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General
MODULE 1. Magnetism
MODULE 2. Particle Physics
MODULE 3. Modern Physics
MODULE 4. Semicoductor Physics
MODULE 5. Superconductivty
MODULE 6. Optics
LESSON 9. Wave Function and its significance
Physical interpretation Wave Function

It is now interesting as well as befitting to define of interprets the significance of wave function \[\Psi\] in terms of observable properties associated with the particle or the system.

It should be defined in such wav that any meaningful question about the result of an experiment performed upon the system can be answered if the wave function is known.

In the beginning it was considered that the wave function is merely an auxiliary mathematical quantity employed to facilitate computation relative to the experimental result.

The first and simple interpretation of \[\Psi\] was given by Schrodinger himself in terms of charge density.

We know that in any electromagnetic wave system if A is the amplitude of the wave, then the energy density i.e energy density per unit volume is equal to A^{2} , so that the number of photons per unit volume i.e photon density is equal to A^{2} , so that the number of photons per unit volume i.e photon density equal to \[\frac{{{A^2}}}{{hv}}\] or the photon density in proportion la to A^{2} as hν is constant.

If the \[\Psi\] is amplitude of matter wave at any point in space then the particle density at that point may be taken as proportion to \[{\Psi ^2}\] . Thus \[{\Psi ^2}\] is a measure of particle density.

When this is multiplied be the charge of the particle, the charge density is obtained. In this way \[{\Psi ^2}\] is a measure of charge density.

It is observed that in some cases \[\Psi\] is appreciably different from zero within some finite region known as wave packets.

It is natural to ask

Where is the particle in relation to wave packet? To explain it, Max Born suggested a new idea about the physical significance of \[\Psi\] which is generally accepted now a day. According to him \[\Psi {\Psi ^2}\] = \[{\left \Psi\right^2}\] gives the probabilities of finding the particle in the state \[\Psi\] i.e \[{\Psi ^2}\] is measure of probability density. The probability of finding a particle volume \[dV = \] \[dxdydz\] is given by \[{\left \Psi\right^2}dxdydz\]

For the total probability of finding the particle somewhere is, of course, unit i.e particle is certainly to be found somewhere in space
∫∫∫ \[{\left \Psi\right^2}dxdydz\] = 1
\[\Psi\] Satisfying above requirement is said to be normalized.
Properties of wave function \[\Psi\]

\[\Psi\] Contains all the measurable information about the particle.

The wave function is complex with real and imaginary parts. The complex conjugate of \[\Psi\] is denoted by \[{\Psi ^*}\]
Thus, if
\[\Psi\] = \[X + iY\]
\[{\Psi ^*}\] = \[X  iY\]
However, is always positive and real as \[\Psi {\Psi ^*}={X^2} + {Y^2}\] because \[{i^2}=1\]

\[\Psi {\Psi ^*}\] Summed up overall space = 1 the wave function must be normalized.

\[\Psi\] is continuous , i.e. its partial derivatives are \[\frac{{\partial \Psi }}{{\partial x}}\] \[\frac{{\partial \Psi }}{{\partial y}}\] \[\frac{{\partial \Psi }}{{\partial z}}\] must be also continuous everywhere.

\[\Psi\] allows energy calculation via the Schrodinger Wave Equation

\[\Psi\] Establishes the probability distribution in three dimensions.

\[\Psi\] Permits calculations of the most probable value of a given variable.

\[\Psi\] For a free particle is in sine wave, implying a precisely determined momentum and a totally uncertain position.

The wave function must be finite everywhere. The particle exists somewhere in the space, therefore, the integral \[\Psi {\Psi ^*}\] overall space must be finite.