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 5, Problem 5.39P
(a)
To determine
Show that with periodic boundary conditions the wave vector satisfies
(b)
To determine
The expressions for
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Problem # 2.
In the two-level system, estimate the emission line full width at half maximum (FWHM) for
spontaneous emission at 650 nm if the spontaneous radiative lifetime of the upper state is about
3,000 nanoseconds.
Consider a single electron confined to a one-dimensional quantum well device of length L = 0.5 nm. The quantum well device acts as a “trap” for the electron
1.What are the boundary conditions for this system? Apply them to show that ψn(x) = Asin(nπx/L), n = 1,2,3,...
(check image)
2.Normalize the wave function to find the constant A.
3. Sketch ψ1, ψ2, and ψ3, as well as |ψ1|2, |ψ2|2, and |ψ3|2, and evaluate the energy levels E1, E2, and E3 in eV.
4. Suppose the particle is in the first excited state. What is the probability of finding the particle between x = L/4 and x = 3L/4?
5. Suppose, instead of one electron, we trap five electrons in the quantum well. Draw an energy-level diagram to show the electron configuration of the ground state. What is the ground state energy?
2. We consider the harmonic oscillator in one dimension as discussed in section 1.3.2. The time-independent Schrödinger
equation is
h² d² √(x)
2m dr2
+ ½ mw²¹² y(x) = Ev(x).
You are given
1
-x²/(2a²),
Vo(x) =
a√T
1
e-r²/(24²)
(2 a√F (Za)
h
with a =
m.w
It is useful to use the results from question 1 below.
(a) Show that
(r) is correctly normalised.
(b) Show that
(r) is correctly normalised.
(c) Show that
o(a) solves the Schrödinger equation, with E= Shw.
(d) Show that ₁ (r) solves the Schrödinger equation, with E-hw.
=
(e) Show that fx dx (x) ₁(x) = 0, i.e. the different eigenfunctions are orthogonal.
(f) Plot o(r) and ₁(r).
e
Chapter 5 Solutions
Introduction To Quantum Mechanics
Ch. 5.1 - Prob. 5.1PCh. 5.1 - Prob. 5.2PCh. 5.1 - Prob. 5.3PCh. 5.1 - Prob. 5.4PCh. 5.1 - Prob. 5.5PCh. 5.1 - Prob. 5.6PCh. 5.1 - Prob. 5.8PCh. 5.1 - Prob. 5.9PCh. 5.1 - Prob. 5.10PCh. 5.1 - Prob. 5.11P
Ch. 5.2 - Prob. 5.12PCh. 5.2 - Prob. 5.13PCh. 5.2 - Prob. 5.14PCh. 5.2 - Prob. 5.15PCh. 5.2 - Prob. 5.16PCh. 5.2 - Prob. 5.17PCh. 5.2 - Prob. 5.18PCh. 5.2 - Prob. 5.19PCh. 5.3 - Prob. 5.20PCh. 5.3 - Prob. 5.21PCh. 5.3 - Prob. 5.22PCh. 5.3 - Prob. 5.23PCh. 5.3 - Prob. 5.24PCh. 5.3 - Prob. 5.25PCh. 5.3 - Prob. 5.26PCh. 5.3 - Prob. 5.27PCh. 5 - Prob. 5.29PCh. 5 - Prob. 5.30PCh. 5 - Prob. 5.31PCh. 5 - Prob. 5.32PCh. 5 - Prob. 5.33PCh. 5 - Prob. 5.34PCh. 5 - Prob. 5.35PCh. 5 - Prob. 5.36PCh. 5 - Prob. 5.38PCh. 5 - Prob. 5.39P
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