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Lecture-XXII Introduction to Quantum Mechanics Why Quantum Mechanics? Universe consists of material particles and radiation. Dynamics of material particles is described by classical mechanics. Radiation is described by electromagnetism. However, both fail at the atomic level. (Several examples in quantum theory) Need of a new mechanics, a theory to explain physics at the atomic level. Quantum Mechanics Electromagnetic radiation is quantized as photons. Compton scattering, etc) Do particles behave like waves? (Photoelectric effect, Double Slit Experiment: with light waves Slit 1 open: φ1 = A1e − i (ωt − kx1 ) Slit 2 open: φ2 = A2 e 2 , I1 = φ1 = φ1φ1* = A12 − i (ωt − kx2 ) 2 , I 2 = φ2 = φ2φ2* = A22 Both the slits open: φ = φ1 + φ2 = A1e −i(ωt −kx ) + A2 e−i(ωt − kx ) 1 2 2 2 I = φ = (φ1 + φ2 ) = A12 + A22 + 2 A1 A2 cos δ , δ = k ( x1 − x2 ) = 2π λ ( x1 − x2 ) Double Slit Experiment: with bullets Imagine an experimental setup in which a machine gun is spraying bullets at a screen in which there are two narrow openings, or slits. Suppose, in the first instance, that this experiment is carried out with only one slit opened, slit 1 say. Slit1 is open: P1 (x) is the probability that a bullet landing at x. Slit2 is open: P2 (x) is the probability that a bullet landing at x. Both the slits open: It simply shows that the two probabilities are added up. Double Slit Experiment: with electrons The wave nature of electrons is also confirmed by Davisson and Germer by passing electron through a thin gold foil. Matter Wave • Wave-particle duality- matter can behave both like particles as well as waves. Louis de Broglie • If a particle of mass m moves with a velocity v then it behaves like a matter wave having a wavelength λ given by, λ= h h = mv p Why are bullets different from electrons? Bullets are far more massive than the electrons. One can observe them as long as one likes but it would not make any difference to them. There are interference wiggles in the case of bullets also. The are so crowded that it is physically impossible to resolve them, one sees an average behaviour. De Broglie wave length Wave 0 Particle Electron diffraction with spies Electrons are no more waves, they are particles now. The Heisenberg’s uncertainty principle Sets limit on what we can observe. Consider our attempt to view the electrons in the double slit system by shining light of wavelength λ on them with photon momentum pphoton = h/λ . If we manage to see an electron it will be because one of these photons has struck it. Clearly the electron momentum will be affected by this interaction with the photon. Let the change in the electron's momentum is ∆pelectron and certainly it is proportional to pphoton. Let us assume the constant of proportionality to be 1 in the present context. Thus, ∆pelectron = h/λ. Now the precision with which we can determine a distance is limited by the size of the wavelength λ of the light which we use to measure the distance. In fact this uncertainty in position ∆x is directly proportional to the wavelength of the light; again, at our present level of accuracy, we can set the proportionality constant to 1. Thus, ∆xelectron ~ λ Therefore: ∆ x . ∆ p ~ λ h λ ∆x∆p ≥ 2 = h Caution: Uncertainty principle is not an explanation to the null result of double slit experiment as we try to to observe it. Rather it is entirely due to the unavoidable disturbance that we make when we observe the experiment. The Uncertainty Principle implies a built-in, unavoidable limit to the accuracy with which we can make measurements. No precise coordinates, no trajectory.