Here we explore further the idea introduced in Impact \(19.2\) that quantum mechanical effects need to be invoked in the description of the electronic properties of metallic nanocrystals, here modelled as three-dimensional boxes. (a) Set up the Schrodinger equation for a particle of mass \(m\) in a threedimensional rectangular box with sides \(L_{1}, L_{2}\), and \(L_{3}\). Show that the Schrodinger equation is separable. (b) Show that the wavefunction and the energy are defined by three quantum numbers. (c) Specialize the result from part (b) to an electron moving in a cubic box of side \(L=5 \mathrm{~nm}\) and draw an energy diagram resembling Fig. \(9.2\) and showing the first 15 energy levels. Note that each energy level may consist of degenerate energy states. (d) Compare the energy level diagram from part (c) with the energy level diagram for an electron in a one-dimensional box of length \(L=5 \mathrm{~nm}\). Are the energy levels become more or less sparsely distributed in the cubic box than in the one-dimensional box?

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 the foundation of all quantum physics, including quantum chemistry, quantum field theory, quantum technology, and quantum information science. At its heart, quantum mechanics deals with phenomena that are inherently probabilistic. Unlike classical physics, where objects have predictable and deterministic positions and velocities, quantum particles are described by wavefunctions that give the probabilities of finding the particle at different locations.

The Schrödinger equation is a key component of quantum mechanics. It is a mathematical formula used to describe how the quantum state of a physical system changes over time. In essence, it is the quantum equivalent of Newton's laws of motion for classical mechanics. It's crucial because it facilitates the understanding of how quantum systems evolve, and it's particularly relevant when discussing the behavior of particles within nanocrystals.

Nanocrystals are tiny particles that, due to their size, have unique optical, electronic, and physical properties that differ from their bulk counterparts. These particles often range from 1 to 100 nanometers in size. One of the intriguing aspects of nanocrystals is the quantum mechanical effects that become important at these small sizes. When materials are reduced to the nanoscale, their electronic and optical properties change significantly.

Quantum Confinement in Nanocrystals

The concept of quantum confinement occurs when the dimensions of the nanocrystal are smaller than the de Broglie wavelength of the electrons. This confinement affects the energy levels of the electrons, leading to the quantization of energy states. In the exercise we reviewed, a metallic nanocrystal is modeled as a three-dimensional box to explain these effects using quantum mechanics.

In quantum physics, the quantization of energy refers to the fact that at a microscopic scale, energy is not continuous, but rather comes in discrete packets called quanta. This concept of quantization is fundamental to understanding the behavior of particles in confined systems, such as electrons in atoms or in nanoscale structures like nanocrystals. In the context of a particle in a box, as examined in the exercise, quantized energy levels result from the boundary conditions imposed on the wavefunction of a particle.

The permitted energies are determined by the size and shape of the box, as evidenced by the energy equation derived from the Schrödinger equation. These energies depend on the quantum numbers, which arise due to the solutions of the separated equations in three dimensions, creating quantized energy states specific to the system's constraints.

The wavefunction is a fundamental concept in quantum mechanics, commonly denoted by the Greek letter \( \psi \). It is a mathematical function that provides the probability amplitude for the position, momentum, or other physical properties of a particle. When the square of the wavefunction is taken, it gives the probability density that a particle will be found at a given location or with a particular momentum.

In the Schrödinger equation presented in the exercise, the wavefunction \( \psi(x,y,z) \) represents the quantum state of a particle in the three-dimensional box. By solving the equation for \( \psi \) under the given boundary conditions, it becomes possible to predict the likelihood of finding the particle within various regions of the box, reflecting the probability-based nature of quantum mechanics.

Quantum numbers are sets of numerical values that describe the unique quantum state of a particle and the properties of the particle associated with its wavefunction. Specifically, in a system where a particle is confined, such as the particle in a three-dimensional box shown in our exercise, quantum numbers relate to the quantized energy levels of the system. There are three key quantum numbers for the three spatial dimensions: \( n_x \), \( n_y \) and \( n_z \), which arise from solving the Schrödinger equation independently for each coordinate.

These integers determine the energy levels available to the particle in the box. As each quantum number increases, the energy of the particle in that dimension also increases. For each set of quantum numbers, there is a corresponding wavefunction and thus an energy state. It is important to note that energy levels can be degenerate, meaning different sets of quantum numbers can result in the same energy level, as visualized in the energy level diagram from the exercise.