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Chapter 28: Problem 5

What is the de Broglie wavelength of an electron moving at speed $\frac{3}{5} c ?$

Short Answer

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Question: Determine the de Broglie wavelength of an electron moving at a speed of \(\frac{3}{5}c\), where \(c\) is the speed of light. Answer: The de Broglie wavelength of the electron moving at the given speed is approximately \(4.23 \times 10^{-11} \,\text{m}\).

Step by step solution

01

Write down the given values and constants

The given values and constants are: - Electron velocity, \(v = \frac{3}{5}c\) - Speed of light, \(c = 3.00 \times 10^8 \,\text{m/s}\) - Rest mass of electron, \(m_0 = 9.11 \times 10^{-31} \text{kg}\) - Planck's constant, \(h = 6.63 \times 10^{-34} \text{J s}\)
02

Calculate the momentum of the electron

To find the momentum of the electron, we'll use the relativistic momentum formula: \(p = \frac{m_0v}{\sqrt{1 - \frac{v^2}{c^2}}}\) Substitute \(v = \frac{3}{5}c\) and other given constants into the formula: \(p = \frac{(9.11 \times 10^{-31} \text{kg})(\frac{3}{5})(3.00 \times 10^8 \text{m/s})}{\sqrt{1 - \frac{(\frac{3}{5}c)^2}{c^2}}}\) Simplify the equation by canceling out \(c\) in the denominator of the fraction: \(p = \frac{(9.11 \times 10^{-31} \text{kg})(\frac{3}{5})(3.00 \times 10^8 \text{m/s})}{\sqrt{1 - (\frac{3}{5})^2}}\) Now, evaluate the expression to find the momentum: \(p \approx 1.566 \times 10^{-23} \,\text{kg m/s}\)
03

Calculate the de Broglie wavelength

Now, use the formula for the de Broglie wavelength and substitute the momentum: \(\lambda = \frac{h}{p}\) \(\lambda = \frac{6.63 \times 10^{-34} \text{J s}}{1.566 \times 10^{-23} \,\text{kg m/s}}\) Evaluate this expression to find the de Broglie wavelength: \(\lambda \approx 4.23 \times 10^{-11} \,\text{m}\) The de Broglie wavelength of the electron moving at the given speed is approximately \(4.23 \times 10^{-11} \,\text{m}\).

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Most popular questions from this chapter

A photoconductor (see Conceptual Question 13 ) allows charge to flow freely when photons of wavelength \(640 \mathrm{nm}\) or less are incident on it. What is the band gap for this photoconductor?
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.]
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Suppose the electron in a hydrogen atom is modeled as an electron in a one- dimensional box of length equal to the Bohr diameter, \(2 a_{0} .\) What would be the ground-state energy of this "atom"? How does this compare with the actual ground-state energy?
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