/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 An electron is confined in a one... [FREE SOLUTION] | 91影视

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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 \(E_{2L}/E_{L} = \frac{1}{4}\).

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Most popular questions from this chapter

Before the discovery of the neutron, one theory of the nucleus proposed that the nucleus contains protons and electrons. For example, the helium-4 nucleus would contain 4 protons and 2 electrons instead of - as we now know to be true- 2 protons and 2 neutrons. (a) Assuming that the electron moves at nonrelativistic speeds, find the ground-state energy in mega-electron- volts of an electron confined to a one-dimensional box of length $5.0 \mathrm{fm}\( (the approximate diameter of the \)^{4} \mathrm{He}$ nucleus). (The electron actually does move at relativistic speeds. See Problem \(80 .)\) (b) What can you conclude about the electron-proton model of the nucleus? The binding energy of the \(^{4} \mathrm{He}\) nucleus - the energy that would have to be supplied to break the nucleus into its constituent particles-is about \(28 \mathrm{MeV} .\) (c) Repeat (a) for a neutron confined to the nucleus (instead of an electron). Compare your result with (a) and comment on the viability of the proton-neutron theory relative to the electron-proton theory.
In a ruby laser, laser light of wavelength \(694.3 \mathrm{nm}\) is emitted. The ruby crystal is \(6.00 \mathrm{cm}\) long, and the index of refraction of ruby is \(1.75 .\) Think of the light in the ruby crystal as a standing wave along the length of the crystal. How many wavelengths fit in the crystal? (Standing waves in the crystal help to reduce the range of wavelengths in the beam.)
(a) What are the electron configurations of the ground states of lithium \((Z=3),\) sodium \((Z=11),\) and potassium \((Z=19) ?\) (b) Why are these elements placed in the same column of the periodic table?
What are the possible values of \(L_{z}\) (the component of angular momentum along the \(z\) -axis) for the electron in the second excited state \((n=3)\) of the hydrogen atom?
An electron confined to a one-dimensional box has a ground-state energy of \(40.0 \mathrm{eV} .\) (a) If the electron makes a transition from its first excited state to the ground state, what is the wavelength of the emitted photon? (b) If the box were somehow made twice as long, how would the photon's energy change for the same transition (first excited state to ground state)?
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