5 introduction to quantum mechanics

*IntroductiontoQuantum MechanicsSOLO HERMELINUpdated: 11.11.13 10.10.14

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Physics The Presentation is my attempt to study and cover the fascinating subject of Quantum Mechanics. The completion of this presentation does not make me an expert on the subject, since I never worked in the field. I thing that I reached a good coverage of the subject and I want to share it. Comments and suggestions for improvements are more than welcomed.*SOLOIntroduction to Quantum Mechanics

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Physics1900 At the end of the 19th century, physics had evolved to the point at which classical mechanics could cope with highly complex problems involving macroscopic situations; thermodynamics and kinetic theory were well established; geometrical and physical optics could be understood in terms of electromagnetic waves; and the conservation laws for energy and momentum (and mass) were widely accepted. So profound were these and other developments that it was generally accepted that all the important laws of physics had been discovered and that, henceforth, research would be concerned with clearing up minor problems and particularly with improvements of method and measurement."There is nothing new to be discovered in physics now. All that remains is more and more precise measurement" - Lord Kelvin1900: This was just before Relativity and Quantum Mechanics appeared on the scene and opened up new realms for exploration. Completeness of a Theory*SOLO

http://en.wikipedia.org/wiki/History_of_physicshttp://amasci.com/weird/end.html*

*Classical TheoriesSOLO

1.1 Newtons Laws of MotionThe Mathematical Principles of Natural Philosophy 1687First Law Every body continues in its state of rest or of uniform motion instraight line unless it is compelled to change that state by forcesimpressed upon it.Second Law The rate of change of momentum is proportional to the forceimpressed and in the same direction as that force.Third Law To every action there is always opposed an equal reaction.

*SOLO1.2 Work and EnergyThe kinetic energy T is defined as:For a constant mass mClassical Theories

*SOLOWork and Energy (continue)Classical Theories

*SOLOWork and Energy (continue)andBut also for a constant mass system Therefore for a constant mass in a conservative field Classical Theories

SOLO1.5 Rotations and Angular MomentumClassical Theories- Angular Rotation Rate of the Body (B) relative to Inertia (I)- ForceAngular Momentum

Relative to C.G.- Momentum

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SOLO1.6 Lagrange, Hamilton, JacobiClassical TheoriesLagrangiamsHamiltonian*

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*SOLO1.4 Basic DefinitionsGiven a System of N particles. The System is completely defined by Particles coordinates and moments: The path of the Particles are defined by Newton Second LawIn Classical Mechanics:Time and Space are two Independent Entities.No limit in Particle VelocitySince every thing is Deterministic we can Measure all quantities simultaneously.

The outcome of all measurements are repeatable and depends only on the accuracy of the measurement device.Causality: Every Effect hase a Cause that preceed it.

Classical Theories

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GRAVITYClassical Theories

Newtons Law of Universal GravityAny two body attract one another with a Force Proportional to the Product of theMasses and inversely Proportional to theSquare of the Distance between them.G = 6.67 x 10-8 dyne cm2/gm2 the Universal Gravitational ConstantInstantaneous Propagation of the Force along the direction between the Masses (Action at a Distance).*

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Newton was fully aware of the conceptual difficulties of his action-at-a-distance theory of gravity. In a letter to Richard Bentley Newton wrote: "It is inconceivable, that inanimate brute matter should, without the mediation of something else, which is not material, operate upon, and affect other matter without mutual contact; as it must do, if gravitation, ...., be essential and inherent in it. And this is one reason, why I desired you would not ascribe innate gravity to me. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another, at a distance through vacuum, without the mediation of anything else, by and through their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking, can ever fall into it."GRAVITYClassical Theories*

*SOLONewton published Opticks1704 Newton threw the weight of his authority on the corpuscular theory. This conviction was due to the fact that light travels in straight lines, and none of the waves that he knew possessed this property. Newtons authority lasted for one hundred years, and diffraction results of Grimaldi (1665) and Hooke (1672), and the view of Huygens (1678) were overlooked. Optics Every point on a primary wavefront serves the source of spherical secondary wavelets such that the primary wavefront at some later time is the envelope o these wavelets. Moreover, the wavelets advance with a speed and frequency equal to that of the primary wave at each point in space.Huygens Principle 1678Light: Waves or ParticlesClassical Theories

