Question

(a) Find the lowest energy level for a particle in a box if the particle is a billiard ball (\(m\) = 0.20 kg) and the box has a width of 1.3 m, the size of a billiard table. (Assume that the billiard ball slides without friction rather than rolls; that is, ignore the \(rotational\) kinetic energy.) (b) Since the energy in part (a) is all kinetic, to what speed does this correspond? How much time would it take at this speed for the ball to move from one side of the table to the other? (c) What is the difference in energy between the \(n\) = 2 and \(n\) = 1 levels? (d) Are quantum-mechanical effects important for the game of billiards?

Short answer

Part (a) energy is extremely small; corresponding speed is negligible. Quantum effects are not significant for billiards.

Step-by-step solution

Define the Problem

We need to find the energy levels of a particle in a one-dimensional box, modeled as a billiard ball sliding without friction. We will apply quantum mechanics to this classical object.

Quantum Mechanics and the Particle in a Box

For a particle in a box, the energy levels are given by \[ E_n = \frac{n^2 h^2}{8mL^2} \] where \( n \) is the quantum number, \( h \) is Planck's constant (\(6.626 \times 10^{-34} \text{ J s}\)), \( m \) is the mass of the particle, and \( L \) is the length of the box. We will use \( n = 1 \) for the lowest energy level (ground state).

Calculate the Ground State Energy

Substituting \( n = 1 \), \( m = 0.20 \text{ kg} \), and \( L = 1.3 \text{ m} \) into the formula for \( E_n \), we have: \[ E_1 = \frac{1^2 \times (6.626 \times 10^{-34})^2}{8 \times 0.20 \times (1.3)^2} \]. Calculate this to find \( E_1 \).

Find Corresponding Speed

The energy (from part a) is purely kinetic, so \( E = \frac{1}{2}mv^2 \). Rearranging, \( v = \sqrt{\frac{2E}{m}} \). Use \( E_1 \) calculated in the previous step to find \( v \).

Calculate Time for Ball Movement

With the speed \( v \) known, the time to travel a distance \( L = 1.3 \text{ m} \) is \( t = \frac{L}{v} \). Calculate \( t \) using the speed found in Step 4.

Determine Energy Difference for n=2 and n=1

The energy difference is given by \( \Delta E = E_2 - E_1 \), where \[ E_2 = \frac{2^2 h^2}{8mL^2} = 4 \times E_1 \]. Calculate \( \Delta E \) as \( 3 \times E_1 \).

Assess Quantum Mechnical Significance

Calculate \( \Delta E \) and compare to typical energy scales for a billiard ball. Quantum effects are significant if these energies are large and cause noticeable differences in phenomena not explained classically.

Key concepts

Particle in a Box

The concept of a "Particle in a Box" is a fundamental model in quantum mechanics that describes how particles such as electrons behave in a confined space. Imagine a box where the walls are so high that the particle cannot escape; this is our one-dimensional box. In this scenario, the particle can only "live" within the box's boundaries, and this constraint leads to quantized, or discrete, energy levels.

• The particle is essentially trapped, having only certain permissible energy states.
• This is because, in quantum mechanics, particles exhibit wave-like properties and only certain wavelengths fit perfectly within the box.
• Each of these wavelengths corresponds to a different energy level, resulting in quantization.

The mathematical formula for the energy levels of a particle in a box is given by \[ E_n = \frac{n^2 h^2}{8mL^2} \] where:

• \( n \) is the principal quantum number indicating the energy level (n=1, 2, 3, ...),
• \( h \) is Planck's constant,
• \( m \) is the mass of the particle, and
• \( L \) is the length of the box.

Since the energy levels are discrete, a particle cannot have any arbitrary amount of energy. It can only exist in one of the defined energy states.

Quantum Energy Levels

Quantum energy levels are akin to a ladder that a particle in a confined space, like our particle in a box, can only "climb" one rung at a time. These energy levels are quantized meaning they take specific discrete values, unlike a classical scenario where energy can vary continuously.

For the particle in a box:

• Each possible state is represented by a quantum number \( n \).
• The energy of each state is proportional to the square of \( n \), given by the equation \[ E_n = \frac{n^2 h^2}{8mL^2} \].
• The lowest possible energy (ground state) is when \( n = 1 \).
• If \( n = 2 \), the energy is four times that of the ground state.
• So, the difference between the first and second energy levels \( (\Delta E) \) becomes \( 3 \times E_1 \).

As particles move between these levels, they absorb or emit precise amounts of energy. This quantum behavior stands in contrast to classical mechanics, where transitions can be continuous. Understanding this concept is crucial, especially in quantum technologies like lasers and semiconductors.

Kinetic Energy

Kinetic energy is the energy of motion. In this case, the particle in a box has its energy entirely expressed through kinetic energy, as we're ignoring any other forms like potential energy. The kinetic energy of a particle is defined as \( E = \frac{1}{2}mv^2 \).

Here:

• \( E \) is the kinetic energy where the particle's motion is translational within the box.
• The term \( m \) signifies the particle's mass.
• The velocity \( v \) is calculated from the energy using the relation \( v = \sqrt{\frac{2E}{m}} \),
where \( E \) is derived from the quantum energy level calculations for the ground state.

In this scenario, as the particle travels from one side of the box to the other, we can gauge the time it takes using the relationship:

• \( t = \frac{L}{v} \),where \( L \) is the box's length.

The principles of kinetic energy apply universally to understanding particle motion and are vital in connecting classical mechanics with the derivations we've made using quantum concepts.

Planck's Constant

Planck's constant \( (h) \) is a fundamental quantity in quantum mechanics pivotal to describing the smallest scales of physics. It sets the scale for the quantization of energy levels in physical systems. The value of Planck's constant is approximately \( h = 6.626 \times 10^{-34} \, \text{J s} \).

In our study of the particle in a box model, Planck's constant ties the particle's properties to its quantized energy levels:

• It appears in the energy level formula: \[ E_n = \frac{n^2 h^2}{8mL^2} \].
• This constant ensures that only certain energy levels are allowed, aligning with the wave-particle duality in quantum mechanics.
• The small magnitude of \( h \) highlights why quantum effects are typically only noticeable at very tiny scales (like atomic and subatomic levels).

Understanding Planck's constant is key to appreciating why energies are quantized in certain systems and is foundational in the development of quantum theory and its applications.