2 S.No. ModuleLectur e No. PPT Slide No. 1 Waves & Particles - Planck’s Quantum theory. L15 2 De Broglie hypothesis, matter waves. L Verification of matter waves L Heisenberg uncertainty principle. L UNIT INDEX UNIT-2
3 5 Schrödinger’s time independent wave equation L Physical significance of wave function L Particle in one dimensional potential box. L817-18
4 Introduction Lecture-1 1.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 E λ =8πһс∕λ 5 [exp(h ט /kT) -1]
5 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/m ט and is called de Broglie wavelength
6 Characteristics of matter waves Since λ=h/m ט, 1.Lighter the particle, greater is the wavelength associated with it. 2.Lesser the velocity of the particle, longer the wavelength associated with it. v=0, 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. Lecture-3
7 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 Difference between matter wave and E.M.wave:: Matter waves E.M.wave 1.Matter wave is associated with moving particle. 2Wavelength depends on the mass of the particle and its velocity λ=h/m ט 3. Can travel with a velocity greater than the velocity of light. 4.Matter wave is not electromagnetic wave. 1.Oscillating charged particle give rise to e.m. wave. 2.Wave length depends on the energy of photon λ=hc/E 3. Travel with velocity of light c=3x10 8 m/s 4.Electric field and magnetic field oscillate perpendicular to each other.
9 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 Lecture-4
10 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/4 π
11 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/4 π
12 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 The one dimensional time independent schrodinger wave equation is given by d 2 Ψ/dx 2 + [2m(E-V)/ ћ 2 ] Ψ=0 (or) d 2 Ψ/dx 2 + [8π 2 m(E-V) / h 2 ] Ψ=0
14 Physical significance of Wave function Ψ Lecture-7 1. The wave functions Ψ n and the corresponding energies E n, 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 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=1 4.It can be considered as probability amplitude since it is used to find the location of the particle.
16 Particle in one dimensional potential 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 n th quantum level,
17 En=(n 2 π 2 ћ 2 )/2mL 2 = n 2 h 2 /8mL 2 When the particle is in a potential well of width L, Ψ n =(√2/L)sin(nπ/L)x & En = n 2 h 2 /8mL 2,n=1,2,3,…. When the particle is in a potential box of sides L x, L y, L z Ψ n =(√8/V)sin(n x π/L x ) x sin (n y π/L y ) ysin (n z π/L z )z. Where n x, n y or n z is an integer under the constraint n2= n x 2 +n y 2 + n z 2.