Wavefront

Sources

*SOLO In 1801 Thomas Young uses constructive and destructive interference of waves to explain the Newtons rings.1801 - 1803 In 1803 Thomas Young explains the fringes at the edges of shadows using the wave theory of light. But, the fact that was belived that the light waves are longitudinal, mad difficult the explanation of double refraction in certain crystals.OpticsYoung Double Slit ExperimentClassical Theories

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*POLARIZATION Arago and Fresnel investigated the interference of polarized rays of light and found in 1816 that tworays polarized at right angles to each other never interface.SOLO Arago relayed to Thomas Young in London the resultsof the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillationsin the optical wave where transverse, or perpendicular to the direction of propagation, and not longitudinal as every proponent of wave theory believed.1816 - 1817Classical Theories

*SOLO

In 1818 Fresnel, by using Huygens concept of secondary wavelets and Youngs explanation of interface, developed the diffraction theory of scalar waves.1818Diffraction - HistoryClassical Theories

*Diffraction SOLO

In 1818 Fresnel, by using Huygens concept of secondary wavelets and Youngs explanation of interface, developed the diffraction theory of scalar waves. According to Huygens Principle second wavelets will start at the aperture and will add at the image point P. Obliquity factor and /2 phase were introduced by Fresnel to explain experiences results.Fresnel Diffraction FormulaFresnel took in consideration the phase of each wavelet to obtain:Classical Theories

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MAXWELLs EQUATIONSSOLO1. AMPRES CIRCUIT LAW (A) 1821 2. FARADAYS INDUCTION LAW (F) 18313. GAUSS LAW ELECTRIC (GE) ~ 1830 4. GAUSS LAW MAGNETIC (GM) James Clerk Maxwell(1831-1879) 1865Electromagnetism

MAXWELL UNIFIED ELECTRICITY AND MAGNETISMClassical Theories

*SOLOELECTROMGNETIC WAVE EQUATIONS For Homogeneous, Linear and Isotropic MediumFor Source lessMedium

Definewherec is the velocity of light in free space.ElectromagnetismClassical Theories

Completeness of a TheorySOLO At the end of the 19th century, physics had evolved to the point at which classical mechanics could cope with highly complex problems involving macroscopic situations; thermodynamics and kinetic theory were well established; geometrical and physical optics could be understood in terms of electromagnetic waves; and the conservation laws for energy and momentum (and mass) were widely accepted. So profound were these and other developments that it was generally accepted that all the important laws of physics had been discovered and that, henceforth, research would be concerned with clearing up minor problems and particularly with improvements of method and measurement."There is nothing new to be discovered in physics now. All that remains is more and more precise measurement" - Lord Kelvin1900:1894: "The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote.... Our future discoveries must be looked for in the sixth place of decimals." - Albert. A. Michelson, speech at the dedication of Ryerson Physics Lab, U. of Chicago 1894 This was just before Relativity and Quantum Mechanics appeared on the scene and opened up new realms for exploration. *Classical Theories

http://en.wikipedia.org/wiki/History_of_physicshttp://amasci.com/weird/end.html*

*QUANTUM THEORIES

http://www.mathematik.uni-muenchen.de/~bohmmech/Poster/post/postE.htmlThe Double Slit ExperimentSOLO*QUANTUM THEORIEShttps://www.youtube.com/watch?v=Q1YqgPAtzho&src_vid=4C5pq7W5yRM&feature=iv&annotation_id=annotation_3921431599

http://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics#Schr.C3.B6dinger_wave_equation*

