Origin of quantum mechanics
- 1. revising Quantum Physics K. Muhammed Abdurahman Physics, M.E.S Ponnani College
- 5. End of 19th century - 1885 Ultimate description of nature…!!!!!!! Physics Classical mechanics Dynamics of material bodies Electro magnetic theory Radiation – matter in terms of particle and radiation in waves Thermo dynamics Interaction between matter and radiation
- 6. In the beginning of 20th century, classical physics seriously challenged on two major fronts: validity of classical physics ceases at 1. at very high speeds (v ̴ c): Relativistic domain: Einstein’s 1905 theory of relativity showed that the validity of Newtonian mechanics ceases 2. microscopic level: Microscopic domain: newly discovered phenomena - atomic and subatomic structures – classical physics fails. the and that new concepts had to be invoked to describe, for instance, the structure of atoms and molecules and how light interacts with them.
- 7. Classical physics fails to explain microscopic phenomena like - • Blackbody radiation • photoelectric effect • Compton scattering • atomic stability • atomic spectroscopy
- 8. Max Planck – quantum theory • In 1900 - Quantum of energy • energy exchange between radiation and its surroundings takes place in discrete (quantized) amounts • energy exchange between an electromagnetic wave of frequency ‘ν’ and matter occurs only in integer multiples of hν E = n. hν h is Planck’s constant
- 9. • Planck’s Quantum theory – provides an accurate explanation of blackbody radiation • Photoelectric effect – experiments by Hertz in 1887 • theoretical support by Einstein in 1905 Quantization of em waves – valid for light also • light itself is made of discrete bits of energy called photons, each of energy hν, ν frequency of the light
- 10. • Rutherford’s discovery of the atomic nucleus in 1911 • Bohr model of the hydrogen atom in 1913. • atoms can be found only in discrete states of energy • interaction of atoms with radiation, i.e., the emission or absorption of radiation by atoms, takes place only in discrete amounts of energy • explanation to atomic stability and atomic spectroscopy Bohr’s hydrogen atom model Rutherford’s atomic model Planck’s quantum concept Einstein’s photons
- 11. • in 1923 Compton scattering - confirmation for the corpuscular aspect of light. • series of breakthroughs—due to Planck, Einstein, Bohr, and Compton—gave both the theoretical foundations as well as the conclusive experimental confirmation for the particle aspect of waves • waves exhibit particle behavior at the microscopic scale. • At this scale, classical physics fails not only quantitatively but even qualitatively and conceptually. • de Broglie in 1923 -not only does radiation exhibit particle- like behavior but, conversely, material particles themselves display wave-like behavior. • confirmed experimentally in 1927 by Davisson and Germer
- 12. • Postulates of Planck and assumptions of Bohr - lacking the ingredients of a theory • do not follow from the first principles of a theory. • need to fit them within the context of a consistent theory - lead Heisenberg and Schrödinger to search for the theoretical foundation underlying new ideas.
- 13. • By 1925 - welded the various experimental findings as well as Bohr’s postulates into a refined theory: quantum mechanics – old quantum physics • providing an accurate reproduction of the existing experimental data • theory turned out to explore and unravel many uncharted areas of the microphysical world.
- 14. Two independent formulations of quantum mechanics Schrödinger’s wave mechanics Heisenberg’s Matrix mechanics
- 15. Heisenberg’s Matrix mechanics describe atomic structure from the observed spectral lines • Expressing dynamical quantities such as x, p, E, & L in terms of matrices, he obtained an eigenvalue problem that describes the dynamics of microscopic systems • the diagonalization of the Hamiltonian matrix yields the energy spectrum and the state vectors of the system. • Matrix mechanics was very successful in accounting for the discrete quanta of light emitted and absorbed by atoms
- 16. Schrödinger’s wave mechanics • generalization of the de Broglie postulate. • more intuitive than matrix mechanics • describes the dynamics of microscopic matter by means of a wave equation, called the Schrödinger equation • instead of the matrix eigenvalue problem of Heisenberg, Schrödinger obtained a differential equation. • The solutions of this equation yield the energy spectrum and the wave function of the system under consideration. • square moduli of the wave functions are probability densities.
- 17. Dirac’s formulation • Schrödinger’s wave formulation and Heisenberg’s matrix approach—were shown to be equivalent. • P A M Dirac • general formulation of QM - deals with kets (state vectors), bras, and operators. • The representation of Dirac’s formalism in a continuous basis—the position or momentum representations— Schrödinger’s wave mechanics. • As for Heisenberg’s matrix formulation, representing Dirac’s formalism in a discrete basis
- 18. Quantum mechanics • theory that describes the dynamics of matter at the microscopic scale. • the only valid framework for describing the microphysical world. • It is vital for understanding the physics of solids, lasers, semiconductor and superconductor devices, plasmas, etc. • founding basis of all modern physics: solid state, molecular, atomic, nuclear, and particle physics, optics, thermodynamics, statistical mechanics, and so on. • the foundation of chemistry and biology also.
