23.3 De Broglie's Wave Equation
The photoelectric effect and the Compton effect both demonstrate the particle characteristics of a photon supporting the notion of wave-particle duality. A natural consequence of this is to ask if a particle, such as an electron, could behave like
a wave. Louis de Broglie introduced this idea in 1923. He proposed, based on Compton's findings, that matter-possessing momentum has a characteristic wavelength according to
.
For an electron travelling much less than the speed of light, this equation becomes
.
The first evidence of matter waves came several years later when physicists C.J. Davison and L.H. Germer accidentally discovered that a beam of electrons can create an interference pattern. Ultimately, this discovery contributed to the development of the modern electron microscope.
ReadRead "Then, Now and Future, The Electron Microscope" on page 727 and "De Broglie's Wave Hypothesis: Strange but True!" on page 729 of your physics textbook. |
As you discovered in the reading about electron microscopes, the magnification of a microscope depends on the inverse of the wavelength that produces the image. The electron microscope is able to generate matter waves with wavelengths much smaller than that of visible light, leading to greater magnification as you can see in the picture of the fly's eye to the right.
Example Problem 1.
What is the wavelength of an electron with a speed of 5.50 × 10 6 m/s? How many times smaller is this wavelength compared to a 400-nm violet light?
Hint: When asked to compare two quantities, answer by stating how many times larger or smaller one quantity is relative to the other.
Given
Required
the wavelength of the electron and the ratio of the wavelengths of the electron and violet light
Analysis and Solution
Paraphrase
The wavelength of the electron is 1.32 × 10 -10 m, and it is 3.03 × 10 3 times smaller than the wavelength of violet light.
Example Problem 2
What is the wavelength of an electron that has a kinetic energy of 3.10 × 10 -16 J?
Given
Required
the wavelength of the electron
Analysis and Solution
Paraphrase
The wavelength of the electron is 2.79 × 10 -11 m.
Self-CheckAnswer the following self-check (SC) questions then click the "Check your work" bar to assess your responses. |
SC 3.
What is the wavelength of an electron that has a speed of 2.25 × 10 7 m/s?
SC 4.
What is the wavelength of an electron that has a kinetic energy of 7.2 × 10 -4 MeV?
SC 5.
What is the speed of an electron that has a wavelength of 7.00 pm?
SC 6.
Contact your teacher if your answers vary significantly from the answers provided here.
SC 3.
Given
Required
the wavelength of the electron
Analysis and Solution
Paraphrase
The wavelength of the electron is 3.23 × 10 -11 m.
SC 4.
Given
Required
the wavelength of the electron
Analysis and Solution
Change the energy in eV to J
Use the kinetic energy to find the velocity.
Use the velocity to find the wavelength.
Paraphrase
The wavelength of the electron is 4.58 × 10 -11 m.
SC 5.
Given
Required
the velocity of the electron
Analysis and Solution
Paraphrase
The speed of the electron is 1.04 × 10 8 m/s.
SC 6
Given
V = 1000 V
Required
the wavelength of the electron
Analysis and Solution
Calculate the change in energy (or work done) of the electron.
Calculate the velocity using .
Calculate the wavelength.
Paraphrase
The wavelength of the electron is 3.88 × 10 -11 m.
ReadRead "De Broglie's Hypothesis-A Key Concept of Quantum Physics" and "Heisenberg's Uncertainty Principle" on pages 730 to 735 of the textbook. |
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