According to the results of the double slit experiment, if experimenters do something to learn which slit the photon goes through, they change the outcome of the experiment and the behavior of the photon. If the experimenters know which slit it goes through, the photon will behave as a particle. If they do not know which slit it goes through, the photon will behave as if it were a wave when it is given an opportunity to interfere with itself. The double-slit experiment is meant to observe phenomena that indicate whether light has a particle nature or a wave nature. Richard Feynman observed that if you wish to confront all of the mysteries of quantum mechanics, you have only to study quantum interference in the two-slit experimentThe Double Slit ExperimentSOLO*QUANTUM THEORIES

http://en.wikipedia.org/wiki/Wheeler's_delayed_choice_experiment*

QUANTUM THEORIESSome trajectories of a harmonic oscillator (a ball attached to a spring) in classical mechanics (AB) and quantum mechanics (CH). In quantum mechanics (CH), the ball has a wave function, which is shown with real part in blue and imaginary part in red. The trajectories C,D,E,F, (but not G or H) are examples of standing waves, (or "stationary states"). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy*

http://en.wikipedia.org/wiki/Wave_function*

Modern Physics1900*

http://www.bubblews.com/news/401138-what-is-quantum-theory*QUANTUM THEORIES

*SOLOhttp://thespectroscopynet.com/educational/Kirchhoff.htmSpectroscopy1868 A.J. ngstrm published a compilation of all visible lines inthe solar spectrum. 1869 A.J. ngstrm made the first reflection grating. Anders Jonas Angstrm a physicist in Sweden, in 1853 had presented theories about gases having spectra in his work: Optiska Underskningar to the Royal Academy of Sciences pointing out that the electric spark yields two superposed spectra. Angstrm also postulated that an incandescent gas emits luminous rays of the same refrangibility as those which it can absorb. This statement contains a fundamental principle of spectrum analysis. http://en.wikipedia.org/wiki/Spectrum_analysis

*J. R. Meyer-Arendt, Introduction to Classical and Modern Optics, Prentice Hall, 3th Ed., 1989, pg.5

*ParticlesSOLO1874George Johnstone Stoney 1826 - 1911 As early as 1874 George Stoney had calculated the magnitude of his electron from data obtained from the electrolysis of water and the kinetic theory of gases. The value obtained later became known as a coulomb. Stoney proposed the particle or atom of electricity to be one of three fundamental units on which a whole system of physical units could be established. The other two proposed were the constant universal gravitation and the maximum velocity of light and other electromagnetic radiations. No other scientist dared conceive such an idea using the available data. Stoney's work set the ball rolling for other great scientists such as Larmor and Thomas Preston who investigated the splitting of spectral lines in a magnetic field. Stoney partially anticipated Balmer's law on the hydrogen spectral series of lines and he discovered a relationship between three of the four lines in the visible spectrum of hydrogen. Balmer later found a formula to relate all four. George Johnstone Stoney was acknowledged for his contribution to developing the theory of electrons by H.A. Lorentz , in his Nobel Lecture in 1902. George Stoney estimates the charge of the then unknown electron to be about 10-20 coulomb, close to the modern value of 1.6021892 x 10-19 coulomb. (He used the Faraday constant (total electric charge per mole of univalent atoms) divided by Avogadro's Number.

*http://www.universityscience.ie/pages/scientists/sci_georgestoney.phphttp://www.acmi.net.au/AIC/TV_HIST_CATHRAY.html

*Physical Laws of RadiometrySOLOStefan-Boltzmann Law Stefan 1879 Empirical - fourth power law Boltzmann 1884 Theoretical - fourth power law For a blackbody:Stefan-Boltzmann Law 187918841893Wiens Displacement Law Wien 1893from which:Nobel Prize 1911

*SOLOJohan Jakob Balmer presented an empirical formula describing the position of the emission lines in the visible part of the hydrogen spectrum.Spectroscopy1885Balmer Formula Balmer was a mathematical teacher who, in his spare time, was obsessed with formulae for numbers. He once said that, given any four numbers, he could find a mathematical formula that connected them. Luckily for physics, someone gave him the wavelengths of the first four lines in the hydrogen spectrum.

