$$ \begin{aligned} &\text { Calculate the wavelength of an electron moving with a velocity of }\\\ &2.05 \times 10^{7} \mathrm{~ms}^{-1} \end{aligned} $$

Short Answer

Expert verified
The wavelength of an electron moving with a velocity of 2.05 x 10^7 m/s is approximately 3.55 x 10^-11 meters.

Step by step solution

01

Recalling the de Broglie Wavelength Formula

To calculate the wavelength of a particle such as an electron, we use the de Broglie wavelength formula, which is given by \[ \lambda = \frac{h}{p} \]where \(h\) is the Planck constant (\(6.626 \times 10^{-34} \text{ m}^2\text{kg/s}\) and \(p\) is the momentum of the particle. The momentum \(p\) is calculated by multiplying the mass \(m\) of the electron (\(9.11 \times 10^{-31}\) kg) by its velocity \(v\).
02

Calculate the Momentum of the Electron

The momentum \(p\) of the electron is given by the product of its mass \(m\) and velocity \(v\): \[ p = mv = (9.11 \times 10^{-31} \text{ kg})(2.05 \times 10^{7} \text{ m/s}) \]Calculate this product to find the momentum of the electron.
03

Calculate the Wavelength using de Broglie's Formula

Insert the calculated momentum and the Planck constant into the de Broglie's equation to find the wavelength \(\lambda\): \[ \lambda = \frac{h}{p} = \frac{6.626 \times 10^{-34} \text{ m}^2 \text{kg/s}}{mv} \]Substitute the momentum value calculated in step 2 to obtain the wavelength.
04

Compute the Final Answer

After substituting all the known values, perform the calculation to determine the electron's wavelength.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

de Broglie's Equation
One of the fundamental concepts in quantum mechanics is de Broglie's hypothesis, which postulates that all moving particles—particularly subatomic particles such as electrons—exhibit wave-like behavior. This is expressed through the de Broglie equation: \[ \lambda = \frac{h}{p} \] where \(\lambda\) is the wavelength associated with the particle, \(h\) is the Planck constant, and \(p\) is the momentum of the particle. The formula bridges the gap between classical physics, which describes particles as matter with mass and velocity, and quantum mechanics, which also considers their wave-like properties.

Understanding this equation allows us to calculate the electron's wavelength when given its velocity and mass, leading to a deeper grasp of the particle-wave duality concept crucial in quantum mechanics.
Electron Momentum
The term 'electron momentum' refers to the product of an electron's mass and its velocity. It's an essential quantum mechanical property that, when combined with de Broglie's hypothesis, provides a way to comprehend the quantum behavior of electrons. The momentum \(p\) for an electron with mass \(m\) and velocity \(v\) is calculated using the formula: \[ p = mv \] Since an electron has an incredibly small mass (approximately \(9.11 \times 10^{-31}\) kg), even moderate velocities can result in a significant demonstration of its wave-like properties. By determining the momentum, one can then use de Broglie's equation to find the associated wavelength.
Planck Constant
The Planck constant (denoted as \(h\)) is a fundamental physical constant central to quantum mechanics. Its value is \(6.626 \times 10^{-34} \text{ m}^2\text{kg/s}\), and it represents the proportionality between the energy (\(E\)) of a photon and the frequency (\(u\)) of its associated electromagnetic wave, encapsulated in Planck's relation: \[ E = hu \] The constant equates the wave aspect of light with its particle aspect, similarly to how it's used in de Broglie's equation to correlate the momentum of particles to their wavelength. As such, it has immense implications for how we understand and calculate phenomena at the quantum scale.
Electron Wavelength
The electron wavelength is the distance over which the wave-like nature of the electron is periodic. It indicates the spread and interference pattern an electron can create, akin to that of light waves passing through a double-slit apparatus. By using the de Broglie equation where \(\lambda\) represents the wavelength, one can determine the scale at which quantum effects become significant for electrons. This wavelength becomes particularly relevant when electrons interact with objects of a similar size, as seen in phenomena like the diffraction patterns in a crystal lattice.
Physics Problem Solving
The approach to physics problem-solving typically involves several key steps such as understanding the problem, identifying the known and unknown variables, finding the right formula, and performing the calculations. For instance, when solving a problem that requires calculating an electron's de Broglie wavelength, one must first understand the concepts of momentum and wave-particle duality. Then, clearly defining the given electron's velocity (\(v\)) and mass (\(m\)), plus Planck's constant (\(h\)), allows the use of de Broglie's wavelength formula to methodically find the solution. With a structured, step-by-step approach, formidable physics problems can be broken down into manageable parts, ultimately demystifying complex quantum mechanical phenomena.

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