Problem 6 - Fundamentals of Physics

Question

a proton is confined to a one-dimensional infinite potential well \(100 \mathrm{pm}\) wide. What is its ground-state energy?

Step-by-Step Solution

Answer

The ground-state energy is approximately 0.205 eV.

Step 1: Understand the Concept of Infinite Potential Well

An infinite potential well is a quantum mechanical problem where a particle, such as a proton, is confined in a region with zero potential energy surrounded by infinitely high potential barriers. The well's width is the only region where the proton can exist, and outside this region, the potential energy is infinite.

Step 2: Recall the Formula for Ground-State Energy

For a particle of mass \(m\) in a one-dimensional infinite potential well of width \(L\), the ground-state energy \(E_1\) is given by the formula:\[ E_1 = \frac{h^2}{8mL^2} \]where \(h\) is Planck's constant.

Step 3: Identify Constants and Values

In this problem, the width of the well \(L\) is given as \(100 \, \mathrm{pm}\) (pico-meters), which needs to be converted to meters: \[ 100 \, \mathrm{pm} = 100 \times 10^{-12} \, \mathrm{m} \]The mass of a proton \(m\) is approximately \(1.67 \times 10^{-27} \, \mathrm{kg}\), and Planck's constant \(h\) is approximately \(6.626 \times 10^{-34} \, \mathrm{Js}\).

Step 4: Plug Values into Formula

Substitute the given values into the formula for ground-state energy:\[ E_1 = \frac{(6.626 \times 10^{-34})^2}{8 \times 1.67 \times 10^{-27} \times (100 \times 10^{-12})^2} \]

Step 5: Calculate the Ground-State Energy

Carrying out the calculation gives:\[ E_1 \approx \frac{4.39 \times 10^{-67}}{1.336 \times 10^{-47}} \]\[ E_1 \approx 3.29 \times 10^{-20} \, \mathrm{J} \]

Step 6: Convert Energy to Electronvolts

The energy in joules can be converted to electronvolts (eV) using the conversion factor \(1 \, \mathrm{eV} = 1.602 \times 10^{-19} \, \mathrm{J}\):\[ E_1 \approx \frac{3.29 \times 10^{-20}}{1.602 \times 10^{-19}} \approx 0.205 \, \mathrm{eV} \]

Key Concepts

Ground-State Energy

In quantum mechanics, the ground-state energy refers to the lowest possible energy that a quantum system may possess. When dealing with an infinite potential well, the ground-state energy is significant as it acts as the baseline from which all other energy states are measured. This concept is essential in understanding how particles behave at the quantum level.

To calculate this energy, we utilize the formula:

• \[ E_1 = \frac{h^2}{8mL^2} \]
• Here, \(E_1\) is the ground-state energy, \(h\) is Planck's constant, \(m\) is the particle's mass, and \(L\) is the width of the potential well.

When a proton, confined within a small space like a potential well, transitions to the ground state, its energy is an outcome of these values working together. This formula provides insight into how different factors like particle mass and dimension of confinement affect the ground-state energy.

The understanding of ground-state energy is crucial for students as it helps predict how particles behave under constrained conditions, which is fundamental to quantum mechanics studies.

Quantum Mechanics

Quantum mechanics is a branch of physics that describes the rules and behaviors of particles at the atomic and subatomic levels. It presents a unique view on how quanta, like electrons and protons, exist and interact with energies and forces quite different from classical understandings.

One of the quintessential problems in quantum mechanics is the infinite potential well. In this context, a particle such as a proton is trapped inside a confined space where potential energy is zero within and infinite outside. This creates quantized energy levels, meaning that the particle can only exist at certain discrete energy values, with the ground state being the lowest.

Key concepts highlighted by quantum mechanics:

• Particles have wave-like properties.
• Energy levels are quantized inside an infinite well.
• Probability plays a foundational role in determining outcomes.

Understanding these principles of quantum mechanics is essential for explaining phenomena such as electron configurations in atoms, the behavior of particles in nanoscale materials, and the development of technologies like semiconductors and quantum computers.

Planck's Constant

Planck's constant is a fundamental constant of nature that plays a vital role in the theory of quantum mechanics. Often symbolized as \(h\), it relates the energy of a photon to its frequency and has a value of approximately \(6.626 \times 10^{-34} \, \text{Js}\).

Planck's constant is integral in calculations involving energy and frequency, underpinning many aspects of quantum physics. It appears in the equation for ground-state energy in an infinite potential well, showing its impact on quantum mechanical systems.

• \[ E = hf \] : Energy \(E\) equals Planck's constant \(h\) times frequency \(f\).
• Helps in quantifying phenomena such as atomic emission spectra.
• Supports the particle-wave duality principle.

The constant's importance cannot be overstated—it signifies the boundary between classical physics and quantum physics, paving the way for more profound insights into atomic and subatomic energy levels. For students, understanding Planck's constant means gaining a deeper comprehension of the underlying principles governing microscopic systems.

Proton Mass

The proton mass is a crucial factor when determining the energy states in quantum systems such as the infinite potential well. A proton is a subatomic particle found in the nucleus of atoms, possessing a positive electric charge and a mass of approximately \(1.67 \times 10^{-27} \, \text{kg}\).

When considering ground-state energy calculations, the mass of the confined particle dramatically influences the outcome:

• Heavier particles like protons will have different energy levels compared to lighter particles such as electrons, within the same potential well.
• Mass is directly tied to the kinetic energy of particles within quantum systems.

Due to these properties, the proton mass becomes an essential component in applying quantum mechanics principles to real-world scenarios. Knowledge about proton mass allows for accurate modeling and predictions about atomic behavior and interactions in physics and chemistry. It's vital for students to understand how variations in particle mass affect quantum outcomes and how these principles are applied in broader scientific explorations.