Schrödinger equation

The Schrödinger equation is one of the fundamental equations of quantum mechanics and describes the spatial and temporal behavior of quantum-mechanical systems. Austrian physicist Erwin Schrödinger first proposed the equation in early 1926.

Mathematically, the Schrödinger equation is an example of an eigenvalue problem whose eigenvectors are called "wavefunctions" or "quantum states" and whose eigenvalues correspond to energy levels. Built into the eigenvectors are the probabilities of measuring all values of all physical observables, meaning that a solution of the Schrödinger equation provides a complete physical description of a system. The equation can be written in terms of Hamiltonians for both classical and quantum mechanical systems; in the latter case, the Hamiltonian functions are replaced by Hamiltonian operators.

The Schrödinger Wave Equation

The wavefunction describes a wave of probability, the square of whose amplitude is equal to the probability of finding a particle at position x and time t. But what is the form of Schrödinger's equation, which describes the time and position evolution of the wavefunction?

We start by assuming that a beam of particles will have a wavefunction of the form

$\ \psi (x,t)=A\exp(i(kx-\omega t-\phi ))$

The square of this function (its probability amplitude) is a constant independent of position and time, which makes sense for a constant beam of particles: there is an equal probability of finding a particle at every point along the beam and any time. Using de Broglie's relations,

$\ p=\hbar /\lambda =\hbar k$

$\ E=\hbar \omega$

Based on the functional form of $\psi$, we see that

$\ p\psi =\hbar k\psi ={\frac {\hbar }{i}}{\frac {\partial \psi }{\partial x}}$

$E\psi =\hbar \omega \psi =-{\frac {\hbar }{i}}{\frac {\partial \psi }{\partial t}}=\hbar i{\frac {\partial \psi }{\partial t}}$

Using the classical relationship between energy and momentum,

$\ E\psi =[{\frac {p^{2}}{2m}}+V(x)]\psi$

Substituting for p and E yields the one-dimensional time-dependent Schrödinger equation,

$\hbar i{\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\psi }{\partial x^{2}}}+V(x)\psi$

When the probability amplitude of the wavefunction is independent of time, it can be shown that energy is constant, and so the equation reduces to

$E\psi =-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\psi }{\partial x^{2}}}+V(x)\psi$

In three dimensions, the second derivative becomes the Laplacian:

$\hbar i{\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V(x,y,z)\psi$

The Hamiltonian

The time-independent S.E. has the form

${\hat {H}}\psi =E\psi$

where

${\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(x,y,z)$

H is an example of a quantum-mechanical operator, the Hamiltonian (classical Hamiltonians also exist). It must be a self-adjoint operator because its eigenvalues E are the discrete, real energy levels of the system. Also, the various eigenfunctions $\psi _{n}$ must be linearly independent and in fact form a basis for the state space of the system. In other words, the state of any system is reducible to a linear combination of solutions of the Schrödinger equation for that system. The Hamiltonian essentially contains all of the energy "sources" of the system, and its eigenstates describe the possible state of the system entirely.

As an example, consider a particle in a one-dimensional box with infinite potential walls and a finite potential a inside the box. The Hamiltonian of this system is

${\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+a$

The first term of the Hamiltonian corresponds to the classical expression for kinetic energy, p²/2m (see above), and the second term is the potential energy as defined. No other sources of energy exist in the system as defined, and so the particle's state must be some linear combination of the eigenstates of H.

References

1. http://walet.phy.umist.ac.uk/QM/LectureNotes/

1. http://en.wikipedia.org/wiki/Schrodinger_equation