(t) × È(7(t)) and 7'(t) are perpen- dicular at every time t = [a, b]. (2) Using Part (1), calculate W := the work done on the particle p by the force as p moves from D = 7(a) to E = √ (b) along C. F (3) Prove that ((t)||²)=27' (t) • F(t), at each time t. (4) Using Parts (2) and (3), and Newton's Second Law, prove that if the magnetic force - ₹(7,7) is the total force on p at every time t, then p moves along C at a constant speed. dtarrow_forward
- Consider two particles m1 and m2. Let mlbe confined to move on a circle of radius R1 in the z-0 plane and centered at x-0, y-0. Let m2 be confined to move on a circle of radius R2 in the z-a plane and centered at x-0, y-0. A massless spring of spring constant C is joining the two particles. www Set up the Lagrangian for the system. Set up the equations of constraints. Set up the Lagrange equations using Lagrange multipliers.arrow_forwardFor Problem 8.16, how do I prove the relations and give the correct expressions?arrow_forwardPart Barrow_forward
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