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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

An electron in a one-dimensional box has ground-state energy 0.010 eV. (a) What is the length of the box? (b) Sketch the wave functions for the lowest three energy states of the electron. (c) What is the wavelength of the electron in its second excited state \((n=3) ?\) (d) The electron is in its ground state when it absorbs a photon of wavelength \(15.5 \mu \mathrm{m}\). Find the wavelengths of the photon(s) that could be emitted by the electron subsequently.
In the Davisson-Germer experiment (Section \(28.2),\) the electrons were accelerated through a \(54.0-\mathrm{V}\) potential difference before striking the target. (a) Find the de Broglie wavelength of the electrons. (b) Bragg plane spacings for nickel were known at the time; they had been determined through x-ray diffraction studies. The largest plane spacing (which gives the largest intensity diffraction maxima) in nickel is \(0.091 \mathrm{nm} .\) Using Bragg's law [Eq. ( \(25-15\) )], find the Bragg angle for the first-order maximum using the de Broglie wavelength of the electrons. (c) Does this agree with the observed maximum at a scattering angle of \(130^{\circ} ?\) [Hint: The scattering angle and the Bragg angle are not the same. Make a sketch to show the relationship between the two angles.]
A beam of neutrons is used to study molecular structure through a series of diffraction experiments. A beam of neutrons with a wide range of de Broglie wavelengths comes from the core of a nuclear reactor. In a time-offlight technique, used to select neutrons with a small range of de Broglie wavelengths, a pulse of neutrons is allowed to escape from the reactor by opening a shutter very briefly. At a distance of \(16.4 \mathrm{m}\) downstream, a second shutter is opened very briefly 13.0 ms after the first shutter. (a) What is the speed of the neutrons selected? (b) What is the de Broglie wavelength of the neutrons? (c) If each shutter is open for 0.45 ms, estimate the range of de Broglie wavelengths selected.
How many electron states of the H atom have the quantum numbers \(n=3\) and \(\ell=1 ?\) Identify each state by listing its quantum numbers.
(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.
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