Question

Consider an electron that is confined to a onedimensional infinite potential well of width \(a=0.10 \mathrm{nm}\), and another electron that is confined by an infinite potential well to a three-dimensional cube with sides of length \(a=0.10 \mathrm{nm} .\) Let the electron confined to the cube be in its ground state. Determine the difference in energy and the excited state of the one- dimensional electron that minimizes the difference in energy with the three- dimensional electron.

Short answer

In this problem, we are asked to find the difference in energy between two electrons: one in a one-dimensional infinite potential well and another in a three-dimensional infinite potential well. We need to identify the excited state of the one-dimensional electron that minimizes the energy difference with the three-dimensional electron in its ground state. To solve this, we first need to find the energy levels for electrons in infinite potential wells: \(E_{1D} = \frac{h^2 n^2}{8ma^2}\) for a one-dimensional well and \(E_{3D} = \frac{h^2 (n_x^2 + n_y^2 + n_z^2)}{8ma^2}\) for a three-dimensional well. Next, we calculate the ground state energy for the electron in the 3D cube and find the energy difference with the one-dimensional electron at different energy levels. Finally, we identify the energy level of the 1D electron that results in the minimum energy difference. Through this process, we can find the excited state of the one-dimensional electron that minimizes the energy difference with the three-dimensional electron in its ground state.

Step-by-step solution

Energy levels of electrons in infinite potential wells

For an electron confined in an infinite potential well, the energy levels can be derived using Schrödinger's equation and boundary conditions. The energy levels for a one-dimensional electron and a three-dimensional electron are given by the formulas: One-dimensional well: \(E_{1D} = \frac{h^2 n^2}{8ma^2}\) Three-dimensional well: \(E_{3D} = \frac{h^2 (n_x^2 + n_y^2 + n_z^2)}{8ma^2}\) Here, \(E_{1D}\) and \(E_{3D}\) are the energy levels of the electron in 1D and 3D wells, respectively. \(n\) corresponds to the energy level in the 1D case, whereas \(n_x\), \(n_y\), and \(n_z\) represent the levels in the x, y, and z directions for the 3D case. \(h\) is Planck's constant, \(m\) is the mass of the electron, and \(a=0.10 \mathrm{nm}\) is the width of the one-dimensional well and side length of the cube.

Calculate ground state energy of 3D electron and energy difference

The ground state corresponds to the lowest energy level, which happens when \(n_x = n_y = n_z = 1\). Calculating \(E_{3D}\) for the ground state: \(E_{3D} = \frac{h^2 (1^2 + 1^2 + 1^2)}{8ma^2} = \frac{3h^2}{8ma^2}\) Now, we need to find the energy difference between the one-dimensional electron and the three-dimensional electron. We will do this by computing \(E_{1D}\) for different energy levels \(n\) and find the energy difference \(|\Delta E| = |E_{1D} - E_{3D}|\).

Identify the energy level of the 1D electron that minimizes energy difference

For this step, we will calculate the energy difference for multiple energy levels (n) of the 1D electron and find the minimum energy difference: 1. Start with n=1 (ground state) and calculate \(E_{1D}\) using the given formula. 2. Compute the energy difference \(|\Delta E|\) for each value of n. 3. Identify the energy level that results in the minimum energy difference. By calculating the energy difference for different energy levels, we can identify the excited state of the one-dimensional electron that minimizes the difference in energy with the three-dimensional electron in its ground state.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. It is a branch of science that delves into the behavior and interactions of energy and matter on very small scales, where the classical laws of physics do not always apply. In this realm, particles exhibit both wave-like and particle-like properties, known as wave-particle duality.

One of the most intriguing aspects of quantum mechanics is the concept of quantization, which implies that certain physical properties, such as energy, are not continuous but rather come in discrete 'packets' called quanta. This is significantly different from what we experience in the everyday world, where things seem to change smoothly and continuously.

Schrödinger's Equation

Schrödinger's equation is a key component of quantum mechanics, providing a way to calculate how quantum systems change over time. This equation is a mathematical formula that describes how the quantum state of a physical system changes in time.

For systems with time-independent potentials, the Schrödinger equation can be simplified to what is known as the time-independent Schrödinger equation. This formulation is especially useful for understanding systems where the potential energy does not change, such as particles in a box or, more formally, electrons in an infinite potential well, the focus of our exercise.

Energy Levels of Electrons

In quantum mechanics, the energy levels of electrons are quantized, meaning electrons can only exist in certain discrete energy states. This is starkly different from classical physics, where electrons could, in principle, have any amount of energy.

The energies of these allowed states are determined by the rules governing the quantum system, often derived through Schrödinger's equation. These quantized energy states are crucial to understand phenomena such as the emission and absorption spectra of atoms and also to solve problems concerning electrons in potential wells.

One-Dimensional Potential Well

A one-dimensional potential well is a hypothetical construct in quantum mechanics representing a region where a particle is confined to move along a single axis between two impenetrable barriers. The infinite potential well is a specific case where the potential energy outside the well is considered infinitely large, effectively 'trapping' the particle within.

The energy levels in a one-dimensional infinite potential well can be determined using the time-independent Schrödinger's equation. The allowed energy levels of an electron in such a well are given by the formula mentioned in the step-by-step solution, signifying the quantized nature of energy in quantum systems.

Three-Dimensional Potential Well

The concept of a potential well can be extended to three dimensions, where an electron is confined within a cube-shaped box with infinite potential barriers on all sides. In a three-dimensional potential well, the energy levels become more complex due to the additional degrees of freedom - the electron can now move along the x, y, and z axes.

The energy states in this case depend on three quantum numbers (n_x, n_y, n_z), one for each dimension, as opposed to just one in the one-dimensional case. This results in a spectrum of energy levels that are richer and more complex than the one-dimensional counterpart. As with the one-dimensional well, the energies can be found by solving the time-independent Schrödinger's equation for the three-dimensional case.