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CODE : 07A1BS05 I B.TECHCSE, IT, ECE & EEE UNIT-2NO. OF SLIDES : 18 2
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UNIT INDEX UNIT-2S.No Module Lectur PPT Slide. e No. No. 1 Waves & Particles - L1 5 Planck’s Quantum theory. 2 De Broglie L2 6-9 hypothesis, matter waves. 3 Verification of matter L3-4 10 waves 4 Heisenberg uncertainty L5 311-12 principle.
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5 Schrödinger’s L6 13-14 time independent wave equation6 Physical L7 15-16 significance of wave function7 Particle in one L8 17-18 dimensional potential box. 4
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Introduction Lecture-11. According to Plank’s quantum theory, energy is emitted in the form of packets or quanta called Photons.2. According to Plank’s law, the energy of photons per unit volume in black body radiation is given by Eλ=8πһс∕λ5[exp(h/טkT) -1] 5
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Waves-particles Lecture-2 According to Louis de Broglie since radiation such as light exhibits dual nature both wave and particle, the matter must also posses dual nature. The wave associated with matter called matter wave has the wavelength λ=h/mט and is called de Broglie wavelength 6
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Characteristics of matter waves Lecture-3Since λ=h/m,ט1. Lighter the particle, greater is the wavelength associated with it.2. Lesser the velocity of the particle, longer the wavelength associated with it.3. For v=0, λ=∞. This means that only with moving particle matter wave is associated.4. Whether the particle is charged or not, matter wave is associated with it. This reveals that these waves are not electromagnetic but a new kind of waves. 7
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6.No single phenomena exhibits both particle nature and wave nature simultaneously.7. While position of a particle is confined to a particular location at any time, the matter wave associated with it has some spread as it is a wave. Thus the wave nature of matter introduces an uncertainty in the location of the position of the particle. Heisenberg’s uncertainty principle is based on this concept. 8
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Difference between matterwave and E.M.wave:: Matter waves E.M.wave1.Matter wave is associated 1.Oscillating charged particlewith moving particle. give rise to e.m. wave.2Wavelength depends on the 2.Wave length depends on themass of the particle and its energy of photonvelocity λ=h/mט λ=hc/E3. Can travel with a velocity 3. Travel with velocity of lightgreater than the velocity of c=3x108 m/slight. 4.Electric field and magnetic4.Matter wave is not field oscillate perpendicular toelectromagnetic wave. each other. 9
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Lecture-4 Davisson and Germer provided experimental evidence on matter wave when they conducted electron diffraction experiments. G.P.Thomson independently conducted experiments on diffraction of electrons when they fall on thin metallic films. x 10
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Heisenberg’s uncertainty principle Lecture-5 “It is impossible to specify precisely and simultaneously the values of both members of particular pair of physical variables that describe the behavior an atomic system”. If ∆x and ∆p are the uncertainties in the measurements of position and momentum of a system, according to uncertainty principle. ∆x∆p≥ h/4π• 11
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9.If ∆E and ∆t are the uncertainties in the measurements of energy and time of a system, according to uncertainty parinciple. ∆E∆t≥ h/4π 12
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Schrödinger wave equation Lecture-6 Schrodinger developed a differential equation whose solutions yield the possible wave functions that can be associated with a particle in a given situation. This equation is popularly known as schrodinger equation. The equation tells us how the wave function changes as a result of forces acting on the particle. 13
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• The one dimensional time independent schrodinger wave equation is given by d2Ψ/dx2 + [2m(E-V)/ ћ2] Ψ=0 (or) d2Ψ/dx2+ [8π2m(E-V) / h2] Ψ=0 14
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Physical significance of Wavefunction Ψ Lecture-71. The wave functions Ψn and the corresponding energies En, which are often called eigen functions and eigen values respectively, describe the quantum state of the particle.2.The wave function Ψ has no direct physical meaning. It is a complex quantity representing the variation of matter wave. It connects the particle nature and its associated wave nature. 15
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3.ΨΨ* or |Ψ|2 is the probability density function. ΨΨ*dxdydz gives the probability of finding the electron in the region of space between x and x+dx, y and y+dy and z and z+dz.If the particle is present∫ ∫ΨΨ*dxdydz=14.It can be considered as probability amplitude since it is used to find the location of the particle. 16
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Particle in one dimensionalpotential box Lecture-8 • Quantum mechanics has many applications in atomic physics. • Consider one dimensional potential well of width L. • Let the potential V=0inside the well and V= ∞outside the well. • Substituting these values in Schrödinger wave equation and simplifying we get the energy of the nth quantum level, 17
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• En=(n2π2ћ2)/2mL2= n2h2/8mL2• When the particle is in a potential well of width L, Ψn=(√2/L)sin(nπ/L)x & En = n2h2/8mL2,n=1,2,3,….• When the particle is in a potential box of sides Lx,Ly,Lz Ψn=(√8/V)sin(nx π/Lx) x sin (ny π/Ly) ysin (nz π/Lz)z.• Where nx, ny or nz is an integer under the constraint n2= nx2+ny2+ nz 2. 18
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