Tru Physics

Schrödinger Equation

—

by

Tru Physics
in

The Schrödinger equation is a foundational equation in quantum mechanics that describes the evolution of quantum states over time. It was formulated by Erwin Schrödinger in 1926 and provides a mathematical framework for understanding the wave-like behavior of particles at the quantum level. The equation plays a crucial role in determining the behavior and properties of quantum systems, including the energy levels of atoms and the distribution of electrons in molecules.

i \hbar \dfrac{\partial}{\partial t} \Psi (\vec{r},t) = - \dfrac{\hbar^2}{2m} \nabla^2 \Psi (\vec{r},t) + V(\vec{r}) \Psi (\vec{r},t)

Schrödinger Equation on chalkboard.

Time-Independent Schrödinger Equation

The time-independent Schrödinger equation is a simpler form of the equation and is used to analyze the stationary states of a quantum system. The time-independent equation is written most generally as:

\hat{H} | \Psi \rangle = E | \Psi \rangle

where \Psi is the wave function of the system, \hat{H} is the Hamiltonian operator (representing the sum of the kinetic and potential energy operators), and E is the energy of the system. The wave function, \Psi, provides information about the probability density of finding a particle in a given location.

Particle in a Box

A classic example to illustrate the time-independent Schrödinger equation is the “particle in a box” problem. It involves a particle confined within an infinitely deep potential well (or “box”) with zero potential energy inside the box and infinite potential energy outside the box. By solving the time-independent Schrödinger equation for this scenario, one can find the quantized energy levels as well as the wave function of the particle.

Time-Dependent Schrödinger Equation

The time-dependent Schrödinger equation is the more general form of the equation that accounts for the temporal evolution of quantum states. In other words, the time-dependent equation takes into account how the wave function changes with time. Most generally, it is given by:

i \hbar \dfrac{d}{d t} | \Psi (t) \rangle = \hat{H} | \Psi (t) \rangle

where i is the imaginary unit, \hbar is the reduced Planck constant, and t is time. This equation is a partial differential equation that connects the Hamiltonian operator to the time evolution of the wave function.

Separation of Variables

A common approach to solving the time-dependent Schrödinger equation is to use the separation of variables technique. By assuming a solution of the form \Psi (x,t) = \psi(x) \phi(t), one can separate the equation into time-independent and time-dependent components. This results in two ordinary differential equations that can be solved independently, which can then be combined to obtain the full solution for \Psi (x,t).

Hamiltonian Operator

The Hamiltonian operator, H, is a central component of the Schrödinger equation. It represents the total energy of the system and can be written as:

H=- \dfrac{\hbar^2}{2m} \nabla^2 + V(x)

where m is the mass of the particle, \nabla^2 is the Laplacian operator (which represents the second derivative with respect to spatial coordinates), and V (x) is the potential energy of the system.

For systems with multiple particles, the Hamiltonian will include kinetic and potential energy terms for each particle. In the case of atoms and molecules, the potential energy terms will typically involve the Coulomb potential between charged particles (e.g., electrons and nuclei).

Further Methods

At a higher level, the study of the Schrödinger equation involves more advanced mathematical techniques and explores various applications in quantum mechanics. Some of these topics include:

  1. Perturbation Theory: A technique used to approximate the solutions of the Schrödinger equation when the Hamiltonian can be separated into a solvable part and a small perturbation. Perturbation theory is particularly useful in studying the effects of external fields or weak interactions between particles on the energy levels and wave functions of quantum systems.
  2. Variational Method: A technique that provides an upper bound on the ground-state energy of a quantum system by minimizing a trial wave function. The variational method can be used to find approximate solutions to the Schrödinger equation when exact solutions are not possible.
  3. Scattering Theory: The study of how particles interact with each other or with external potentials, often using the Schrödinger equation to describe the behavior of particles before, during, and after collisions. Scattering theory is essential in understanding various physical phenomena, such as the interaction of particles in accelerators or the scattering of light by atoms.
  4. Quantum Mechanics in Higher Dimensions: Solving the Schrödinger equation for systems with more than one spatial dimension requires advanced mathematical techniques, such as Green’s functions or the use of spherical harmonics. Higher-dimensional quantum mechanics is crucial for understanding the behavior of more complex systems, such as atoms and molecules.
  5. Numerical Methods: As many quantum systems do not have exact analytical solutions, numerical methods, such as the finite difference method, finite element method, or the use of basis sets, are often employed to solve the Schrödinger equation. These methods can provide accurate approximations of the energy levels and wave functions for a wide range of systems, enabling the study of various physical and chemical properties.
  6. Relativistic Quantum Mechanics: The Schrödinger equation is a non-relativistic equation, which means it does not account for the effects of special relativity. To study particles moving at speeds close to the speed of light, relativistic quantum mechanics, such as the Klein-Gordon equation or the Dirac equation, must be used.

The Schrödinger equation is a fundamental equation in quantum mechanics that provides a powerful framework for studying the behavior and properties of quantum systems. Its applications range from simple systems like the particle in a box to more complex scenarios, such as atomic and molecular structure or scattering processes. A deep understanding of the Schrödinger equation and its associated mathematical techniques is essential for any physicist working in the field of quantum mechanics.

Want to study more? Visit our Index here!

Do you prefer video lectures over reading a webpage? Follow us on YouTube to stay updated with the latest video content!

Comments

Have something to add? Leave a comment!