*Beiser, Perspectives of Modern Physics, McGraw Hill-Kogakusha, International Student Edition, 1961, Ch. 6

T. Hey, P. Walters, The Quantum Universe, Cambridge University Press, 1987, pp.39-40http://en.wikipedia.org/wiki/Balmer_series

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SOLOSpectroscopy1887Johannes Robert Rydberg 1854 - 1919Rydberg generalized Balmers hydrogen spectral lines formula.

*http://en.wikipedia.org/wiki/Johannes_Rydberghttp://en.wikipedia.org/wiki/Rydberg_constanthttp://faculty.rmwc.edu/tmichalik/lyman.htmhttp://de.wikipedia.org/wiki/Friedrich_Paschenhttp://www.findagrave.com/cgi-bin/fg.cgi?page=gr&GRid=86737504

*PhotoelectricitySOLO In 1887 Heinrich Hertz, accidentally discovered the photoelectric effect.Hertz conducted his experiments that produced radio waves. By chance he noted that a piece of zinc illuminated by ultraviolet light became electrically charged. Without knowing he discovered the Photoelectric Effect.1887http://en.wikipedia/wiki/Photoelectric_effecthttp://en.wikipedia/wiki/Heinrich_Hertz

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*Spectroscopy SOLOZeeman Effect Pieter Zeeman observed that the spectral lines emitted by an atomic source splited when the source is placed in a magnetic field. In most atoms, there exists several electron configurations that have the same energy, so that transitions between different configuration correspond to a single line. 1896 Because the magnetic field interacts with the electrons, this degeneracy is broken giving rice to very close spectral lines.http://en.wikipedia.org/wiki/Zeeman_effectNobel Prize 1902

*A. Beiser, Perspectives of Modern Physics, McGraw Hill-Kogakusha, International Student Edition, 1961,

*Physical Laws of Radiometry SOLOWien Approximation to Black Body Radiation Wien's Approximation (also sometimes called Wien's Law or the Wien Distribution Law) is a law of physics used to describe the spectrum of thermal radiation (frequently called the blackbody function). This law was first derived by Wilhelm Wien in 1896. The equation does accurately describe the short wavelength (high frequency) spectrum of thermal emission from objects, but it fails to accurately fit the experimental data for long wavelengths (low frequency) emission.Comparison of Wien's Distribution law with the RayleighJeans Law and Planck's law, for a body of 8 mK temperatureThe Wien s Law may be written aswhere I(,T) is the amount of energy per unit surface area per

unit time per unit solid angle per unit frequency emitted at a frequency . T is the temperature of the black body. h is Planck's constant. c is the speed of light. k is Boltzmann's constant

1896

http://en.wikipedia.org/wiki/Wien_radiation_lawhttp://en.wikipedia.org/wiki/Wien_approximation*

*SOLOParticles J.J. Thomson showed in 1897 that the cathode rays are composed of electrons and he measured the ratio of charge to mass for the electron. Discovery of the Electron1897 The total charge on the collector (assuming all electrons are stick to the cathode collector and no secondary emissions is: The energy of the particles reaching the cathode is:

*Physical Laws of Radiometry SOLORayleighJeans Law In 1900, the British physicist Lord Rayleigh derived the 4 dependence of the RayleighJeans law based on classical physical arguments.[3] A more complete derivation, which included the proportionality constant, was presented by Rayleigh and Sir James Jeans in 1905. The RayleighJeans law revealed an important error in physics theory of the time. The law predicted an energy output that diverges towards infinity as wavelength approaches zero (as frequency tends to infinity) and measurements of energy output at short wavelengths disagreed with this prediction.Rayleigh considered the radiation inside a cubic cavity of length L and temperature T whose walls are perfect reflectors as a series of standing electromagnetic waves. At the walls of the cube, the parallel component of the electric field and the orthogonal component of the magnetic field must vanish. Analogous to the wave function of a particle in a box, one finds that the fields are superpositions of periodic functions. The three wavelengths 1, 2 and 3, in the three directions orthogonal to the walls can be:19001905

http://en.wikipedia.org/wiki/Rayley_Jeans_lawBeiser, Perspectives of Modern Physics, McGraw-Hill, International Student Edition, 1969*

