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MODULE 1. Magnetism
MODULE 2. Particle Physics
MODULE 3. Modern Physics
MODULE 4. Semicoductor Physics
MODULE 5. Superconductivty
MODULE 6. Optics
5 April - 11 April
12 April - 18 April
19 April - 25 April
26 April - 2 May
LESSON 9. Wave Function and its significance
Physical interpretation Wave Function
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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.
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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.
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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.
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The first and simple interpretation of \[\Psi\] was given by Schrodinger himself in terms of charge density.
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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.
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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.
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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.
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It is observed that in some cases \[\Psi\] is appreciably different from zero within some finite region known as wave packets.
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It is natural to ask
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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 \[{\left| \Psi\right|^2}dxdydz\]
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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 \[\Psi\]
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\[\Psi\] Contains all the measurable information about the particle.
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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 \[{i^2}=1\]
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\[\Psi {\Psi ^*}\] Summed up overall space = 1 the wave function must be normalized.
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\[\Psi\] 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.
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\[\Psi\] allows energy calculation via the Schrodinger Wave Equation
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\[\Psi\] Establishes the probability distribution in three dimensions.
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\[\Psi\] Permits calculations of the most probable value of a given variable.
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\[\Psi\] For a free particle is in sine wave, implying a precisely determined momentum and a totally uncertain position.
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The wave function must be finite everywhere. The particle exists somewhere in the space, therefore, the integral \[\Psi {\Psi ^*}\] overall space must be finite.