M. Sc Physics, 2nd Semester Quantum Mechanics-I (PH-515) Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

Contents of Course: Linear Vector Space and Matrix Mechanics Stationary State Approximate Methods Angular Momentum

Books Recommended: Quantum Mechanics, concept and applications by Nouredine Zetili Introduction to Quantum Mechanics by D.J. Griffiths Quantum Mechanics, Vol 1, Cohen Tannoudji The Principle of Quantum Mechanics by P A M Dirac Feynmann Lectures, Vol 3, Quantum Physics Quantum Mechanics by Eugen Merzbacher

Linear Vector Space and Matrix Mechanics Chapter 1 Lecture 1.1 About the Birth of Quantum Mechanics

In classical physics we can define the state of physical system (macroscopic system) with arbitrary accuracy and can study its future behaviour completely. For e.g. Consider particle of mass m subject to force F(x,t) and moving along x-axis. Once we know the position x(t), we can find velocity (v = dx/dt), momentum (p=mv) or any other dynamical variable. We can use the Newton law of motion to study the system.

Everything was fine with the classical physics upto the end of 19th centaury. However, with the advent of advance technology, as one goes to the study of physical systems at microscopic level the classical physics runs into difficulties and contradictions. This led to the birth of quantum mechanics. Quantum mechanics is the description of the behaviour of matter and light in all its details and about the description of phenomena at atomic scale.

We list here few experimental observations which led to the birth of quantum mechanics. Black body radiations and ultraviolet catastrophe Photoelectric effect Compton effect Discrete spectral line Atomic stability Specific heat of solids

Black Body Radiations: A blackbody is a perfect absorber as well as a perfect emitter of radiation. When solids are heated they emit the radiations with continuous distribution of frequency whereas when gases are heated then we get a discrete distribution of spectrum. Figure on next slide shows the experimentally observed spectral energy density of blackbody radiation at different temperatures as a function of frequency.

Using the classical physics Wilhelm Wein (in 1889) and Rayleigh (in 1900) tried to explain the observed spectrum of radiations. Wein failed to explain the spectrum at low frequency whereas the Rayleigh-Jeans Law was failed at high end frequencies (ultraviolet catastrophe). Max-Planck introduced the concept of quantum of energy in 1900. Experimental facts about the black body Radiations are explained by Planck using the concept that the energy exchange between radiation and its surroundings takes place in discrete or quantized amounts. See figure on next slide.

14th December 1900, Birthday of Quantum Mechanics

Photoelectric effect: This effect was experimentally observed in 1887 by Hertz and theoretically explained by Einstein in 1905 using the idea of quantization of light.

Compton Effect: Scattering of X-rays by electron and explained by considering the X-rays as particle (photon) and then collision of X-rays photon with electron.

Wave-particle duality is not restricted to radiations Wave Aspect of Particles: Wave-particle duality is not restricted to radiations but must be universal. All material particles must also possess wave-particle duality. Each material particle of momentum p behaves as a group of waves whose wavelength and wave vector are given by

Davisson Germer Experiment: This experiment prove the wave nature of the material particle.

In optics, when wavelength is very small compared with size of object, we deal with geometrical optics or rays optics. Wave nature of light is not detectable in this case. However, when we cannot neglect the wavelength, one deal with the physical optics. Similarly, when the de Broglie wavelength of object are comparable to size of objects, then these are detectable and cannot neglected. However, when the size of objects exceed considerably as compared to the wavelength, then the wave behaviour is not detectable.

Particle Vs Waves: Double slit experiment: The experiment is used to illustrate the particle and wave properties from classical and quantum physics point of view. Experimental set-up consist of: Source S (Source can be of material particles or waves) A wall with two slits S1 and S2 A back screen equipped with counters that record whatever arrive on it from slits

In Fig 1 Source S fire bullets (macroscopic particle). I1, I2 and I are the intensities of bullets observed on the when S1 or S2 or Both slits are open. Fig 1

Fig 2 In Fig 2 S is Source of waves. Note that when both the slits are open then the total intensity is not just sum of I1 and I2. A oscillating term has to be added. This is because of interference. Fig 2

Classically wave exhibit interference pattern but particles do not. When two non-interacting stream of particles combine in the same region of space their intensities add. When wave combine their amplitude add but their intensities do not.

Fig 3 In Fig 3 S is Source of electron. Note that when both the slits are open then the total intensity is not just sum of I1 and I2. A oscillating term has to be added. This is because of interference which is a property of waves. It means the material particles electron behave as wave. . Fig 3

In Fig 4 S is Source of electron In Fig 4 S is Source of electron. Here a light source is placed behind the wall containing S1 and S2. Now note that interference pattern is destroyed and the total intensity is I = I1+I2 Fig 4

Note that in fig (4), when we use the light source to watch the electrons, the interference pattern destroys. In quantum Mechanics it means measurements interfere with the states Of microscopic objects. We slow down the light source, so that some of the electrons are not seen but we heard on counter. The profile corresponding to unseen electron (but we heard on counter ) shows interference pattern. For the electrons which show the interference pattern, it is not possible to know that through which slit they pass. This means the microscopic world is indterministic.

In classical physics, we can follow the path of macroscopic Particle. But in case of microscopic particles this is not possible. This led Heisenberg to postulate uncertainty principle: It is impossible to design an apparatus which allows us to determine the slit that the electron went through without disturbing the electron enough to destroy the interference pattern. Conclusion: The electron behave both as wave as well as particle. When we keep watching them with the help of light source, they behave as particle. But when we see only the pattern of their distribution on screen after many measurement, they behave as wave.

Atomic Structure: Using the idea of Rutherford’s atomic model, Planck’s quantum concept and Einstein’s Photon, Bohr in 1913 introduced model of hydrogen atom. Atoms can be found in discrete states of energy and emission or absorption of radiations by atom takes place only in discrete amount of hv.

Above outlined new ideas in physics led to the Schrodinger and Heisenberg to formulate a consistent theory known as Quantum mechanics. In 1925 Heisenberg formulated Matrix Mechanics version of Quantum mechanics. In this, the energy, position, momentum, etc are expressed in terms of matrices and Eigen-value equations are solved to describe the dynamics of microphysical system. Second formulation, known as Wave Mechanics was Introduced by Schrodinger in 1926 generalizing de Broglie postulates. In this the dynamics of microphysical systems are described by the Wave equations known as Schrodinger’s equation.

Dirac suggested a more general formalism of quantum Mechanics in which one deal with the abstract objects Such as kets, bras and operators . The representation of Dirac formalism in continuous basis i.e. in position or momentum representation give back Schrödinger’s Wave mechanics. By representing the Dirac formalism in Discrete basis we get Heisenberg’s matrix mechanics.