Problem 41 - College Physics

Question

\(\cdot\) How fast would an electron have to move so that its de Broglie wavelength would be 1.00 \(\mathrm{mm}\) ?

Step-by-Step Solution

Answer

The electron's velocity should be approximately 7.28 × 10² m/s.

Step 1: Understand de Broglie Wavelength

The de Broglie wavelength equation is given by \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is the Planck's constant \( 6.626 \, \times \, 10^{-34} \, \text{m}^2 \cdot \text{kg} \cdot \text{s}^{-1} \), \( m \) is the mass of the electron \( 9.109 \, \times \, 10^{-31} \, \text{kg} \), and \( v \) is the velocity of the electron. Our goal is to find the velocity \( v \) for a wavelength of 1.00 mm or \( 1.00 \, \times \, 10^{-3} \, \text{m} \).

Step 2: Rearrange the Formula for Velocity

Rearrange the de Broglie equation to solve for velocity \( v \), resulting in \( v = \frac{h}{m\lambda} \). This formula will allow us to calculate the necessary velocity by plugging in known values for the other variables.

Step 3: Plug in Known Values

Substitute the known values into the formula: \[ v = \frac{6.626 \, \times \, 10^{-34} \, \text{m}^2 \cdot \text{kg} \cdot \text{s}^{-1}}{9.109 \, \times \, 10^{-31} \, \text{kg} \times 1.00 \, \times \, 10^{-3} \, \text{m}} \]

Step 4: Calculate the Velocity

Carry out the calculation from Step 3:\[v = \frac{6.626 \, \times \, 10^{-34}}{9.109 \, \times \, 10^{-31} \times 1.00 \, \times \, 10^{-3}} \approx 7.28 \, \times \, 10^{2} \, \text{m/s}\]This results in a velocity of approximately \( 7.28 \, \times \, 10^{2} \, \text{m/s} \).

Key Concepts

Electron Velocity

In physics, the concept of **electron velocity** is essential when discussing phenomena like de Broglie wavelengths. Velocity refers to how fast the electron is moving. In the context of de Broglie wavelengths, we calculate the velocity of an electron required to achieve a specific wavelength. This velocity gives us insight into the particle-wave nature of electrons.
To determine this velocity, we use the formula derived from the de Broglie equation:

• Rearrange the de Broglie equation to solve for electron velocity \(v\), resulting in: \(v = \frac{h}{m\lambda}\).
• Here, \(h\) represents Planck's constant, \(m\) is the electron's mass, and \(\lambda\) is the required wavelength.

Using this formula, we can insert values for Planck's constant, the mass of the electron, and the desired wavelength to find the electron's required velocity. This concept of velocity is foundational in understanding how electrons might behave similarly to waves under certain conditions.

Planck's constant

**Planck's constant** is a fundamental constant in quantum mechanics, symbolized by \(h\). It plays a crucial role in the quantization of energy levels. With a value of \(6.626 \times 10^{-34} \, \text{m}^2 \cdot \text{kg} \cdot \text{s}^{-1}\), it helps bridge the world of macroscopic classical physics and the microscopic quantum physics.
Planck's constant is significant in the de Broglie wavelength formula:

• It is present in the formula \(\lambda = \frac{h}{mv}\), where \(\lambda\) is the wavelength, \(m\) is mass, and \(v\) is velocity.
• The constant helps to calculate wave-like properties of particles like electrons, which are typically not noticeable in everyday macroscopic scales.

Thus, Planck's constant is indispensable in equations that explain how particles exhibit wave and particle duality, a cornerstone concept of quantum mechanics.

Mass of Electron

The **mass of an electron** is a fundamental property that affects many calculations in quantum mechanics. In the de Broglie wavelength equation, it is one of the key variables used to determine the wavelength or velocity of an electron.
The precise mass of an electron is \(9.109 \times 10^{-31} \, \text{kg}\). This tiny mass means electrons move in ways vastly different from macroscopic objects, displaying quantum behaviors.
In the velocity calculation:

• Mass \(m\) appears in the denominator of the velocity formula \(v = \frac{h}{m\lambda}\). This indicates that the lighter the electron, the higher the velocity required to maintain a specific de Broglie wavelength.
• Its involvement in the formula shows the important relationship between mass and wavelength in determining electron properties.

Understanding the mass of an electron helps in comprehending its extremely high velocities even at relatively modest energy levels due to its smallness in mass.

Wavelength Calculation

**Wavelength calculation** in the context of the de Broglie wavelength involves analyzing how an electron's motion translates into a wave property. This calculation is crucial for predicting how electrons behave in various physical contexts, blending their particle and wave characteristics.
The de Broglie wavelength is determined by the formula:

• \(\lambda = \frac{h}{mv}\), linking wavelength \(\lambda\), Planck's constant \(h\), mass \(m\), and velocity \(v\).
• In cases where the wavelength is known, this equation can be rearranged to solve for **velocity** (\(v = \frac{h}{m\lambda}\)) as shown in the step by step solution.

To achieve a specific de Broglie wavelength, wavelength calculation allows us to understand necessary conditions like velocity, playing a pivotal role in quantum mechanical practices and theoretical predictions.