Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 28: Problem 63

An electron is confined in a one-dimensional box of length \(L\). Another electron is confined in a box of length 2 \(L\). Both are in the ground state. What is the ratio of their energies $E_{2 l} / E_{L} ?$

Short Answer

Expert verified
Answer: The ratio of the energies of the electrons is \(E_{2L}/E_{L} = \frac{1}{4}\).

Step by step solution

01

Calculate the energy of the electron in a box of length L

Using the formula for the energy of a particle in a one-dimensional box, we can calculate the ground-state energy (\(E_1\)) of an electron in a box of length \(L\): $$ E_{L} = \frac{1^2 \hbar^2 \pi^2}{2mL^2} $$
02

Calculate the energy of the electron in a box of length 2L

Similarly, we can calculate the ground-state energy (\(E'_1\)) of an electron in a box of length \(2L\): $$ E_{2L} = \frac{1^2 \hbar^2 \pi^2}{2m(2L)^2} $$
03

Find the ratio of their energies \(E_{2L} / E_{L}\)

Dividing the energy of the electron in the box of length \(2L\) by the energy of the electron in the box of length \(L\), we get: $$ \frac{E_{2L}}{E_{L}} = \frac{\frac{1^2 \hbar^2 \pi^2}{2m(2L)^2}}{\frac{1^2 \hbar^2 \pi^2}{2mL^2}} $$
04

Simplify the expression

Simplifying the expression, we get: $$ \frac{E_{2L}}{E_{L}} = \frac{\frac{1^2 \hbar^2 \pi^2}{2m(2L)^2}}{\frac{1^2 \hbar^2 \pi^2}{2mL^2}} = \frac{1^2 \hbar^2 \pi^2 \cdot 2mL^2}{2m(2L)^2\cdot 1^2 \hbar^2 \pi^2} = \frac{1}{2^2} = \frac{1}{4} $$ Therefore, the ratio of the energies of the electrons is \(E_{2L}/E_{L} = \frac{1}{4}\).

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) Show that the ground-state energy of the hydrogen atom can be written \(E_{1}=-k e^{2} /\left(2 a_{0}\right),\) where \(a_{0}\) is the Bohr radius. (b) Explain why, according to classical physics, an electron with energy \(E_{1}\) could never be found at a distance greater than \(2 a_{0}\) from the nucleus.
What is the de Broglie wavelength of a basketball of mass \(0.50 \mathrm{kg}\) when it is moving at \(10 \mathrm{m} / \mathrm{s} ?\) Why don't we see diffraction effects when a basketball passes through the circular aperture of the hoop?
A light-emitting diode (LED) has the property that electrons can be excited into the conduction band by the electrical energy from a battery; a photon is emitted when the electron drops back to the valence band. (a) If the band gap for this diode is \(2.36 \mathrm{eV},\) what is the wavelength of the light emitted by the LED? (b) What color is the light emitted? (c) What is the minimum battery voltage required in the electrical circuit containing the diode?
An electron is confined to a one-dimensional box. When the electron makes a transition from its first excited state to the ground state, it emits a photon of energy 1.2 eV. (a) What is the ground-state energy (in electronvolts) of the electron? (b) List all energies (in electronvolts) of photons that could be emitted when the electron starts in its second excited state and makes transitions downward to the ground state either directly or through intervening states. Show all these transitions on an energy level diagram. (c) What is the length of the box (in nanometers)?
An electron is confined to a one-dimensional box of length \(L\) (a) Sketch the wave function for the third excited state. (b) What is the energy of the third excited state? (c) The potential energy can't really be infinite outside of the box. Suppose that \(U(x)=+U_{0}\) outside the box, where \(U_{0}\) is large but finite. Sketch the wave function for the third excited state of the electron in the finite box. (d) Is the energy of the third excited state for the finite box less than, greater than, or equal to the value calculated in part (b)? Explain your reasoning. [Hint: Compare the wavelengths inside the box.] (e) Give a rough estimate of the number of bound states for the electron in the finite box in terms of \(L\) and \(U_{0}\).
See all solutions

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Read Explanation

Fluids

Read Explanation

Rotational Dynamics

Read Explanation

Force

Read Explanation

Medical Physics

Read Explanation

Magnetism and Electromagnetic Induction

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