Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 15

Calculate the leading term in the third-order perturbed energy of a hydrogen atom and a point charge \(Z e\) at a fixed distance \(R\), for large \(R\).

Short Answer

Expert verified
\( E^{(3)}_{lead} \propto \frac{Z^3 e^6}{R^3} \).

Step by step solution

01

- Identify the problem

Calculate the leading term in the third-order perturbed energy of a hydrogen atom interacted with a point charge \( Z e \) at a fixed distance \( R \). Consider the scenario for large \( R \).
02

- Write down the perturbation potential

The perturbation potential \( V \) between the hydrogen atom and the point charge \( Z e \) at a distance \( R \) is given by: \[ V = \frac{Ze^2}{R} \].
03

- Use third-order perturbation theory

The third-order correction to the energy, \( E^{(3)} \), can generally be written in terms of the matrix elements of the perturbation potential. Begin with the general formula for third-order perturbation energy in non-degenerate perturbation theory:\[ E^{(3)} = \sum_{k, l, m eq n} \frac{|\langle n| V | k \rangle \langle k| V | l \rangle \langle l| V | n \rangle|}{(E_0 - E_k)(E_0 - E_l)(E_0 - E_m)}. \]
04

- Consider large \( R \) simplification

For large \( R \), the electron in the hydrogen atom experiences the perturbation potential as weak. Therefore, the matrix elements of \( V \) can be expanded in terms of \( R \). Since \( V \propto \frac{1}{R} \), third-order energy correction terms will go as \( \frac{1}{R^3} \).
05

- Evaluate the leading term

Evaluate the leading term by noting that the third-order term, as derived from the third-order perturbation formula, can be approximated in the limit of large \( R \). Thus, the leading term in the third-order perturbed energy is:\[ E^{(3)}_{lead} \propto \frac{Z^3 e^6}{R^3}. \]

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

hydrogen atom
The hydrogen atom consists of a single proton and a single electron. It is the simplest atom and serves as a fundamental example in quantum mechanics.
The electron orbits the proton in quantized energy levels, described by the Schrödinger equation.
This results in discrete energy levels and a well-known spectrum.
  • The potential energy of the electron in the hydrogen atom is given by the Coulomb potential: \[ V(r) = -\frac{e^2}{4 \pi \epsilon_0 r}, \]
  • where \( e \) is the electron charge, and \( r \) is the distance from the proton.
  • The ground state energy of the hydrogen atom is \[ E_0 = -13.6 \text{eV}. \]
Understanding the hydrogen atom is crucial for exploring more complex atomic structures and interactions with external perturbations.
perturbation theory
Perturbation theory is a mathematical approach used in quantum mechanics to find an approximate solution to a system that cannot be solved exactly.
It examines how slight changes (perturbations) in the Hamiltonian affect the system's eigenvalues and eigenstates.
  • The perturbed Hamiltonian is \[ H = H_0 + V, \]
  • where \( H_0 \) is the unperturbed Hamiltonian, and \( V \) is the perturbation.
  • Perturbation theory is often used in power series, expanding the energy and wave functions as follows: \[ E = E_0 + E^{(1)} + E^{(2)} + E^{(3)} + ... \]

Here, \( E^{(n)} \) is the \( n \)th-order energy correction.
This allows us to approximate the effects of small interactions or disturbances on the system.
point charge interaction
A point charge interaction involves the interaction between a charged particle and another charge at a fixed location.
It is described by Coulomb's Law, which gives the potential energy between two point charges.
  • For charges \( Ze \) and \( e \), separated by distance \( R \), the interaction potential is: \[ V = \frac{Ze^2}{4 \pi \epsilon_0 R}. \]

In this problem, the hydrogen atom interacts with an external point charge at distance \( R \).
This leads to changes in the energy levels of the hydrogen atom, calculated using perturbation theory.
large distance approximation
The large distance approximation is employed when the distance \( R \) between interacting bodies is significantly large.
In such cases, interactions become weak, and perturbation terms become simpler to handle.
This approximation allows the expansion of the potential in terms of \( \frac{1}{R} \).
  • For large \( R \), the interaction potential \( V \) becomes small: \[ V \propto \frac{1}{R}. \]
  • Third-order correction terms then scale as: \[ E^{(3)} \propto \frac{1}{R^3}. \]

Hence, in the large distance limit, only the leading order terms need to be considered, simplifying calculations significantly.
This approximation is widely used in physics to handle cases where exact solutions are impractical.

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 system that has three unperturbed states can be represented by the perturbed hamiltonian matrix $$ \left[\begin{array}{lll} E_{1} & 0 & a \\ 0 & E_{1} & b \\ a^{*} & b^{*} & E_{2} \end{array}\right] $$where \(E_{2}>E_{1}\). The quantities \(a\) and \(b\) are to be regarded as perturbations that are of the same order and are small compared with \(E_{2}-E_{1} .\) Use the second-order nondegenerate perturbation theory to calculate the perturbed eigenvalues (is this procedure correct?). Then diagonalize the matrix to find the exact eigenvalues. Finally, use the second- order degenerate perturbation theory. Compare the three results obtained.

A one-dimensional harmonic oscillator is perturbed by an extra potential energy \(b x^{3} .\) Calculate the change in each energy level to second order in the perturbation.

Apply the WKB method to the one-dimensional motion of a particle of mass \(m\) in a potential that equals \(-V_{0}\) at \(x=0\), changes linearly with \(x\) until it vanishes at \(x=\pm a\), and is zero for \(|x|>a .\) Find all the bound energy levels obtained in this approximation if \(m V_{0} a^{2} / \hbar^{2}=40\).

A trial function \(\psi\) differs from an eigenfunction \(u_{E}\) by a small amount, so that \(\psi=u_{B}+\epsilon \psi_{1}\), where \(u_{E}\) and \(\psi_{1}\) are normalized and \(\epsilon \ll 1 .\) Show that \(\langle H\rangle\) differs from \(E\) only by a term of order \(\epsilon^{2}\), and find this term.

A one-dimensional harmonic oscillator of charge \(e\) is perturbed by an electric field of strength \(E\) in the positive \(x\) direction. Calculate the change in each energy level to second order in the perturbation, and calculate the induced electric dipole moment. Show that this problem can be solved exactly, and compare the result with the perturbation approximation. Repeat the calculation for a three-dimensional isotropic oscillator. Show that, if the polarizability \(\alpha\) of the oscillator is defined as the ratio of the induced electric dipole moment to \(E\), the change in energy is exactly \(-\frac{1}{2} \alpha E^{2}\).
See all solutions

Recommended explanations on Physics Textbooks

Nuclear Physics

Read Explanation

Physical Quantities And Units

Read Explanation

Waves Physics

Read Explanation

Measurements

Read Explanation

Medical Physics

Read Explanation

Solid State Physics

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