*Physical Laws of Radiometry SOLORayleighJeans Law (continue ) The RayleighJeans law agrees with experimental results at large wavelengths (or, equivalently, low frequencies) but strongly disagrees at short wavelengths (or high frequencies). This inconsistency between observations and the predictions of classical physics is commonly known as the ultraviolet catastrophe.Comparison of RayleighJeans law and Planck's lawThe term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest, although the concept goes back to 1900 with the first derivation of the 4 dependence of the RayleighJeans law;SolutionMax Planck solved the problem by postulating that electromagnetic energy did not follow the classical description, but could only oscillate or be emitted in discrete packets of energy proportional to the frequency, as given by Planck's law. This has the effect of reducing the number of possible modes with a given energy at high frequencies in the cavity described above, and thus the average energy at those frequencies by application of the equipartition theorem. The radiated power eventually goes to zero at infinite frequencies, and the total predicted power is finite. The formula for the radiated power for the idealized system (black body) was in line with known experiments, and came to be called Planck's law of black body radiation. Based on past experiments, Planck was also able to determine the value of its parameter, now called Planck's constant. The packets of energy later came to be called photons, and played a key role in the quantum description of electromagnetism.

http://en.wikipedia.org/wiki/Rayley_Jeans_lawBeiser, Perspectives of Modern Physics, McGraw-Hill, International Student Edition, 1969*

*Physical Laws of Radiometry SOLO**Wiens Law 1896 RayleighJeans Law1900 - 1905Comparison of RayleighJeans law with Wien approximation and Planck's law, for a body of 8 mK temperature

http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html*

*Physical Laws of Radiometry SOLOPlancks Law 1900 Planck derived empirically, by fitting the observed black body distribution to a high degree of accuracy, the relationBy comparing this empirical correlation with the Rayleigh-Jeans

formula Planck concluded that the error in

classical theory must be in the identification of the average oscillator energy as kT and therefore in the assumption that the oscillator energy is distributed continuously. He then posed the following question:If the average energy is defined as

how is the actual oscillator energies distributed?

*Physical Laws of Radiometry SOLOIf the average energy is defined as

how is the actual oscillator energies distributed?Planck deviated appreciable from the concepts of classical physics by assuming that the energy of the oscillators, instead of varying continuously, can assume only certain discrete valuesLet determine the average energyFrom Statistical Mechanics we know that the probability of a system assuming energy between and +d is proportional to exp (-/kT) dwhere n is an integer (n = 0, 1, 2, ), and h =6.6260.10-14 W.sec2is a constant introduced empirically by Planck , the Plancks Constant.

*Physical Laws of Radiometry SOLOPlancks Postulate: The energy of the oscillators, instead of varying continuously, can assume only certain discrete valueswhere n is an integer (n = 0, 1, 2, ). We say that the oscillators energy is Quantized.The average energy is

*Physical Laws of Radiometry SOLOPlanks Law Planks Law applies to blackbodies; i.e. perfect radiators. The spectral radial emittance of a blackbody is given by:Planks Law 1900

*SOLOParticles J.J. Thomson showed in 1897 that the cathode rays are composed of electrons and he measured the ratio of charge to mass for the electron. In 1904 he suggested a model of the atom as a sphere of positive matter in which electrons are positioned by electrostatic forces.Thomson Atom Model1904

Plum Pudding Model

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*PhotoelectricitySOLOEinstein and Photoelectricity Albert Einstein explained the photoelectric effect discovered by Hertz in 1887 by assuming that the light is quantized (using Plank results) in quantities that later become known as photons.1905 The kinetic energy Ek of the ejected electron is: where: To eject an electron the frequency of the incoming EM wave v must be above a threshold v0 (depends on metallic surface).Increasing the Intensity of the EM Wave will increase the number of electrons ejected, but not their energy.