- 20. Particle Aspect of Radiation • Classical physics: • Particle: energy and momentum – E & p • Wave: amplitude and wave vector – A & k • How radiation interacts with matter…..
- 21. Black body radiation How radiation interacts with matter….. • When heated, a solid object glows and emits thermal radiation. As T increases, the object becomes red, then yellow, then white. • The thermal radiation emitted by glowing solid objects consists of a continuous…!! • the radiation emitted by gases has a discrete distribution spectrum. • continuous character of the radiation emitted by a glowing solid object constituted one of the major unsolved problems • ?- problem - specifying the proper theory of thermodynamics that describes how energy gets exchanged between radiation and matter
- 22. • ideal “blackbody” – material object that absorbs all of the radiation falling on it, and hence appears as black • When an object is heated, it radiates electromagnetic energy as a result of the thermal agitation of the electrons in its surface. • Intensity – function of frequency and temperature. • An object in thermal equilibrium with its surroundings radiates as much energy as it absorbs. • blackbody - perfect absorber & perfect emitter • radiation emitted by a blackbody when hot is called blackbody radiation
- 23. • By the mid-1800s, number of experimental data about blackbody radiation for various objects. • at equilibrium, energy density shows a maximum at a given frequency, which increases with T • the peak of the radiation spectrum occurs at a frequency that is proportional to the temperature • blackbody spectrum was not so easy. • Earlier work - Wilhelm Wien and Rayleigh • total intensity the Stefan–Boltzmann law
- 25. Wien’s energy density distribution energy density per unit frequency of the emitted blackbody radiation: Rayleigh’s energy density distribution understanding the nature of the electromagnetic radiation inside the cavity • harmonic oscillations of a large number of electrical charges, electrons, that are present in the walls of the cavity. • standing waves - equivalent to harmonic oscillators • Radiation- consist of standing waves having a temperature T with nodes at the metallic surfaces.
- 27. • In thermal equilibrium, the electromagnetic energy density inside the cavity is equal to the energy density of the charged particles in the walls of the cavity • Only for low frequencies • diverges for high values of freq. means the cavity contains an infinite amount of energy. • high frequencies - in the ultraviolet range • a real catastrophical failure of classical physics indeed! • Failure - the average energy is continuous ultraviolet catastrophe
- 28. • energy exchange between radiation and matter must be discrete. Wien’s displacement law • the wavelength that corresponds to the maximum of the Planck energy density • to determine the wavelength corresponding to the maximum intensity if the temperature of the body is known. • to determine the temperature of the radiating body if the wavelength of greatest intensity is known. • to estimate the temperature of stars (or of glowing objects) from their radiation Planck’s energy density distribution
- 29. Photoelectric Effect • threshold frequency— depends on the properties of the metal • Instantaneous process. • number of electrons ejected increases with the intensity of the light but does not depend on the light’s frequency. • KE of the photoelectrons depends on the frequency but not on the intensity of the beam; • KE of the photoelectrons increases linearly with the incident frequency.
- 30. • According to classical physics any (continuous) amount of energy can be exchanged with matter an electron would keep on absorbing energy • Einstein introduced concept of photon - • W – work function & threshold frequency ν0. • Stopping potential – Vs
- 34. Compton Effect Scattering of radiation as particles • classical physics - the incident and scattered radiation should have the same wavelength • scattering of X-rays by free electrons – Wavelength of the scattered radiation is larger than the wavelength of the incident radiation. This can be explained only by assuming that the X-ray photons behave like particles. • Incident radiation as a stream of particles—photons— colliding elastically with individual electrons • Elastic collision – conservation of energy and momentum
- 36. Pair Production & pair annihilation - relativistic process • Quantum theory of Schrödinger and Heisenberg is limited to non-relativistic phenomena • Dirac – QM + Special theory of relativity • Relativistic QM - predicted existence of positron. Pair production: • pair production cannot occur in empty space. • photon must interact with an external field Coulomb field of an atomic nucleus to absorb some of its momentum
- 37. • pair annihilation - inverse of pair production, • an electron and a positron collide, they annihilate each other and give rise to electromagnetic radiation • collision of a positron with an electron produces a positronium (hydrogen-like atom) • direct consequence of the mass–energy equation of Einstein E = mc2 • pure energy can be converted into mass and vice versa
- 38. Wave Aspect of Particles de Broglie’s Hypothesis: Matter Waves • In 1923 de Broglie - wave–particle duality is not restricted to radiation, but must be universal: • all material particles possess a dual wave– particle behavior • each material particle of momentum p behaves as a group of waves (matter waves) whose wavelength λ and wave vector k are governed by the speed and mass of the particle
- 39. Davisson–Germer Experiment Experimental Confirmation of de Broglie’s Hypothesis • Mono-energetic (54eV) beam of electrons scattered at Nickel surface. • Maximum intensity at θ = 50o. • instead of the diffuse distribution pattern - material particles, the reflected electrons formed diffraction patterns - identical with Bragg’s X-ray diffraction by a grating
- 41. • Motion of an electron of momentum p must be described by means of a plane wave where A is a constant, k is the wave vector of the plane wave, and ω is its angular frequency • The wave’s parameters, k and ω, are related to the electron’s momentum p and energy E by means of de Broglie’s relations
- 42. Particles versus Waves Classical View of Particles and Waves • Particle: position vector - r(t) • Wave: amplitude A and phase factor φ – • Physical meaning of ψ • Intensity , • Double slit experiments – in the view of both classical and quantum.
