Problem 23 - University Physics with Modern Physics

Question

An electron is in a box of width \(3.0 \times 10^{-10} \mathrm{m} .\) What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the \(n=1\) level; (b) the \(n=2\) level; (c) the \(n=3\) level? In each case how does the wavelength compare to the width of the box?

Step-by-Step Solution

Answer

n=1: λ > L, n=2: λ = L, n=3: λ < L; each momentum is nh/(2L).

Step 1: Understanding the Problem

We need to find the de Broglie wavelength and the momentum of an electron in a box for different energy levels. The box width is given as \(3.0 \times 10^{-10} \text{ m}\). We will use the formulas for the wavelength and momentum of particles confined in a 1D box.

Step 2: Wavefunction and Energy Levels

The de Broglie wavelength, \(\lambda\), of an electron in a box can be derived from its quantum mechanical energy levels. The relation for energy levels is \(E_n = \frac{n^2 h^2}{8mL^2}\), where \(n\) is the energy level, \(h\) is Planck's constant, \(m\) is the mass of the electron, and \(L\) is the length of the box.

Step 3: De Broglie Wavelength Formula

The de Broglie wavelength is given by \(\lambda = \frac{h}{p}\), where \(p\) is the momentum. From the energy expression, we find that \(p = \frac{nh}{2L}\). Therefore, \(\lambda = \frac{2L}{n}\).

Step 4: Calculation for n=1

For \(n=1\), the wavelength \(\lambda = \frac{2L}{1} = 2L\). Substitute \(L = 3.0 \times 10^{-10} \text{ m}\), to find \(\lambda_{n=1} = 6.0 \times 10^{-10} \text{ m}\). The momentum \(p = \frac{h}{2L}\).

Step 5: Calculation for n=2

For \(n=2\), the wavelength \(\lambda = \frac{2L}{2} = L\). Substitute \(L = 3.0 \times 10^{-10} \text{ m}\), to find \(\lambda_{n=2} = 3.0 \times 10^{-10} \text{ m}\). The momentum \(p = \frac{2h}{2L} = \frac{h}{L}\).

Step 6: Calculation for n=3

For \(n=3\), the wavelength \(\lambda = \frac{2L}{3}\). Substitute \(L = 3.0 \times 10^{-10} \text{ m}\), to find \(\lambda_{n=3} = 2.0 \times 10^{-10} \text{ m}\). The momentum \(p = \frac{3h}{2L}\).

Step 7: Comparison with Box Width

Compare the calculated wavelengths for \(n=1, 2, 3\) with the width of the box: \(\lambda_{n=1} > L; \lambda_{n=2} = L; \lambda_{n=3} < L\). Here, \(L\) is the width \(3.0 \times 10^{-10} \text{ m}\).

Key Concepts

Quantum Energy Levels

In quantum mechanics, particles like electrons do not exist at arbitrary energy values. Instead, they can only occupy certain discrete quantities called energy levels. These are denoted by the quantum number \(n\). Each level corresponds to a specific energy the electron can have when confined in a certain space, like the box in our example.
The formula for these energy levels in a one-dimensional box is \(E_n = \frac{n^2 h^2}{8mL^2}\), where \(n\) is an integer representing the energy level, \(h\) is Planck's constant, \(m\) is the mass of the electron, and \(L\) is the width of the box.
As \(n\) increases, the energy of the level increases. This means higher energy levels correspond to electrons that have more energy and are moving faster within the box, leading to shorter wavelengths.

Wavefunction in a Box

The concept of a wavefunction describes the electron’s behavior while inside the box. In quantum mechanics, particles exhibit wave-like properties.
When an electron is in a "box," its behavior can be described by a wavefunction, which mathematically represents the probability of finding the electron at a particular location.
The wavefunction must satisfy the boundary conditions of the box, meaning that at the walls of the box the wavefunction must go to zero. This results in certain patterns or "modes" the wavefunction can take. Each mode corresponds to a different energy level, aligning with the quantum numbers \(n=1, 2, 3,\dots\).

• For \(n=1\), the wavefunction has one half-wave; it is the simplest standing wave that fits within the box.
• For \(n=2\), the wavefunction has one full wave, and so on.

Electron Momentum

Momentum in quantum mechanics takes a slightly different approach compared to classical physics. It's defined in terms of de Broglie wavelength, which relates to how we describe particles as waves.
For our box scenario, the momentum \(p\) of the electron is related to its energy and wavelength by the formula \(p = \frac{nh}{2L}\).
As electrons gain higher energy levels, their momentum increases because they move faster. This means their associated wavelengths become shorter, according to the de Broglie relationship \(\lambda = \frac{h}{p}\).

• Higher momentum corresponds to a higher energy level (larger \(n\)).
• It results in a wave that fits within the box more times.

Quantum Mechanics

Quantum mechanics is the science that deals with the behavior of particles on the smallest scales — like electrons inside atoms or in a box. This branch of physics explains phenomena that cannot be directly observed and calculates properties such as energy levels, wavefunctions, and probabilities.
Fundamental concepts of quantum mechanics include quantization of energy, wave-particle duality, and the uncertainty principle.

• Quantization means energy levels are discrete like the rungs of a ladder, not continuous.
• Wave-particle duality reflects how particles like electrons have both particle features and wave-like behaviors.
• The uncertainty principle limits the precision of simultaneous measurements of certain pairs of properties, such as position and momentum.

Understanding these principles is crucial for solving problems like the electron in a box, where visual and intuitive concepts of classical mechanics fall short.

Particle in a 1D Box

When we talk about a particle in a "1D box," we're looking at a fundamental model in quantum mechanics. Imagine a single particle, like an electron, constrained to move back and forth in a single straight path, much like a tiny train on a track of fixed length L.
The particle can't leave the box and is only allowed to bounce between the two ends.
In this model:

• The walls of the box provide boundary conditions for the electron's wavefunction.
• The model helps us calculate the allowed energy levels and wavefunctions.
• It's used to approximate real systems and gives insight into more complex quantum systems.

This approach lays the foundation for understanding more advanced quantum systems and phenomena, demonstrating the core principles of quantization and wave behavior.