Engineering Physics 3(2+1)

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.