## 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 A2 , so that the number of photons per unit volume  i.e photon density is equal to A2 , 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 A2 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.