Overview of the Schroedinger Equation in Physics and Chemistry

The Schroedinger equation is a quantum mechanical differential equation that describes the energetics of atoms and molecules.  In addition, its solution of atomic and molcular wavefunctions gives the shape and energy of the various atomic and molecular orbitals of the molecule under investigation. The Schroediger equation is a linear differential equation which involves complex numbers.  It was formulated by the Austrian physicist, Erwin Schroedinger, in the early years of the 19th century.

It was formulated after the discovery by the French physicist De-broglie that matter can have wave properties.  The Schroedinger equation includes wave as well as particles parameters.  Its Hamltonian is a differential form of the classical Hamiltonian, written in quantum mechanical operators form.  It involves energy as well as momentum operators. 

Energy in the classical sense has a quantum mechanical formthat is represented by a complex operator which includes a derivative with respect to time.  The kinetic energy in the classical sense is written as P*2/ 2m, where P is the particle momentum and m is its mass.  The momentum of classical mechanics has a quantum mechanical analogue that has a complex structure, written as an operator.

Due to the presence of potential energy term in the quantum mechanical Hamiltonian, it is difficult to solve the Schroedinger equation accurately.  The quantum mechanical Hamiltonian which involves the Schroedinger equation satisfies an eigen value relation which is written in the following operator form H(psi)= E(psi),  where psi is the wave function.  The solution of this differential equation for a particular system gives the energy of the system in addition to the wave function that describes its orbitals.

The wave function that describes electrons is usually complex wave function that involves imaginary numbers.  The Hamiltonian energies usually have real values that do not involve imaginary numbers.  Hamiltonian operators in particular and operators in general that have real energy eigen values are called Hermitian operators.   This is in contrast to non-Hermitian operators which do not represent physical variables and hence have complex energy values when solving the Schroedinger equation for these systems. 

The Schroedinger equation is a non-relativistic description of motion and energetics of electrons.  It does not involve for example spin functions which are manifestations of relativistic effect that arise due to the solution of another quantum mechanical equation for particles that are called fermions and which have have non integer spin values.  This equation is called Dirac equation. 

As an example, spin-orbit coupling in the hydrogen atom is not accounted for by the solution of the regular Schroedinger equation which does not involve relativistic form of the Hamiltonian.  L-S coupling or spin-orbit coupling arises naturally from the solution of Dirac equation for the hydrogen atom.   

Due to its formidable mathematics, the solution of the Schroedinger equation for many physical systems is not possible to perform accurately.  This is so due to the existence of potential energy perturbations that does not allow for exact solution of the equation.  An example is the spin-orbit coupling in the hydrogen atom which is a relativistic effect that adds an extra potential energy term to the Hamiltonian due to the interaction of the electron spin and angular magnetic moment with a magnetic field that arises due to the relative nuclear motion of the nulceus around the electron. 

The Schroedinger equation can be solved exactly for simple systems such as the hydrogen atom and the particle in a box which is a modeling system for the electrons in the atom which are held by a force that restrict their motion.