*A. Beiser, Perspectives of Modern Physics, McGraw Hill-Kogakusha, International Student Edition, 1961, pg. 210 and Ch.10

Special Relativity Theory*

1905 EINSTEINS SPECIAL THEORY OF RELATIVITY

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EINSTEINs SPECIAL THEORY OF RELATIVITY (Continue)First Postulate:

It is impossible to measure or detect the Unaccelerated Translation Motion of a System through Free Space or through any Aether-like Medium.Second Postulate:

Velocity of Light in Free Space, c, is the same for all Observers, independent of the Relative Velocity of the Source of Light and the Observers. Second Postulate (Advanced):

Speed of Light represents the Maximum Speed of transmission of any Conventional Signal. Special Relativity Theory*

*SOLOConsequence of Special Theory of RelativityThe relation between the mass m of a particle having a velocity u and its rest mass m0 is: Special Relativity TheoryEINSTEINs SPECIAL THEORY OF RELATIVITY (Continue)The Kinetic Energy of a free moving particle having a momentum p = m u, a velocity u and its rest mass m0 is: The velocity of a photon is u = c, therefore, from the first equation, it has a rest mass And has a Kinetic Energy and Total Energy of Therefore if v is the photon Frequency and is photon Wavelength, we have

Locality and NonlocalitySOLO A Light Cone is the path that a flash of light, emanating from a single Event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through space-time. The Light Cone Equation isEvents Inside the Light ConeEvents Outside the Light Cone Einsteins Theory of Special Relativity Postulates that no Signal can travel with a speed higher than the Speed of Light c.Thousands of experiments performed with Particles (Photons, Electrons. Neutrons,) complied to this Postulate. However no experiments could be performed with Sub-particles, so, in my opinion the confirmation of this Postulate is still an open issue.Light Cone*

http://en.wikipedia.org/wiki/CHSH_inequalityV.J. Stenger, The Unconscious Quantum Metaphysics in Modern Physics and Cosmology, Prometheus Books, 1995*

Locality and NonlocalitySOLO According to Einstein only Events within Light Cone (shown in the Figure) can communicate with an event at the Origin, since only those Space-time points can be connected by a Signal traveling with the Speed of Light c or less. We call those Events Local although they may be separated in Space-time.Locality The Postulates of Relativity require that all frames of reference to be equivalent. So, if the Events are Local in any realizable frame of reference, they must be Local in all equivalent Frame of Reference. Two Space-time Points within Light Cone are called timelike.NonlocalityTwo Space-time Points outside Light Cone are said to have Spacelike Separation.Nonlocality connected Points outside the Light Cone. They have Space-time separation. Simultaneously Events (Time = 0), in any given Reference Frame , cannot be causally connected unless the signal between them travels at superluminal speed. Some physicists use the term Holistic instead of Nonlocal.Holistic = Nonlocal *

http://en.wikipedia.org/wiki/CHSH_inequalityV.J. Stenger, The Unconscious Quantum Metaphysics in Modern Physics and Cosmology, Prometheus Books, 1995*

*SOLO1908Geiger-Marsden Experiment. Geiger-Marsden working with Ernest Rutherford performed in 1908 the alpha-particle scattering experiment. H. Geiger and E. Marsden (1909), On a Diffuse Reflection of the -particle, Proceedings of the Royal Society Series A 82:495-500 A small beam of -particles was directed at a thin gold foil.According to J.J. Thomson atom-model it was anticipated thatmost of the -particles would go straight through the gold foil, while the remainder would at most suffer only slight deflections. Geiger-Marsden were surprised to find out that, while most of the -particles were not deviated, some were scattered through very large angles after passing the foil.QUANTUM THEORIES

*http://sol.sci.uop.edu/~jfalward/natureofatom/natureofatom.html

*ParticlesSOLOElectron Charge R.A. Millikan measured the charge of the electronby equalizing the weight m g of a charged oil drop with an electric field E.1909

