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Chapter 35: Problem 36

The ground-state energy for an electron in infinite square well \(A\) is equal to the energy of the first excited state for an electron in well B. How do the wells' widths compare?

Short Answer

Expert verified
The width of well B is twice as wide as the width of well A.

Step by step solution

01

Understanding the energy levels

The ground state energy (the smallest non-zero energy) of an electron in a potential well is achieved at quantum number \( n = 1 \). So, for well \( A \), we can use the energy formula for quantum mechanical system: \( E_A = (1^2 h^2) / (8mA^2) \). Similarly, the first excited state (the second smallest energy) of an electron in a potential well is achieved at quantum number \( n = 2 \). So, for well \( B \), we can again use the energy formula for quantum mechanical system: \( E_B = (2^2 h^2) / (8mB^2) \).
02

Setting up the equation

Given that the energy for ground state of well \( A \) is equal to the first excited state of well \( B \), we can set the two energies equal: \( (1^2 h^2) / (8mA^2) = (2^2 h^2) / (8mB^2) \).
03

Simplifying the equation

Notice that on both sides of the equation, \( h^2 / 8m \) cancels out, leaving us with \( 1/A^2 = 4/B^2 \).
04

Solving for the widths

Taking the square root of both sides of the equation gives us the relationship between the widths of the two wells: \( 1/A = 2/B \), which can be simplified to \( B = 2A \).

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Most popular questions from this chapter

An electron is in a narrow molecule 4.4 nm long. a situation that approximates a one-dimensional infinite square well. If the electron is in its ground state, what is the maximum wavelength of electromagnetic radiation that can cause a transition to an excited state?

What's the essential difference between the energy-level structures of infinite and finite square wells?

The ground-state wave function for a quantum harmonic oscillator has a single central peak. Why is this at odds with classical physics?

If the dot behaves as a perfectly cubical 3 -D square well, the first excited state is a. non degenerate. b. twofold degenerate. c. threefold degenerate. d. You can't tell without knowing the energy.

In terms of de Broglie's matter-wave hypothesis, how does making the sides of a box different lengths remove the degeneracy associated with a particle confined to that box?
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