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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 2: 2.14P (page 51)

A particle is in the ground state of the harmonic oscillator with classical frequency $蝇$ , when suddenly the spring constant quadruples, so $蝇' = 2 蝇$ , without initially changing the wave function (of course, $唯$ will now evolve differently, because the Hamiltonian has changed). What is the probability that a measurement of the energy would still return the value $\frac{鈩� 蝇}{2}$ ? What is the probability of getting $鈩� 蝇$ ?

Short Answer

Expert verified

The probability of getting $\frac{1}{2} 鈩� 蝇$ is zero, and the probability of $鈩� 蝇$ is 0.943.

Step by step solution

01

The given values of the question

The particle is in ground state and its classical frequency is $蝇$ . when the spring becomes entangled in a never-ending tangle $蝇' = 2 蝇$ .

02

The probabilities of ℏω2 and ℏω

The new allowed energies are ${E_{n}}^{'} = (n + \frac{1}{2}) 鈩� 蝇^{'} = 2 (n + \frac{1}{2}) 蝇 = 鈩� 蝇, 3 鈩� 蝇, 5 鈩� 蝇, . . . .$

so the probability of getting $\frac{1}{2} 鈩� 蝇$ is zero.

The probability of getting $鈩� 蝇$ is $P_{0} = {|c_{0}|}^{2}$ , where $c_{0} = 鈭� 唯 (x, 0) 蠄_{0}^{'} d x$ with

Substitute the known values in $P_{0} = {|c_{0}|}^{2}$

Thus, the probability of $鈩� 蝇$ is 0.943.

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Most popular questions from this chapter

This is a strictly qualitative problem-no calculations allowed! Consider the "double square well" potential (Figure 2.21). Suppose the depth $V_{0}$ and the width a are fixed, and large enough so that several bound states occur.

(a) Sketch the ground state wave function $唯_{1}$ and the first excited state localid="1658211858701" $唯_{2}$ (i) for the case b = 0 (ii) for $b 鈮� a$ and (iii) for $b 鈮� a$

(b) Qualitatively, how do the corresponding energies $(E_{1} and E_{2})$ and vary, as b goes from 0 to ? Sketch $E_{1} (b)$ and $E_{2} (b)$ on the same graph.

(c) The double well is a very primitive one-dimensional model for the potential experienced by an electron in a diatomic molecule (the two wells represent the attractive force of the nuclei). If the nuclei are free to move, they will adopt the configuration of minimum energy. In view of your conclusions in (b), does the electron tend to draw the nuclei together, or push them apart? (Of course, there is also the internuclear repulsion to consider, but that's a separate problem.)

Show that $[A e^{i k x} + B e^{- i k x}]$ and $[C c o s (k x) + D s i n (k x)]$ are equivalent ways of writing the same function of $x$ , and determine the constants $C$ and $D$ in terms of $A$ and $B$ , and vice versa.

A particle in the infinite square well (Equation 2.22) has the initial wave function 唯 (x, 0) = A sin3(蟺x/a) (0 鈮� x 鈮� a). Determine A, find 唯(x, t), and calculate 銆坸銆塧s a function of time. What is the expectation value of the energy? Hint: sinn胃 and cosn胃 can be reduced, by repeated application of the
trigonometric sum formulas, to linear combinations of sin(m胃) and cos(m胃), with m = 0, 1, 2, . . ., n.

Find $鉄� x 鉄�, 鉄� p 鉄�, <x^{2}), <p^{2}>, 鉄� T 鉄�$ , for the nth stationary state of the harmonic oscillator, using the method of Example $2.5$ . Check that the uncertainty principle is satisfied.

Consider the double delta-function potential $V (x) = - 伪 [未 (x + a) + 未 (x - a)]$ Whereand are positive constants

(a) Sketch this potential.

(b) How many bound states does it possess? Find the allowed energies, for $伪 = 魔 / m a$ and for $伪 = 魔^{2} / 4 m a$ , and sketch the wave functions.

See all solutions

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