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 4.5, Problem 4.44P
To determine
The proof that the wave function given by equation 4.197 satisfies Schrodinger’s equation.
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4.10 A particle is bound in a one-dimensional potential V(x), where
V(x) is symmetric, i.e., V(x) = V(−x).
= −x,
(a) Suppose that (x) is a solution of the Schrödinger equation
with energy E. Make the change of variables y
and show
that (y) is also a solution of the Schrödinger equation with energy
E.
(b) Since the solutions of the Schrödinger equation for a fixed value
of E are unique (up to multiplication by a constant), the result
from part (a) implies that √(x) = c(-x), where c is an unknown
constant. Use this result to show that (x) must be either even
[(x) = (x)] or odd [(-x) = v(x)].
(c) For a particle bound in a one-dimensional symmetric potential,
so that V(x) = V (x), show that all of the following are true:
(i) * is a symmetric function,
(ii) (x) = 0,
(iii) (p)
= 0.
I have an electron that I want to put in a rigid box. How small do I need to make the box so that the speed of my electron in its ground state inside the box will be equal to the speed of light? Include a sketch of U(x) and ?(x).
Sketch the situation, defining all your variables.
Consider the function
v(1,2) =(
[1s(1) 3s(2) + 3s(1) 1s(2)]
[x(1) B(2) + B(1) a(2)]
Which of the following statements is incorrect concerning p(1,2) ?
a.
W(1,2) is normalized.
Ob.
The function W(1,2) is symmetric with respect to the exchange of the space and the spin coordinates of the two electrons.
OC.
y(1,2) is an eigenfunction of the reference (or zero-order) Hamiltonian (in which the electron-electron repulsion term is ignored) of Li with
eigenvalue = -5 hartree.
d.
The function y(1,2) is an acceptable wave function to describe the properties of one of the excited states of Lit.
Oe.
The function 4(1,2) is an eigenfunction of the operator S,(1,2) = S;(1) + S,(2) with eigenvalue zero.
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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- Problem 2.2 Show that E must exceed the minimum value of V (x), for every normalizable solution to the time-independent Schrödinger equation. What is the classical analog to this statement? Hint: Rewrite Equation 2.5 in the form d² 2m [V(x) - E]; dx² if E < Vmin, then and its second derivative always have the same sign-argue that such a function cannot be normalized. h² d² 2m dx² + Vy = Ev. (2.5)arrow_forwardProblem 3.36. Consider an Einstein solid for which both N and q are much greater than 1. Think of each oscillator as a separate "particle." (a) Show that the chemical potential is N+ - kT ln N (b) Discuss this result in the limits N > q and N « q, concentrating on the question of how much S increases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?arrow_forwardThe following problem arises in quantum mechanics (see Chapter 13, Problem 7.21). Find the number of ordered triples of nonnegative integers a, b, c whose sum a+b+c is a given positive integer n. (For example, if n = 2, we could have (a, b, c) = (2, 0, 0) or (0, 2, 0) or (0, 0, 2) or (0, 1, 1) or (1, 0, 1) or (1, 1, 0).) Hint: Show that this is the same as the number of distinguishable distributions of n identical balls in 3 boxes, and follow the method of the diagram in Example 5.arrow_forward
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