Problem 69 - University Physics with Modern Physics

Question

(a) The wave nature of particles results in the quantum-mechanical situation that a particle confined in a box can assume only wavelengths that result in standing waves in the box, with nodes at the box walls. Use this to show that an electron confined in a one-dimensional box of length \(L\) will have energy levels given by $$E_{n}=\frac{n^{2} h^{2}}{8 m L^{2}}$$ (Hint: Recall that the relationship between the de Broglie wave-length and the speed of a nonrelativistic particle is \(m v=h / \lambda\) . The energy of the particle is \(\frac{1}{2} m v^{2} . )\) (b) If a hydrogen atom is modeled as a one- dimensional box with length equal to the Bohr radius, what is the energy (in electron volts) of the lowest energy level of the electron?

Step-by-Step Solution

Answer

The energy of the lowest level is approximately 13.6 eV.

Step 1: Understand the Standing Wave Condition

When a particle is confined in a one-dimensional box, it can only form standing waves that fit perfectly within the box. The condition for standing waves is that the length of the box must be an integer multiple of half wavelengths. Thus, we have: \( L = n \frac{\lambda}{2} \), where \( n \) is an integer (1, 2, 3,...).

Step 2: Express Wavelength in terms of Box Length

Rearrange the standing wave condition to express \( \lambda \). We get \( \lambda = \frac{2L}{n} \).

Step 3: Relate Wavelength to Momentum

According to the de Broglie relationship, the wavelength \( \lambda \) is related to the momentum \( p \) by \( p = \frac{h}{\lambda} \). Substituting the expression for \( \lambda \), we have \( p = \frac{nh}{2L} \).

Step 4: Express Energy using Kinetic Energy Formula

The kinetic energy of a particle is given by \( KE = \frac{1}{2}mv^2 \). From the momentum equation \( p = mv \), we know \( v = \frac{p}{m} \). Substituting \( p = \frac{nh}{2L} \) into \( v = \frac{p}{m} \), we get \( v = \frac{nh}{2mL} \).

Step 5: Substitute Velocity in Kinetic Energy Formula

Substitute the expression of \( v \) into the kinetic energy formula: \[ KE = \frac{1}{2}m\left(\frac{nh}{2mL}\right)^2 = \frac{n^2h^2}{8mL^2} \]. This expression matches the given energy level formula \( E_n = \frac{n^2h^2}{8mL^2} \).

Step 6: Calculate Energy for Hydrogen Atom in a Box

For part (b), use the Bohr radius \( a_0 = 5.29 \times 10^{-11} \) m as the length \( L \). The energy of the lowest level (\( n=1 \)) is given by \( E_1 = \frac{1^2 h^2}{8mL^2} \). Substitute \( h = 6.626 \times 10^{-34} \) Js and \( m = 9.109 \times 10^{-31} \) kg. Calculate \[ E_1 = \frac{(6.626 \times 10^{-34})^2}{8 \times 9.109 \times 10^{-31} \times (5.29 \times 10^{-11})^2} \].

Step 7: Convert Energy to Electron Volts

Convert the energy from Joules to electron volts using the conversion factor 1 eV = \( 1.602 \times 10^{-19} \) J. Calculate the energy in electron volts \( E_1 \) obtained in the previous step.

Key Concepts

de Broglie wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics, illustrating the wave-particle duality of matter. According to de Broglie, every particle exhibits wave-like properties. For a particle with momentum \( p \), its wavelength \( \lambda \) is given by the de Broglie equation \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant. This equation shows that as a particle's momentum increases, its associated wavelength decreases.

In practical terms, the de Broglie wavelength becomes significant at atomic and subatomic scales, like that of electrons. The notion is essential in understanding phenomena such as electron diffraction and quantum mechanics' inconsistency with classical physics. It's this wave property of particles that allows them to form standing wave patterns in confined systems.

Understanding the de Broglie wavelength is crucial to exploring the quantum world, where particles do not behave like the solid objects in our everyday experiences. Instead, they present a dual nature that influences their behavior in constrained spaces.

particle in a box

The 'particle in a box' model is a simplified quantum system that helps to conceptualize how particles behave when confined to a restricted space. In this scenario, the particle is free to move within a box but cannot escape its walls. This model is pivotal in quantum mechanics for demonstrating quantization—the idea that a particle can only occupy specific energy levels within a given system.

Within a one-dimensional box, quantization arises from the necessity for the particle to form standing wave patterns with nodes at the box walls. Mathematically, this condition leads to quantized wavelengths \( \lambda_n = \frac{2L}{n} \), where \( n \) is an integer, and \( L \) is the length of the box.

• The quantization implies that only certain energy levels \( E_n \) are possible, given by \( E_n = \frac{n^2h^2}{8mL^2} \).
• It effectively shows that discrete energy levels are an inherent feature of the quantum world.

The particle in a box model thus serves as a foundational concept that not only facilitates calculations but also underscores the unique characteristics of quantum systems.

energy levels

Energy levels in quantum mechanics refer to the fixed values of energy that a quantum system can have. This discreteness is a departure from classical physics, where energy can be any value. In a system like the 'particle in a box', these energy levels are derived based on the requirement for the particle to form standing waves with specific wavelengths.

The energy level formula \( E_n = \frac{n^2h^2}{8mL^2} \) for a particle in a one-dimensional box indicates that energy levels increase with the square of the quantum number \( n \).

• Lowest possible energy happens when \( n = 1 \), known as the ground state energy.
• Higher numbers correspond to excited states, where the particle has more energy.

Understanding these quantized energy levels is critical for explaining phenomena such as atomic spectra and the behavior of electrons in atoms and solids. The concept is fundamental to various applications, including electronic devices and spectroscopy.

hydrogen atom

The hydrogen atom is a key model in quantum mechanics, as it is the simplest atom consisting of one proton and one electron. By utilizing quantum mechanics principles, we can explain the hydrogen atom's energy levels, which manifest in its emission spectrum. These levels are understood by applying models like Bohr's, which equate electron wave patterns in circular orbits to ground and excited states through quantum numbers.

In quantum mechanics, if we model the hydrogen atom as a particle in a box with the box length equivalent to the Bohr radius (\( a_0 \)), we can estimate the energy of its lowest state. The Bohr radius is approximately \( 5.29 \times 10^{-11} \) meters, representing the most probable distance of the electron from the nucleus in its ground state.

By calculating the energy using the particle in a box model, we observe that the electron's energy levels are quantized, consistent with observed hydrogen spectral lines. These quantized levels provide insight into electron transitions, essential for understanding how light is absorbed or emitted by atoms.