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Chapter 28: Problem 32

The particle in a box model is often used to make rough estimates of ground- state energies. Suppose that you have a neutron confined to a one-dimensional box of length equal to a nuclear diameter (say \(10^{-14} \mathrm{m}\) ). What is the ground-state energy of the confined neutron?

Short Answer

Expert verified
Answer: The ground-state energy of the confined neutron is approximately \(9.813 \times 10^{-14} \mathrm{J}\).

Step by step solution

01

List the given values

We know the following values: - Length of the box (\(L\)): \(10^{-14} \mathrm{m}\) - Planck constant (\(h\)): \(6.626 \times 10^{-34} \mathrm{J \cdot s}\) - Mass of the neutron (\(m\)): \(1.675 \times 10^{-27} \mathrm{kg}\) - Ground state energy level (\(n\)): 1
02

Plug the values into the ground-state energy formula

Now, we need to plug the given values into the ground-state energy formula: \(E_n = \dfrac{n^2 \cdot h^2}{8 \cdot m \cdot L^2}\)
03

Calculate the ground-state energy

Substitute the values of \(n\), \(h\), \(m\), and \(L\) into the equation: \(E_1 = \dfrac{1^2 \cdot (6.626 \times 10^{-34} \mathrm{J \cdot s})^2}{8 \cdot (1.675 \times 10^{-27} \mathrm{kg}) \cdot (10^{-14} \mathrm{m})^2}\) Now, calculate the ground-state energy: \(E_1 = \dfrac{(6.626 \times 10^{-34})^2}{8 \cdot (1.675 \times 10^{-27}) \cdot (10^{-14})^2} \mathrm{J}\) \(E_1 = 9.813 \times 10^{-14} \mathrm{J}\) So, the ground-state energy of the confined neutron is approximately \(9.813 \times 10^{-14} \mathrm{J}\).

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Most popular questions from this chapter

A free neutron (that is, a neutron on its own rather than in a nucleus) is not a stable particle. Its average lifetime is 15 min, after which it decays into a proton, an electron, and an antineutrino. Use the energy-time uncertainty principle \([\mathrm{Eq} .(28-3)]\) and the relationship between mass and rest energy to estimate the inherent uncertainty in the mass of a free neutron. Compare with the average neutron mass of \(1.67 \times 10^{-27} \mathrm{kg} .\) (Although the uncertainty in the neutron's mass is far too small to be measured, unstable particles with extremely short lifetimes have marked variation in their measured masses.)
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What are the de Broglie wavelengths of electrons with the following values of kinetic energy? (a) \(1.0 \mathrm{eV}\) (b) $1.0 \mathrm{keV} .
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