Question

Calculate the ground state energy for an electron confined to a cube with sides equal to twice the Bohr radius \((R=0.0529 \mathrm{nm}) .\) Determine the spring constant that would give this same ground state energy for a harmonic oscillator.

Short answer

Answer: The spring constant for a harmonic oscillator that would give the same ground state energy as an electron confined to the given cube is approximately \(1.158 \times 10^{-1} \ N/m\).

Step-by-step solution

Calculate the dimensions of the cube

We are given that each side of the cube is twice the Bohr radius, so we need to find the length of the side. The Bohr radius, \(R\), is given as \(0.0529 \ nm\) (nanometers). \(side\_length = 2 * R = 2 * 0.0529 \ nm = 0.1058 \ nm\)

Calculate the ground state energy of an electron in a cube using particle-in-a-box model

Using the particle in a box model, the energy levels of an electron can be calculated by the following formula: \(E_n = \frac{n^2 * h^2}{8 * m * L^2}\) where \(n\) is the principal quantum number (for the ground state, \(n = 1\)), \(h\) is Planck's constant (\(6.626 \times 10^{-34} \ Js\)), \(m\) is the mass of the electron (\(9.109 \times 10^{-31} \ kg\)), and \(L\) is the side length of the cube (which we calculated in Step 1). \(E_1 = \frac{1^2 * (6.626 \times 10^{-34})^2}{8 * (9.109 \times 10^{-31}) * (0.1058 \times 10^{-9})^2} = 1.483 \times 10^{-19} \ J\) We now have the ground state energy for the electron confined within the cube.

Calculate the ground state energy for a quantum harmonic oscillator

The ground state energy for a quantum harmonic oscillator is given by the following equation: \(E_0 = \frac{1}{2} h \omega\) where \(\omega\) is the angular frequency of the oscillator and is related to the spring constant, \(k\), via the equation: \(\omega = \sqrt{\frac{k}{m}}\) We are given the ground state energy for the confined electron already, so our task is to determine the spring constant that would give us the same ground state energy as the confined electron.

Calculate the spring constant for the harmonic oscillator

Knowing that the ground state energies for both systems must be equal, we can set to equal: \(\frac{1}{2} h \omega = E_1\) Now, we can use the relation between \(\omega\) and the spring constant to rewrite the equation in terms of \(k\): \(\frac{1}{2} h \sqrt{\frac{k}{m}} = E_1\) Next, we can solve for the spring constant, \(k\): \(k = \frac{4 * m * E_1^2}{h^2}\) Plugging in the values, we get: \(k = \frac{4 * (9.109 \times 10^{-31} \ kg) * (1.483 \times 10^{-19} \ J)^2}{(6.626 \times 10^{-34} \ Js)^2} = 1.158 \times 10^{-1} \ N/m\) Hence, the required spring constant is approximately \(1.158 \times 10^{-1} \ N/m\).

Key concepts

Ground State Energy

In the fascinating realm of quantum mechanics, the ground state energy refers to the lowest possible energy that a quantum mechanical physical system may have. Unlike classical systems that can continuously lose energy, quantum systems are confined to quantized energy levels. This means that even at their lowest energy level, known as the ground state, they still possess some energy. This intrinsic energy is generally attributed to the Uncertainty Principle, indicating that particles like electrons cannot have zero kinetic energy because their position and momentum cannot both be precisely determined.
Understanding ground state energy is crucial, especially when analyzing physical systems like atoms. It provides insights into the system's stability and reactivity. In our specific example, calculating the ground state energy of an electron in a confined box helps compare this to more complex quantum systems like the quantum harmonic oscillator. This comparison unveils how different quantum limits influence the behavior and characteristics of particles within varied quantum mechanical models.

• Always present in quantum systems.
• Reflects energy uncertainty due to the Heisenberg Uncertainty Principle.
• Key for understanding the stability of quantum systems.

Particle in a Box Model

The particle in a box model is a fundamental quantum mechanics concept that provides a simplified framework to understand quantum systems. Imagine an electron confined within a perfectly rigid box with impenetrable walls; this is essentially the essence of the model.
The walls of the box act as boundaries, preventing the particle from existing outside of them, thus confining it to a finite space. In our exercise, the cube represents this 'box' where an electron is trapped, and we're interested in the lowest energy level it can attain, which is the ground state energy.
The energy levels for this system are discrete, meaning only specific energy values are allowed. This quantization is expressed using the formula: \[E_n = \frac{n^2 h^2}{8 m L^2}\]where:

• \(n\) is the quantum number associated with energy levels.
• \(h\) is Planck's constant.
• \(m\) is the mass of the particle, in this case, an electron.
• \(L\) is the length of the cube's side.

With \(n = 1\) for the ground state, the formula helps us compute the minimal energy level, cementing the understanding of quantum states and illustrating the essence of quantization in physics.

Harmonic Oscillator

The harmonic oscillator model is another cornerstone in quantum mechanics. It's used to describe systems that undergo simple harmonic motion, such as molecules vibrating. In quantum terms, it's essential for modeling vibrations in diatomic molecules and can be further extended to complex organic molecules.
Like the particle in a box model, the quantum harmonic oscillator also displays quantized energy levels. While the particle in a box models confinement through geometric space, the harmonic oscillator considers a potential energy that varies with displacement, commonly modeled by a spring system.
The ground state energy in a quantum harmonic oscillator can be described by the equation:\[E_0 = \frac{1}{2} h \omega\]where:

• \(\omega\) represents the angular frequency of the oscillator.
• Derived from a spring constant \(k\) and mass \(m\), where \(\omega = \sqrt{\frac{k}{m}}\).

This equation indicates that even at its most stable (non-moving) state, the system has energy. In our problem, finding a spring constant that would yield the same ground state energy as the electron in the cube links two elemental quantum systems, unraveling the intricate yet beautifully structured world of quantum mechanics.

Bohr Radius

The Bohr radius is a fundamental physical constant pivotal in describing atomic structures, particularly within the Bohr model of the atom. In its most basic application, the Bohr radius \(a_0 = 0.0529 \, \text{nm}\), denotes the typical distance between the electron and the nucleus in a hydrogen atom at its ground state.
This constant emerges from a perfect balance of the electric forces keeping the electron in a stable orbit around the nucleus with the angular momentum quantization principles.
In our exercise, the Bohr radius plays a crucial role in determining the dimensions of the 'box' or 'cube' in which the electron is confined, as the side length is calculated to be twice the Bohr radius. This showcases a practical application of the Bohr radius beyond philosophical constructs into tangible exercises, helping students visualize and connect abstract quantum concepts with real calculations. Understanding this radius thus becomes vital for comprehending not only atomic models but also for grappling with the complexities of quantum mechanics at fundamental levels.

• Defines basic scale of atomic structures.
• Crucial for calculating electron confinement scenarios.
• Connects classical and quantum atomic models effectively.