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, Problem 11.37P
(a)
To determine
The probability that the system is still in the upper state after the two measurements, and the probability if the first measurement was not taken.
(b)
To determine
The probability that the system is still in the upper state at time
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3. This problem to be handed in for grading by 11:59 pm, Fri. April 10th
The radial probability density, P(r), is defined such that the probability to find the electron
between spheres of radius r and r+ dr is given by P(r)dr. In spherical coordinates with a
radially symmetric potential P(r) = p2 Rn1(r)|* . Here the r² term is present because we are
working in spherical coordinates. See sections 7.3 and 7.4 of Krane.
(a) For an electron in the 2s state of the hydrogen atom, find the probability to find the
electron inside the proton. Assume that the proton is a sphere of radius R =
8.414 x 1016 m. Show your work. Radial wave functions can be found in Table
7.1 of Krane.
(b) For an electron in the 2p state of the hydrogen atom, find the probability to find the
electron inside the proton. Assume that the proton is a sphere of radius R
8.414 x 10-16 m. Show your work.
(c) Evaluate these two probabilities for muonic hydrogen.
12
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
4. Calculate the mean kinetic energy, , for a harmonic oscillator in the ground
1.3.5...(2n-1)
O/2 , and
00
state. Given that Sx2n exp(-ax²) dx =
00
2nan
L, exp(-ax²) dx = E)+/ .
00
(hv/4 or ½ Eo)
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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