- 43. 1. S is a source of streams of bullets
- 44. 2. S is a source of waves
- 45. • Phase term - responsible for the interference pattern • Classically, waves exhibit interference patterns, particles do not. • When two non-interacting streams combine in the same region of space, for particle stream - intensities add; for waves stream - amplitudes add - but their intensities do not
- 46. Quantum View of Particles and Waves
- 48. • When both slits are open, we see a rapid variation in the distribution, an interference pattern • motion gets modified when one watches them • measurements interfere with the states of microscopic objects • the microphysical world is indeterministic • 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
- 49. Wave–Particle Duality: Complementarity • Microscopic systems are neither pure particles nor pure waves, they are both. • The particle and wave manifestations do not contradict or preclude one another, but, are just complementary. • Both concepts are complementary in describing the true nature of microscopic systems. • Being complementary features of microscopic matter, particles and waves are equally important for a complete description of quantum systems.
- 50. Heisenberg’s Uncertainty Principle • Classical physics is thus completely deterministic Δx . Δpx ≥ ћ/2 • If the x-component of the momentum of a particle is measured with an uncertainty Δpx , then its x-position cannot, at the same time, be measured more accurately than Δx=ћ/(2.Δpx) • Energy and time form a pair of complementary variable ΔE . Δt ≥ ћ/2 • two measurements of the energy of a system and if these measurements are separated by a time interval Δt, the measured energies will differ by an amount ΔE which can in no way be smaller than ћ/(2.Δt)
- 52. Atomic Transitions and Spectroscopy Rutherford Planetary Model of the Atom • Rutherford - atom consist of electrons orbiting around a positively charged massive center - nucleus • classical physics - two deficiencies: – atoms are unstable - Maxwell’s electromagnetic theory – electron accelerates - radiates energy - collapses onto the nucleus – atoms radiate energy over a continuous range of frequencies - as the electron orbit collapses, its orbiting frequency increases continuously – continuous spectra • completely disagree with experiment……..
- 53. Bohr Model of the Hydrogen Atom • Rutherford’s planetary model, Planck’s quantum hypothesis, and Einstein’s photon concept - Bohr proposed H-atom model • To account the observed spectrum of the hydrogen atom and its stability. • discrete set of circular stable orbits, stationary states, are allowed • The allowed orbits correspond to those for which the orbital angular momentum of the electron is an integer multiple of h L = nћ Bohr quantization rule of the angular momentum • The radiation corresponding to the electron’s transition is carried out by a photon of energy hν
- 54. Obtain En and radii rn …… • Bohr’s quantization condition leads to a discrete set of energies En and radii rn • electrostatic force applied on electron by the proton = centripetal force • Bohr radius, a0 = 0.053 nm. • Speed of electron is
- 55. • the ratio between the speed of the electron in the first Bohr orbit and the speed of light is equal to a dimensionless constant α, - fine structure constant • Total energy of electron is • R - Rydberg constant = 13.6 eV
- 56. Spectroscopy of the Hydrogen Atom • radiation emitted or absorbed by – glowing solid objects – Continuous – gas - discrete spectrum • Radiation - transition of the electron from a state n to another m has a well defined (sharp) frequency
- 57. • For instance, the Lyman series, which corresponds to the emission of ultraviolet radiation - transitions from excited states n = 2, 3, 4, 5, …to the ground state n = 1 • Balmer series - transitions to the first excited state, n = 2 • The atom emits visible radiation as a result of the Balmer transitions. • Paschen, n = 3 with n > 3; Brackett, n = 4 with n > 4; Pfund, n = 5 with n > 5; and so on. They correspond to the emission of infrared radiation