Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 15

Calculate the second-order energy correction to the ground state of a particle in a one-dimensional box for a perturbation of the form \(H^{(1)}=-\varepsilon \sin (\pi x / L)\) by using the closure approximation. Infer a value of \(\Delta E\) by comparison with the numerical calculation in Example \(6.4 .\) These two problems \((6.14 \text { and } 6.15)\) show that the parameter \(\Delta E\) depends on the perturbation and is not simply a characteristic of the system itself.

Short Answer

Expert verified
The process involves calculating the second-order energy correction, which in this case comes from the interaction between the particle's ground state and the perturbation. The difference energy \(\Delta E\) is then found by comparing this with numerically calculated values. The solution confirms that \(\Delta E\) depends on the form of the perturbation, underlining the insights from perturbation theory in quantum mechanics.

Step by step solution

01

Set the problem

First, establish the properties of the system. In this case, we have a particle in a one-dimensional box. The box has length L. The perturbation to the system is given by \(H^{(1)}=-\varepsilon \sin (\pi x / L)\). The ground state of the system is given by |n>=|1>, with the unperturbed Hamiltonian \(H^{(0)}|n>=E_n^{(0)}|n>\), where \(E_{n}^{(0)}=n^{2} \frac {\pi^{2} \hbar^{2}} {2mL^{2}}\).
02

Compute the first-order Energy Correction

First, note that since \(H^{(1)}\) is independent of time, we can use the time-independent perturbation theory. The first-order energy correction for the state \(|1>\) is given by \(E_{1}^{(1)}=<1|H^{(1)}|1>\). Since the first order perturbation vanishes, there is no first order change in energy.
03

Compute the second-order Energy Correction

The second-order correction to the energy of the ground state is given by \(E_{1}^{(2)}=\sum _{n\neq1} \frac{||^{2}}{E_{1}^{(0)} - E_{n}^{(0)}}\). Computing this expression requires calculating the matrix element \, where n is an integer other than 1. Indeed, applying integral techniques and the orthogonality of the sine functions will yield the expression for the second order correction.
04

Calculate the Delta E value

The value of \(\Delta E\) can be obtained by comparing the \(E_{1}^{(2)}\) from this example with that obtained in a numerical calculation provided in Example 6.4. The comparison will give a sense of how these two problems are related and how the parameter \(\Delta E\) is dependent on the perturbation, not just the system itself.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A hydrogen atom in a \(2 \mathrm{s}^{1}\) configuration passes into a region where it experiences an electric field in the \(z\) -direction for a time \(\tau .\) What is its electric dipole moment during its exposure and after it emerges? Hint. Use eqn 6.62 with \(\omega_{21}=0 ;\) the dipole moment is the expectation value of \(-e z\) \\[\text { use } \int \psi_{2 \mathrm{s}} z \psi_{2 \mathrm{p}} \mathrm{d} \tau=3 a_{0}\\]

Show group-theoretically that when a perturbation of the form \(H^{(1)}=a z\) is applied to a hydrogen atom, the 1 s-orbital is contaminated by the admixture of \(n \mathrm{p}_{z^{-}}\) orbitals. Deduce which orbitals mix into (a) \(2 p_{x}\) -orbitals, (b) \(2 \mathrm{p}_{z}\) -orbitals (c) \(3 d_{x y}\) -orbitals.

Repeat the last problem but set \(H_{\mathrm{s}, \mathrm{s}}=\gamma\) and \(S_{\mathrm{s}^{\prime}} \neq 0\) Evaluate the overlap integrals between 1 s-orbitals on centres separated by \(R ;\) use $$S=\left\\{1+\frac{R}{a_{0}}+\frac{1}{3}\left(\frac{R}{a_{0}}\right)^{2}\right\\} \mathrm{e}^{-R / a_{0}}$$ Suppose that \(\beta / \gamma=S_{\mathrm{s}, \mathrm{s}_{2}} / S_{\mathrm{s}, \mathrm{s}} .\) For a numerical result, take \(R=80 \mathrm{pm}, a_{0}=53 \mathrm{pm}\)

\(\mathrm{A}\) biradical is prepared with its two electrons in a singlet state. A magnetic field is present, and because the two electrons are in different environments their interaction with the field is \(\left(\mu_{\mathrm{B}} / \hbar\right) \mathscr{B}\left(g_{1} s_{1 z}+g_{2} s_{2 z}\right)\) with \(g_{1} \neq g_{2} .\) Evaluate the time-dependence of the probability that the electron spins will acquire a triplet configuration (that is, the probability that the \(S=1, M_{s}=0\) state will be populated). Examine the role of the energy separation \(h J\) of the singlet state and the \(M_{S}=0\) state of the triplet. Suppose \(g_{1}-g_{2}\) \(\approx 1 \times 10^{-3}\) and \(J \approx 0 ;\) how long does it take for the triplet state to emerge when \(\mathscr{B}=1.0 \mathrm{T}\) ? Hint. Use eqn 6.63 ; take \(|0,0\rangle=\left(1 / 2^{1 / 2}\right)(\alpha \beta-\beta \alpha)\) and \(|1,0\rangle=\left(1 / 2^{1 / 2}\right)(\alpha \beta+\beta \alpha) .\) See Problem 4.26 for the significance of \(\mu_{\mathrm{B}}\) and \(g\)

Find the complete dependence of the \(A\) and \(B\) coefficients on atomic number for the \(2 \mathrm{p} \rightarrow 1\) s transitions of hydrogenic atoms. Calculate how the stimulated emission rate depends on \(Z\) when the atom is exposed to black-body radiation at \(1000 \mathrm{K} .\) Hint. The relevant density of states also depends on \(Z\).
See all solutions

Recommended explanations on Chemistry Textbooks

Nuclear Chemistry

Read Explanation

Inorganic Chemistry

Read Explanation

Kinetics

Read Explanation

Making Measurements

Read Explanation

Chemical Reactions

Read Explanation

Chemical Analysis

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free