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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

Expert verified
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 marble of mass \(10 \mathrm{g}\) is confined to a box \(10 \mathrm{cm}\) long and moves at a speed of \(2 \mathrm{cm} / \mathrm{s} .\) (a) What is the marble's quantum number \(n ?\) (b) Why can we not observe the quantization of the marble's energy? [Hint: Calculate the energy difference between states \(n\) and \(n+1 .\) How much does the marble's speed change?]
A fly with a mass of \(1.0 \times 10^{-4} \mathrm{kg}\) crawls across a table at a speed of \(2 \mathrm{mm} / \mathrm{s} .\) Compute the de Broglie wavelength of the fly and compare it with the size of a proton (about $\left.1 \mathrm{fm}, 1 \mathrm{fm}=10^{-15} \mathrm{m}\right)$.
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.
(a) Show that the number of electron states in a subshell is \(4 \ell+2 .\) (b) By summing the number of states in each of the subshells, show that the number of states in a shell is \(2 n^{2} .\) [Hint: The sum of the first \(n\) odd integers, from 1 to \(2 n-1,\) is \(n^{2} .\) That comes from regrouping the sum in pairs, starting by adding the largest to the smallest: \(1+3+5+\dots+(2 n-5)+(2 n-3)+(2 n-1)\) \(=[1+(2 n-1)]+[3+(2 n-3)]+[5+(2 n-5)]+\cdots\) \(=2 n+2 n+2 n+\cdots=2 n \times \frac{n}{2}=n^{2}\)
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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