Question

A proton is in a box of width \(L\). What must the width of the box be for the groundlevel energy to be 5.0 MeV, a typical value for the energy with which the particles in a nucleus are bound? Compare your result to the size of a nucleus-that is, on the order of 10\(^{-14}\) m.

Short answer

The width must be approximately \( 1.44 \times 10^{-14} \text{ m} \). This is similar to the size of a nucleus.

Step-by-step solution

Understand the Problem

We need to calculate the width of a box (potential well) where a proton would have a ground-level energy of 5.0 MeV. This situation is described by the quantum mechanical model for a particle in a box, specifically a 1-dimensional infinite potential well.

Formula for Energy of Particle in a Box

The energy levels for a particle in a one-dimensional box are given by the formula \( E_n = \frac{n^2 h^2}{8mL^2} \) where \( E_n \) is the energy, \( n \) is the quantum number (\( n = 1 \) for ground state), \( h \) is Planck's constant, \( m \) is the mass of the proton, and \( L \) is the width of the box.

Substitute and Rearrange for Box Width

For the ground state, \( n = 1 \), and the energy \( E_1 = 5.0 \text{ MeV} = 5.0 \times 10^6 \times 1.602 \times 10^{-13} \text{ J} \) (converting from MeV to Joules). Rearrange the formula for \( L \) to get \( L = \sqrt{\frac{h^2}{8mE_1}} \).

Plug in Constants

Substitute the known values: Planck's constant \( h = 6.626 \times 10^{-34} \text{ Js} \), mass of proton \( m = 1.673 \times 10^{-27} \text{ kg} \), and \( E_1 = 5.0 \times 10^6 \times 1.602 \times 10^{-13} \text{ J} \) into the equation for \( L \).

Calculate Width of the Box

Perform the calculations. Compute \( L \) using the rearranged formula. This yields \( L \approx 1.44 \times 10^{-14} \text{ m} \).

Compare to Nucleus Size

The calculated box width is \( 1.44 \times 10^{-14} \text{ m} \), which is on the same order of magnitude as the size of an atomic nucleus, typically around \( 10^{-14} \text{ m} \).

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that addresses the behavior of particles at the smallest scales, such as electrons, protons, and even light. Unlike classical mechanics, which works well for large objects, quantum mechanics operates at the scale of atoms and subatomic particles.
At these tiny scales, particles do not follow the simple laws of motion established by Isaac Newton. Instead, they are governed by probability and uncertainty. This means we can only predict the likelihood of a particle being in a particular place at a particular time, not its exact position or path.

• Particles can exhibit wave-like properties, as described by the wave-particle duality.
• Heisenberg's uncertainty principle is a core idea in quantum mechanics which states that the more precisely we know a particle's position, the less precisely we can know its momentum, and vice versa.
• Quantum mechanics also introduces the concept of a quantum state, which is a mathematical entity that provides a probability distribution for every possible outcome of a measurement on the system.

Infinite Potential Well

An infinite potential well, often referenced as a 'particle in a box' problem, is a simplistic yet valuable model in quantum mechanics. It depicts a particle that is confined to a region with infinitely high barriers, which it can never escape.
This model helps in understanding how particles behave when trapped in a confined space, like electrons in an atom or protons in a nucleus.

• The well is considered 'infinite' because the potential energy outside the box is infinitely large; this keeps the particle always inside the box.
• Inside the well, the particle moves freely because the potential energy is zero.
• This model helps calculate energy levels and predicts the quantization of energy, showing that particles can only exist at specific energy levels.

Ground State Energy

The ground state energy is the lowest possible energy that a particle can have when placed in an infinite potential well. It is the most stable state of the particle.
In the particle in a box model, the ground state energy corresponds to the particle's energy level when the quantum number is 1 ( = 1).

• The formula used to find the ground state energy considers factors like the particle's mass, the size of the box, and Planck's constant.
• In our exercise, the ground state energy is provided as 5.0 MeV for the proton in the nuclear scale box.
• Quantization of energy means that at the ground state, the particle has a non-zero minimum energy instead of being at rest (zero energy).

Planck's Constant

Planck's constant is a fundamental constant in quantum mechanics that plays a crucial role in the quantization of energy.
Symbolized by \( h \), it appears in several quantum formulas and relates the energy of photons to their frequency.

• Max Planck introduced this constant to solve problems in black body radiation, marking the beginning of quantum physics.
• Its value is approximately \( 6.626 \times 10^{-34} ext{ Js} \).
• In the context of the infinite potential well, Planck's constant helps determine the scale of the quantum energy levels.

The presence of Planck's constant in the energy formulas signifies the discrete nature of energy levels at the quantum scale.