Question

Consider an electron whose de Broglie wavelength is equal to the wavelength of green light (550 nm). a) Treating the electron non relativistically what is its speed? b) Does your calculation confirm that a non relativistic treatment is sufficient? c) Calculate the kinetic energy of the electron in electron-volts.

Short answer

Answer: The non-relativistic speed of the electron is approximately 1.37 × 10^6 m/s, and its kinetic energy is around 5.35 eV. Since the speed of the electron is much smaller compared to the speed of light, the non-relativistic treatment is sufficient.

Step-by-step solution

Find the speed of the electron

We are given the de Broglie wavelength λ = 550 nm = 550 × 10^(-9) m. We will use the de Broglie wavelength formula to find the speed of the electron: λ = h/(mv) We know the Planck's constant h = 6.626 × 10^(-34) Js and the mass of the electron m = 9.109 × 10^(-31) kg. So we can rearrange the formula for the velocity of the electron: v = h/(mλ) Now, plug in the values to calculate v: v = (6.626 × 10^(-34) Js) / ((9.109 × 10^(-31) kg) × (550 × 10^(-9) m)) v ≈ 1.37 × 10^6 m/s

Check if the non-relativistic treatment is sufficient

To check whether the non-relativistic treatment is sufficient, we need to compare the speed of the electron to the speed of light c = 3 × 10^8 m/s. If the speed of the electron is much smaller compared to the speed of light, then the non-relativistic treatment is indeed sufficient. In this case, v ≈ 1.37 × 10^6 m/s is much smaller than c, so we can conclude that the non-relativistic treatment is sufficient.

Calculate the kinetic energy of the electron in electron-volts

We can use the non-relativistic kinetic energy (KE) formula to find the KE of the electron: KE = (1/2)mv^2 Now, plug in the given values of m and v to calculate KE: KE = (1/2)(9.109 × 10^(-31) kg)(1.37 × 10^6 m/s)^2 KE ≈ 8.575 × 10^(-19) J To convert the kinetic energy into electron-volts, we use the conversion factor: 1 eV = 1.602 × 10^(-19) J. So, KE in eV = (8.575 × 10^(-19) J) / (1.602 × 10^(-19) J/eV) ≈ 5.35 eV. Thus, the kinetic energy of the electron is approximately 5.35 electron-volts.

Key concepts

Electron Speed Calculation

Understanding the speed of an electron is crucial when you want to comprehend the behavior of particles at a quantum level. To begin with, you always start with the de Broglie hypothesis which states that every moving particle exhibits wave-like behavior and has a wavelength, known as the de Broglie wavelength. This wavelength is inversely proportional to the momentum of the particle.
Using the formula for de Broglie wavelength,
\[\begin{equation}\text{de Broglie wavelength, } \( \lambda = \frac{h}{mv} \) \text{where} \( h \) \( \text{is Planck's constant,} \) \( m \) \( \text{is the mass of the electron, and} \) \( v \) \( \text{is the electron's velocity,} \end{equation}\]
you can calculate the electron’s speed after rearranging the formula:\[\begin{equation} v = \frac{h}{m\lambda} \text{.}\end{equation}\]You’d insert the known values for \( h \), \( m \) and \( \lambda \) to find that the speed of an electron \( v \) is quite fast but still significantly less than the speed of light.

Non-Relativistic Treatment of Particles

Quantum particles like electrons can be treated non-relativistically when their speeds are much lower than the speed of light. It’s an approximation that simplifies our calculations and is often used in quantum mechanics for systems where velocities are small relative to the speed of light.

When Is Non-Relativistic Treatment Valid?

In our case, we compare the electron's velocity with the speed of light. It turns out the electron’s speed, \( v \) is much less than \( c \) (speed of light), which validates our non-relativistic treatment. This implies that the relativistic effects - those arising from Einstein's theory of relativity, which becomes significant at close to the speed of light - can be safely ignored for this scenario.

Electron Kinetic Energy in Electron-Volts

The kinetic energy (KE) of an electron can be calculated non-relativistically using the classic formula \( KE = \frac{1}{2}mv^2 \) where \( m \) is the mass and \( v \) is the velocity of the electron. The result is initially in joules because the units of mass and velocity are kilograms and meters per second respectively.
To better suit the scale of particles in quantum mechanics, it's common practice to use electron volts (eV) instead, which is a unit of energy more convenient for the small energies involved at the atomic and subatomic level.

Conversion to Electron Volts

Since 1 electron volt is defined as the amount of kinetic energy gained or lost by an electron when it moves through an electric potential difference of one volt, we can convert joules to electron volts using \( 1 \text{eV} = 1.602 \times 10^{-19} \text{J} \).By using the calculated KE in joules and the conversion factor, the kinetic energy of the electron for our exercise comes out to be 5.35 eV. This provides a more tangible sense of energy scale in the context of electron behavior and interactions.