Quantum Physics Lecture Notes PDF

AI&DS Today’s Agenda PH 1001 T Lec # 18 22.10.24 1 Dr Prita Nair Topic 3 QUANTUM PHYSICS Wave-particle duality (De Broglie’s hypothesis)  AI&DS Learning Objectives PH 1001 T Lec # 18 22....

AI&DS Today’s Agenda PH 1001 T Lec # 18 22.10.24 1 Dr Prita Nair Topic 3 QUANTUM PHYSICS Wave-particle duality (De Broglie’s hypothesis)  AI&DS Learning Objectives PH 1001 T Lec # 18 22.10.24 2 Dr Prita Nair At the end of this session you should be able to:  Write down the expression for De Broglie’s wavelength and calculate the same for a given particle  Understand when wave nature of particle become more evident  Derive the expression for time dependent and time independent Schrödinger equations in one dimension and extend it to 3D  State when to use time dependent and time Next class independent Schrödinger equations  Explain the physical significance of wave function of a particle  AI&DS Historical Perspective PH 1001 T Lec # 18 22.10.24 3 Dr Prita Nair A little bit of history after the acceptance of the concept of quantization of radiation 1921- Stern & Gerlach experiment-space quantization of angular momentum and spin 1923- Compton Scattering 1924- De Broglie’s hypothesis –Matter waves  1925- Heisenberg’s matrix formulation of quantum mechanics 1926- Erwin Schrödinger–Wave formulation of Q.Mech  1926- Einstein-Bohr debates 1927- Davisson Germer experiment, J.J Thompson 1941- Amplitude formulation by Feynmann 'I personally like to regard a probability wave as a real thing, certainly as more than a tool for mathematical calculations.... how could we rely on probability predictions if we do not refer to something real and objective?' (Max Born on Quantum Theory) 'I think it is safe to say that no one understands quantum mechanics'. (Richard Feynman on Quantum Mechanics)  AI&DS De Broglie’s wavelength PH 1001 T Lec # 18 22.10.24 4 Dr Prita Nair According to De Broglie’s hypothesis , any particle of mass m moving with a velocity v, can be associated with a wave (matter wave). Energy of the particle in motion according to Einstein’s mass energy relation E = mc2 (m- relativistic mass) Energy of a wave (quantum of energy) = h = hc /  Energy of a moving particle of mass m =energy of its associate wave (matter wave) i.e. mc2 = hc / dB Wavelength associated with the particle i.e wavelength of the matter wave called the De Broglie wavelength dB = h / mc = h / p  AI&DS Historical Perspective PH 1001 T Lec # 18 22.10.24 5 Dr Prita Nair 'This suggested that electrons too could not be considered simply as particles, but that frequency (wave properties) must be assigned to them also.' (Louis de Broglie on Quantum Theory Determination of the stable motion of electrons in the atom introduces integers, and up to this point discrete 'orbital' states of electrons) the only phenomena involving integers in physics were those of interference and of normal modes of vibration. This fact suggested to me the idea that electrons too could not be considered simply as particles, but that frequency (wave properties) must be assigned to them also. (Louis de Broglie, 1929, Nobel Prize Speech) Explanation for quantization of angular momentum n nh nh n 2 r = n ; r = = = = 2 2p 2mv mv Angular momentum is quantized  mvr = n  AI&DS Wave nature –particle nature PH 1001 T Lec # 18 22.10.24 6 Dr Prita Nair The wave and particle aspects are complementary and can never appear together in the same experiment. This means that the particle nature and wave nature are separately linked. The wave nature of matter introduces an uncertainty in the location of the position of the particle When will wave nature dominate? particle nature wave nature  AI&DS PH 1001 T Lec # 18 22.10.24 7 Dr Prita Nair Wave packet How do we associate a wave nature to a particle? What could represent both wave and particle? Fits the “wave” description “Localized” in space- particle Answer is “Wave Packet” non zero amplitude in only one location/neighbourhood How do you construct a wave packet? that represents the postition of the particle Superposition- Adding up waves of different different wavelengths and phase  If several waves of different wavelengths (frequencies) and phases are superposed together, one would get a resultant which is a localized wave packet  AI&DS PH 1001 T Lec # 18 22.10.24 8 Dr Prita Nair Phase velocity  ( x, t ) = Ae −i (t − ) Group velocity 2 −i (t − x) med doesn't exactly represent the speed @ which it moves = Ae it represents the rate at which the −i (t − kx ) = phase of that moves Ae group velocity as KE= ½ mv²  - angular frequency k –propagation constant – represents phase change per unit distance https://www.gregegan.net/APPLETS/20/20.html  AI&DS PH 1001 T Lec # 18 22.10.24 9 Dr Prita Nair A wave packet is a group of waves with slightly different wavelengths interfering with one another in a way that the amplitude of the group (envelope) is non-zero only in the neighbourhood of the particle Phase and group velocities What is the velocity of the wave packet?  The velocities of the individual waves which superpose to produce the wave packet representing the particle are different - the wave packet as a whole has a different velocity from the waves that comprise it  Phase velocity: The rate at which the phase of the each wave constituting the wave-packet propagates in space  Group velocity: The rate at which the envelope of the wave packet propagates  For matter waves, group velocity is the particle velocity.  AI&DS Matter Waves PH 1001 T Lec # 18 22.10.24 10 Dr Prita Nair Some points to remember regarding matter waves Matter waves exist only when the particle is in motion. Matter waves are associated with all particles, charged and uncharged. Heavier the particle, smaller the wavelength of the matter wave. Group velocity of the matter wavelet is the velocity of the particle > 3 x 10 Phase velocity of the matter wave is larger than speed of light. Wave and particle nature are complimentary and only one of them is found to dominate in each of the phenomena observed experimentally. The wave nature of the particle also gives rise to the uncertainty in the simultaneous measurement of certain pairs of observable phenomenon like position - momentum or energy -time  AI&DS PH 1001 T Lec # 18 22.10.24 11 Dr Prita Nair Case Study: De Broglie wavelength for an electron moving in a potential V Expression for kinetic energy of an electron moving with velocity v Ek=(m-m0)c2 which reduces to Ek = ½mv2 in the limit (v2/c2