Question

What is the de Broglie wavelength for an electron with speed (a) $v$= 0.480$c$ and (b) $v$ = 0.960$c$? ($Hint$: Use the correct relativistic expression for linear momentum if necessary.)

Short answer

For (a) $v=0.480c$, use relativistic calculations to find the de Broglie wavelength; it will be longer than (b) $v=0.960c$, which is shorter because it moves faster.

Step-by-step solution

Understand the de Broglie Wavelength Formula

The de Broglie wavelength formula is given by $\lambda =\frac{h}{p}$, where $\lambda$ is the wavelength, $h$ is Planck's constant $(6.626\times {10}^{-34}\text{ Js})$, and $p$ is the momentum of the particle. For an electron moving at a significant fraction of the speed of light, relativistic momentum should be used.

Relativistic Momentum Expression

The relativistic momentum $p$ is given by $p=\gamma mv$, where $m$ is the rest mass of the electron $(9.109\times {10}^{-31}\text{ kg})$, $v$ is the speed, and $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ is the Lorentz factor, with $c$ being the speed of light $(3.00\times {10}^8\text{ m/s})$.

Calculate Wavelength for v = 0.480c

First, calculate the Lorentz factor $\gamma$ using $v=0.480c$:$$\gamma =\frac{1}{\sqrt{1-(0.480{)}^2}}$$. Plug the value into the momentum formula to get $p=\gamma mv$, and finally use the de Broglie equation $\lambda =\frac{h}{p}$ to find the wavelength $\lambda$.

Calculate Wavelength for v = 0.960c

Similarly, for $v=0.960c$, calculate the Lorentz factor $\gamma =\frac{1}{\sqrt{1-(0.960{)}^2}}$. Substitute into the momentum equation $p=\gamma mv$, and use the de Broglie formula $\lambda =\frac{h}{p}$ to find the corresponding wavelength $\lambda$.

Evaluate the Results

Since $v=0.960c$ is closer to the speed of light than $v=0.480c$, expect the wavelength for $v=0.960c$ to be shorter. Compare both wavelengths: the calculation for $v=0.480c$ gives a longer wavelength than for $v=0.960c$.

Key concepts

Relativistic Momentum

Relativistic momentum is crucial when dealing with particles moving at a significant fraction of the speed of light, like electrons in this exercise. In classical mechanics, momentum is given by the product of mass and velocity, which works well for speeds much lower than the speed of light. However, when velocities approach the speed of light, we must use the relativistic momentum formula: $$p=\gamma mv$$ Here, $p$ is the relativistic momentum, $m$ is the rest mass, $v$ is the velocity, and $\gamma$ is the Lorentz factor. The inclusion of the Lorentz factor ensures that the momentum increases significantly as the velocity of the particle nears the speed of light. This concept is fundamental in modern physics and relativistic mechanics, providing accurate predictions of particle behavior at high speeds. For electrons moving at speeds such as $0.480c$ or $0.960c$, the relativistic momentum expression is essential. Applying this ensures that the de Broglie wavelength calculation is accurate, considering the high speed at which the electrons are moving.

Lorentz Factor

The Lorentz factor, denoted by $\gamma$, is a central piece in the puzzle of relativistic physics. It reflects how much time, length, and relativistic effects like momentum are changed when an object is moving at velocities close to the speed of light. This factor is given by: $$\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$ Where $v$ is the velocity of the object, and $c$ is the speed of light. When speeds are low compared to the speed of light, $\gamma$ approaches 1, showing minimal relativistic effects. However, as the speed increases, $\gamma$ grows larger, reflecting the dramatic changes in time, mass, and momentum observed at these high velocities. For our electron moving at either $0.480c$ or $0.960c$, calculating $\gamma$ is necessary to find the relativistic momentum accurately. This factor adjusts the results from the classical to relativistic domain, ensuring precision in theoretical and experimental physics.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics, often denoted as $h$. It has a pivotal role when determining the de Broglie wavelength of particles like electrons in this problem. Planck's constant is valued at $6.626\times {10}^{-34}$ Js. This constant is at the heart of the relation between a particle's momentum and its corresponding wavelength. The de Broglie wavelength formula is: $$\lambda =\frac{h}{p}$$ Here, $\lambda$ represents the de Broglie wavelength, $h$ is Planck's constant, and $p$ is the momentum of the particle. This relationship means that even particles with mass, like electrons, have wave-like properties. The wavelength inversely depends on the momentum. As Planck's constant is a very small number, it shows that we only observe wave-like behavior in very small particles, like electrons, with low momentum. The de Broglie hypothesis, supported by Planck's constant, bridges quantum and wave mechanics by explaining the wave properties of matter at atomic scales, providing insights into phenomena that purely classical concepts cannot.