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Chapter 28: Problem 28

An electron is confined to a box of length \(1.0 \mathrm{nm} .\) What is the magnitude of its momentum in the \(n=4\) state?

Short Answer

Expert verified
Answer: The magnitude of the momentum of the electron in the n=4 state is \(1.33 \times 10^{-24} \ \mathrm{kg\: m/s}\).

Step by step solution

01

Find the wavelength

Using the quantum number concept, if n=4, there will be four half wavelengths within the box. Therefore, the wavelength of the electron can be determined by the length of the box and the number of half wavelengths: \(\lambda = \frac{2L}{n}\) where n = 4 and L is the length of the box, which is 1.0 nm. Plug in the values and calculate the wavelength: \(\lambda = \frac{2 * 1.0}{4} = 0.5 \ \mathrm{nm}\)
02

Convert wavelength to meters

Since we need to find the momentum in SI units, we need to convert the wavelength from nm to meters: \(\lambda = 0.5 \ \mathrm{nm} * 10^{-9} \ \frac{\mathrm{m}}{\mathrm{nm}} = 0.5 * 10^{-9} \ \mathrm{m}\)
03

Calculate the momentum

Now, we can use the de Broglie wavelength formula to find the momentum: \(p = \frac{h}{\lambda}\) where \(h\) is the Planck's constant, \(6.63 \times 10^{-34} \ \mathrm{Js}\), and \(\lambda\) is the wavelength in meters. Plug in the values and calculate the momentum: \(p = \frac{6.63 \times 10^{-34}}{0.5 * 10^{-9}} = 1.33 \times 10^{-24} \ \mathrm{kg\: m/s}\) So, the magnitude of the momentum of the electron in the n=4 state is \(1.33 \times 10^{-24} \ \mathrm{kg\: m/s}\).

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Most popular questions from this chapter

If diffraction were the only limitation on resolution, what would be the smallest structure that could be resolved in an electron microscope using 10-keV electrons?
The particle in a box model is often used to make rough estimates of energy level spacings. Suppose that you have a proton confined to a one-dimensional box of length equal to a nuclear diameter (about \(10^{-14} \mathrm{m}\) ). (a) What is the energy difference between the first excited state and the ground state of this proton in the box? (b) If this energy is emitted as a photon as the excited proton falls back to the ground state, what is the wavelength and frequency of the electromagnetic wave emitted? In what part of the spectrum does it lie? (c) Sketch the wave function \(\psi\) as a function of position for the proton in this box for the ground state and each of the first three excited states.
List the number of electron states in each of the subshells in the \(n=7\) shell. What is the total number of electron states in this shell?
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.
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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