Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 4, Problem 4.75P
To determine
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Assume 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.
dt
Problem 3.7
(a) Suppose that f(x) and g(x) are two eigenfunctions of an operator Q, with
the same eigenvalue q. Show that any linear combination of f and g is itself
an eigenfunction of Q. with eigenvalue q.
(b) Check that f(x) = exp(x) and g(x) = exp(-x) are eigenfunctions of the
operator d?/dx², with the same eigenvalue. Construct two linear combina-
tions of f and g that are orthogonal eigenfunctions on the interval (-1, 1).
Prove that the vector V(x y .z)= 3y*z*i+ 4x°z²j+3x*y°(-k) is solenoidal.
k
4.2 Show that the vector field V(x, y,z)= o is irrotational using Vx V in cylindrical coordinates.
Chapter 4 Solutions
Introduction To Quantum Mechanics
Ch. 4.1 - Prob. 4.1PCh. 4.1 - Prob. 4.3PCh. 4.1 - Prob. 4.4PCh. 4.1 - Prob. 4.5PCh. 4.1 - Prob. 4.6PCh. 4.1 - Prob. 4.7PCh. 4.1 - Prob. 4.8PCh. 4.1 - Prob. 4.9PCh. 4.1 - Prob. 4.10PCh. 4.1 - Prob. 4.11P
Ch. 4.2 - Prob. 4.12PCh. 4.2 - Prob. 4.13PCh. 4.2 - Prob. 4.14PCh. 4.2 - Prob. 4.15PCh. 4.2 - Prob. 4.16PCh. 4.2 - Prob. 4.17PCh. 4.2 - Prob. 4.18PCh. 4.2 - Prob. 4.19PCh. 4.2 - Prob. 4.20PCh. 4.3 - Prob. 4.21PCh. 4.3 - Prob. 4.22PCh. 4.3 - Prob. 4.23PCh. 4.3 - Prob. 4.24PCh. 4.3 - Prob. 4.25PCh. 4.3 - Prob. 4.26PCh. 4.3 - Prob. 4.27PCh. 4.4 - Prob. 4.28PCh. 4.4 - Prob. 4.29PCh. 4.4 - Prob. 4.30PCh. 4.4 - Prob. 4.31PCh. 4.4 - Prob. 4.32PCh. 4.4 - Prob. 4.33PCh. 4.4 - Prob. 4.34PCh. 4.4 - Prob. 4.35PCh. 4.4 - Prob. 4.36PCh. 4.4 - Prob. 4.37PCh. 4.4 - Prob. 4.38PCh. 4.4 - Prob. 4.39PCh. 4.4 - Prob. 4.40PCh. 4.4 - Prob. 4.41PCh. 4.5 - Prob. 4.42PCh. 4.5 - Prob. 4.43PCh. 4.5 - Prob. 4.44PCh. 4.5 - Prob. 4.45PCh. 4 - Prob. 4.46PCh. 4 - Prob. 4.47PCh. 4 - Prob. 4.48PCh. 4 - Prob. 4.49PCh. 4 - Prob. 4.50PCh. 4 - Prob. 4.51PCh. 4 - Prob. 4.52PCh. 4 - Prob. 4.53PCh. 4 - Prob. 4.54PCh. 4 - Prob. 4.55PCh. 4 - Prob. 4.56PCh. 4 - Prob. 4.57PCh. 4 - Prob. 4.58PCh. 4 - Prob. 4.59PCh. 4 - Prob. 4.61PCh. 4 - Prob. 4.62PCh. 4 - Prob. 4.63PCh. 4 - Prob. 4.64PCh. 4 - Prob. 4.65PCh. 4 - Prob. 4.66PCh. 4 - Prob. 4.70PCh. 4 - Prob. 4.72PCh. 4 - Prob. 4.73PCh. 4 - Prob. 4.75PCh. 4 - Prob. 4.76P
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