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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

What are the possible values of \(L_{z}\) (the component of angular momentum along the \(z\) -axis) for the electron in the second excited state \((n=3)\) of the hydrogen atom?
The particle in a box model is often used to make rough estimates of energy level spacings. For a metal wire \(10 \mathrm{cm}\) long, treat a conduction electron as a particle confined to a one-dimensional box of length \(10 \mathrm{cm} .\) (a) Sketch the wave function \(\psi\) as a function of position for the electron in this box for the ground state and each of the first three excited states. (b) Estimate the spacing between energy levels of the conduction electrons by finding the energy spacing between the ground state and the first excited state.
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