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Chapter 21: Three Dimensional Quantum Mechanics

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21.1 Introduction to Three Dimensional Quantum Mechanics

While we have primarily discussed one-dimensional quantum mechanics up to this point, real-world quantum systems often exist in three dimensions. The principles of quantum mechanics can be extended to describe the behavior of particles in three-dimensional space, leading to a richer understanding of physical systems and enabling the study of more complex phenomena.

21.2 The Schrödinger Equation in Three Dimensions

The three-dimensional time-independent Schrödinger equation is given by:

-\dfrac{\hbar^2}{2m}\nabla^2\psi(\vec{r})+V(\vec{r})\psi(\vec{r})=E\psi(\vec{r})

where \nabla^2 is the Laplace operator, \psi(\vec{r}) is the wavefunction, V(\vec{r}) is the potential energy, E is the total energy, and \vec{r} represents the position vector in three-dimensional space.

In Cartesian coordinates, the Laplace operator is expressed as:

\nabla^2 = \dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}+\dfrac{\partial^2}{\partial z^2}

21.3 Central Potentials

In many physical systems, the potential energy depends only on the distance from the origin, which is known as a central potential. For central potentials, V(\vec{r}) = V(r), and it is convenient to use spherical coordinates (r, \theta, \phi) to describe the system. The Laplace operator in spherical coordinates is:

\nabla^2 = \dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2\dfrac{\partial}{\partial r}\right) + \dfrac{1}{r^2\sin{\theta}}\dfrac{\partial}{\partial\theta}\left(\sin{\theta}\dfrac{\partial}{\partial\theta}\right) + \dfrac{1}{r^2\sin^2{\theta}}\dfrac{\partial^2}{\partial\phi^2}

21.4 Separation of Variables and Angular Momentum

The three-dimensional Schrödinger equation can often be solved by employing the technique of separation of variables, which involves expressing the wavefunction as a product of radial and angular functions:

\psi(\vec{r}) = R(r)Y(\theta,\phi)

In central potentials, the angular part of the wavefunction is described by spherical harmonics, which are eigenfunctions of the angular momentum operators:

Y_{lm}(\theta, \phi) = Y_l^m(\theta, \phi)

Here, l is the orbital quantum number, and m is the magnetic quantum number. The radial part of the wavefunction, R(r), depends on the specific form of the potential energy.

21.5 Hydrogen Atom

The hydrogen atom is one of the most well-studied systems in three-dimensional quantum mechanics. The potential energy for a hydrogen atom is given by the Coulomb potential:

V(r) = -\dfrac{e^2}{4\pi\epsilon_0r}

The solution to the Schrödinger equation for the hydrogen atom yields energy eigenstates characterized by three quantum numbers: n, l, and m. The energy levels of the hydrogen atom depend only on the principal quantum number n, and are given by:

E_n = -\dfrac{13.6 \text{ eV}}{n^2}

The wavefunctions of the hydrogen atom, known as hydrogen-like orbitals, are given by the product of radial and angular parts:

\psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_{lm}(\theta, \phi)

These orbitals provide valuable insights into the behavior of electrons in atoms and form the foundation for understanding atomic structure and chemical bonding.

Chapter Summary

In this chapter, we extended the principles of quantum mechanics to three-dimensional systems, providing a more complete understanding of the behavior of particles in physical systems. The three-dimensional Schrödinger equation allowed us to study central potentials and separate the wavefunction into radial and angular components. The hydrogen atom served as a key example, with its energy levels and wavefunctions offering crucial insights into atomic structure and chemistry.

Continue to Chapter 22: Introduction to Condensed Matter Physics

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