Problem 29 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Consider an atomic nucleus to be a rigid box of width \(2.0 \times 10^{-14} \mathrm{~m} .\) What would be the ground-state energy for (a) an electron, (b) a neutron, and ( \(c\) ) a proton in this nucleus?

Step-by-Step Solution

Answer

(a) Electron: 94.2 MeV, (b) Neutron: 6.31 MeV, (c) Proton: 6.31 MeV.

Step 1: Understanding the Problem

We need to determine the ground-state energy for an electron, neutron, and proton confined in a rigid box representing a nucleus with a width of \(2.0 \times 10^{-14} \text{ m}\). The problem involves calculating the energy levels for particles in a one-dimensional potential well.

Step 2: Using Quantum Mechanics Formula

The ground-state energy in a one-dimensional box is given by the formula:\[E_1 = \frac{h^2}{8mL^2}\]where \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ J s}\), \(m\) is the particle's mass, and \(L\) is the width of the box. We will use this formula to calculate the ground-state energy for each particle.

Step 3: Calculating for an Electron

First, calculate the energy for an electron:\[m_e = 9.11 \times 10^{-31} \text{ kg}\]Substitute values into the formula:\[E_1 = \frac{(6.626 \times 10^{-34})^2}{8 \times 9.11 \times 10^{-31} \times (2.0 \times 10^{-14})^2}\]\[E_1 \approx 1.51 \times 10^{-14} \text{ J} = 94.2 \text{ MeV}\]

Step 4: Calculating for a Neutron

Next, calculate the energy for a neutron:\[m_n = 1.675 \times 10^{-27} \text{ kg}\]Substitute values into the formula:\[E_1 = \frac{(6.626 \times 10^{-34})^2}{8 \times 1.675 \times 10^{-27} \times (2.0 \times 10^{-14})^2}\]\[E_1 \approx 1.01 \times 10^{-20} \text{ J} = 6.31 \text{ MeV}\]

Step 5: Calculating for a Proton

Finally, calculate the energy for a proton:\[m_p = 1.673 \times 10^{-27} \text{ kg}\]Substitute values into the formula:\[E_1 = \frac{(6.626 \times 10^{-34})^2}{8 \times 1.673 \times 10^{-27} \times (2.0 \times 10^{-14})^2}\]\[E_1 \approx 1.01 \times 10^{-20} \text{ J} = 6.31 \text{ MeV}\]

Key Concepts

Ground-State Energy

The ground-state energy is the lowest possible energy that a quantum mechanical system can have. It's important in the study of quantum systems like atoms, molecules, and even subatomic particles within a nucleus. In many ways, ground-state energy defines the baseline activity level of the entire system.

In quantum mechanics, particles do not remain static even at absolute zero temperature because they still possess this intrinsic energy. For a particle confined in a potential, such as in the simplified 'particle in a box' situation, the ground-state energy depends on various factors including the particle's mass and the dimensions of the confining space.

This energy is crucial because it influences other properties of atoms or molecules, for example, determining how they might interact in a reaction. When calculating ground-state energies for different particles like electrons, neutrons, and protons, it's about substituting their respective masses into the quantum equations that describe this minimum energy state. While the calculations can seem complex, understanding the meaning behind the ground-state simplifies these interactions.

Particle in a Box Model

The 'particle in a box' model is a basic concept in quantum mechanics that illustrates, in simple terms, how confined particles behave. Imagine a tiny particle, such as an electron, trapped in a perfectly rigid box. The box has sharp walls that the particle cannot penetrate; therefore, the particle's movement is restricted to bouncing back and forth inside.

This model hasn't just been a fundamental teaching tool; it also provides insights into quantum systems. Particles that are confined this way exhibit quantized energy levels, meaning their energy can only exist in specific increments rather than a continual range.

To determine a particle's energy level within this model, you can use:

• Planck’s constant (\(h\)).
• Particle’s mass (\(m\)).
• Width of the box (\(L\)).

For instance, when the box represents an atomic nucleus, you substitute specifics like particle masses for electrons, neutrons, and protons to find their respective ground-state energies.

Atomic Nucleus Confinement

When considering particles in an atomic nucleus, it is a very small region where various nuclear particles are packed tightly. Imagine how electrons, protons, and neutrons behave in this minuscule space. Unlike free particles, those within a nucleus are bound and move in defined ways similar to the 'particle in a box' model.

The atomic nucleus isn’t an actual box, but the model helps to visualize the limitations on movement and energy that particles undergo. Confinement impacts particles significantly since the dimensions and conditions dictate how particles distribute their energy. For example, constraint within the nucleus means the particles can only occupy certain energy levels depending on their mass and the dimensions of confinement.

This confinement model also explains why particles like protons and neutrons in the nucleus have much higher ground-state energies compared to electrons, due to their comparatively larger masses. Thus, understanding atomic nucleus confinement is central to nuclear physics, providing essential insights into how subatomic particles function.