/*! 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 32 The particle in a box model is o... [FREE SOLUTION] | 91影视

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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) Find the magnitude of the angular momentum \(\overrightarrow{\mathbf{L}}\) for an electron with \(n=2\) and \(\ell=1\) in terms of \(\hbar .\) (b) What are the allowed values for \(L_{2} ?\) (c) What are the angles between the positive z-axis and \(\overline{\mathbf{L}}\) so that the quantized components, \(L_{z},\) have allowed values?
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) Make a qualitative sketch of the wave function for the \(n=5\) state of an electron in a finite box \([U(x)=0\) for \(00\) elsewherel. (b) If \(L=1.0 \mathrm{nm}\) and \(U_{0}=1.0\) keV, estimate the number of bound states that exist.
An electron in an atom has an angular momentum quantum number of \(2 .\) (a) What is the magnitude of the angular momentum of this electron in terms of \(\hbar ?\) (b) What are the possible values for the \(z\) -components of this electron's angular momentum? (c) Draw a diagram showing possible orientations of the angular momentum vector \(\overrightarrow{\mathbf{L}}\) relative to the z-axis. Indicate the angles with respect to the z-axis.
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.
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