Question

A dust particle of mass \(1.00 \cdot 10^{-16}\) kg and diameter \(5.00 \mu \mathrm{m}\) is confined to a box of length \(15.0 \mu \mathrm{m}\). a) How will you know whether the particle is at rest? b) If the particle is not at rest, what will be the range of its velocity? c) At the lower limit of the velocity range, how long will it take to move a distance of \(1.00 \mu \mathrm{m} ?\)

Short answer

Answer: To determine if the particle is at rest, we can calculate its kinetic energy. If the kinetic energy is zero, the particle is at rest. To compute the particle's range of possible velocities, we can follow these steps: 1. Calculate the de Broglie wavelength using the formula \(\lambda = \frac{h}{p}\). 2. Determine the range of possible de Broglie wavelengths by considering multiples of half the box length: \(\lambda_n = \frac{2L}{n}\). 3. Find the range of allowed momenta by rewriting the de Broglie wavelength formula: \(p_n = \frac{h}{\lambda_n}\). 4. Calculate the range of velocities using the relation \(v_n = \frac{p_n}{m}\). To find the time it takes to move a certain distance, use the relationship time = \(\frac{distance}{velocity}\).

Step-by-step solution

Determine if the particle is at rest

To determine if the particle is at rest, we can check its kinetic energy (KE), as a particle at rest will have zero energy. The formula to calculate kinetic energy is: KE = 1/2 * m * v^2, where m is the mass, and v is the velocity. If KE = 0, the particle is at rest.

Calculate the de Broglie wavelength

The de Broglie wavelength allows us to describe the wave nature of particles, which will help us estimate the velocity range of our particle. The formula for the de Broglie wavelength is given by: \(\lambda = \frac{h}{p}\), where \(\lambda\) is the de Broglie wavelength, h is the Planck's constant (\(6.626 \cdot 10^{-34}\) Js), and p is the momentum of the particle (which is the product of mass and velocity, p = m * v).

Determine the range of possible de Broglie wavelengths

Since the particle is confined to a box of length \(15.0 \mu \mathrm{m}\), we can assume that any possible de Broglie wavelength it can have should be a multiple of half the box length (from a node to an antinode in a standing wave). Therefore, the range of possible de Broglie wavelengths can be given by: \(\lambda_n = \frac{2L}{n}\), where \(\lambda_n\) is the possible de Broglie wavelength, L is the length of the box, and n is a positive integer (n = 1, 2, 3, ...).

Find the range of allowed momenta

With the possible de Broglie wavelengths, we can find the range of allowed momenta, as the de Broglie wavelength and momentum are inversely proportional. Therefore, we can rewrite the de Broglie wavelength formula as: \(p_n = \frac{h}{\lambda_n}\), where \(p_n\) is the range of possible momenta for the particle.

Calculate the range of velocities

We can now use the allowed momenta to find the range of possible velocities for the particle. Using the relation p = m * v, we can solve for v: \(v_n = \frac{p_n}{m}\), where \(v_n\) is the possible range of velocities for the particle (as long as it's not at rest).

Calculate the time to move a given distance

Finally, at the lower limit of the velocity range, we can calculate how long it would take for the particle to move a distance of \(1.00 \mu \mathrm{m}\). We can use the relationship: distance = velocity * time, and solve for time: time = \(\frac{distance}{velocity}\), where time is the amount of time it will take for the particle to move the given distance.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It introduced new concepts to physics, like the wave-particle duality, which states that every particle exhibits both wave-like and particle-like properties.

A cornerstone of this theory is the de Broglie hypothesis, which posits that matter waves are associated with moving particles. It is this wave-like behavior that allows us to talk about the de Broglie wavelength, particularly important when discussing the kinetic motion of particles at the quantum scale, as mentioned in the textbook exercise.

Quantum mechanics also introduces the idea that certain quantities, like the energy of particles, are quantized. This means that they can only take on specific, discrete values, which plays a critical role in understanding phenomena like the particle in a box model.

Kinetic Energy of Particles

The kinetic energy of particles is a measure of the energy due to their motion. It can be calculated using the formula \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the particle and \( v \) is its velocity. In quantum mechanics, the concept of kinetic energy is related to the momentum of particles and, by extension, to their de Broglie wavelength.

Higher kinetic energy implies greater motion and vice versa. In the provided exercise, we analyze whether a dust particle is at rest (zero kinetic energy) or moving (positive kinetic energy). When discussing particles at the quantum level, even the energy associated with their motion is subject to the principles of quantum theory, leading to discrete energy levels rather than a continuous range.

Particle in a Box Model

The 'particle in a box' model is a simplified quantum system that helps us understand how quantization works for a single particle confined within a rigidly-defined space. The particle can be thought of as a wave that forms standing waves within the box, similar to a string that is held fixed at both ends.

The allowed energy states of the particle are discrete and related to the wavelength of the particle. Using the de Broglie hypothesis, we can equate the wavelength with the particle's momentum, and this leads to specific, quantized energy levels. This model explains why the particle cannot have any arbitrary energy and why certain phenomena, like the electron orbitals in an atom, have particular, fixed structures.

In the exercise, we used this model to determine the possible wavelengths and therefore velocity ranges of the dust particle confined to a microscopic box. Students should realize through this model, energy, and position of quantum particles are restricted to certain values, an insight that aligns with the foundational principles of quantum mechanics.