*http://chem.ch.huji.ac.il/~eugeniik/history/millikan.html

*SOLORutherford Atom Model1911 Ernest Rutherford finds the first evidence of protons. To explain the Geiger-Marsden Experiment of 1908 he suggested in 1911 that the positively charged atomic nucleus contain protons.Nobel Prize 1908ChemistryHans Wilhelm Geiger 1882 1945Nazi Physicist Sir Ernest Marsden1889 1970 Rutherford assumed that the atom model consists of a smallnucleus, of positive charge, concentrated at the center, surroundedby a cloud of negative electrons. The positive -particles that passed close to the positive nucleus were scattered because of the electricalrepealing force between the positive charged -particle and the nucleus .QUANTUM MECHANICS

*http://sol.sci.uop.edu/~jfalward/natureofatom/natureofatom.html

*1913SOLONiels Bohr presents his quantum model of the atom. Nobel Prize 1922QUANTUM MECHANICSBohr Quantum Model of the Atom.

*Beiser, Perspectives of Modern Physics, McGraw Hill-Kogakusha, International Student Edition, 1961, Ch.6

*1913SOLONobel Prize 1922Explanation of Bohr Hydrogen Model In 1911, Bohr travelled to England. He met with J. J. Thomson of the Cavendish Laboratory and Trinity College, Cambridge, and New Zealand's Ernest Rutherford, whose 1911 Rutherford model of the atom had challenged Thomson's 1904 Plum Pudding Model.[Bohr received an invitation from Rutherford to conduct post-doctoral work at Victoria University of Manchester. He adapted Rutherford's nuclear structure to Max Planck's quantum theory and so created his Bohr model of the atom.[Bohr Model of the Hydrogen Atom consists on a electron, of negative charge, orbiting a positive charge nucleus. The Forces acting on the orbiting electron arem electron massv electron orbital velocityr orbit radiuse electron chargeQUANTUM MECHANICS

*Beiser, Perspectives of Modern Physics, McGraw Hill-Kogakusha, International Student Edition, 1961, Ch.5-6http://en.wikipedia.org/wiki/Niels_Bohr

*1913SOLONobel Prize 1922Explanation of Bohr Hydrogen Model (continue 1)The Conditions for Orbit Stability are

The Total Energy E, of the Electron, is the sum of the Kinetic Energy T and the Potential Energy VTo get some quantitative filing let use the fact that to separate the electron from the atom we need 13.6 eV (this is an experimental result), then E = -13.6 eV = 2.2x10-18 joule.ThereforeQUANTUM MECHANICS


1913SOLONobel Prize 1922Explanation of Bohr Hydrogen Model (continue 2)The problem with this Model is, since the electron accelerates with a =v2/r,according to Electromagnetic Theory it will radiate energy given byLarmor Formula (1897)As the electron loses energy the Total Energy becomes more negative and the radius decreases, and since P is proportional to 1/r4, the electron radiates energy faster and faster as it spirals toward the nucleus.Bohr had to add something to explain the stability of the orbits.He knew the results of the discrete Hydrogen Spectrum lines and the quantization of energy that Planck introduced in 1900 to obtain the Black Body Radiation Equation.QUANTUM MECHANICS*

*Beiser, Perspectives of Modern Physics, McGraw Hill-Kogakusha, International Student Edition, 1961, Ch.5-6http://en.wikipedia.org/wiki/Larmor_formula

1913SOLONobel Prize 1922Explanation of Bohr Hydrogen Model (continue 3) To understand Bohr novelty let look at an Elastic Wire that vibrates transversally. At Steady State the Wavelengths always fit an integral number of times into the Wire Length. This is true if we bend the Wire and even if we obtain a Closed Loop Wire. If the Wire is perfectly elastic the vibration will continue indefinitely.This is Resonance. Bohr noted that the Angular Momentum of the Orbiting electron in the Atom Hydrogen Model had the same dimensions as the Plancks Constant. This led him to postulate that the Angular Momentum of the Orbiting Electrons must be multiple of Plancks Constant divided by 2 .QUANTUM MECHANICS*


1913SOLONobel Prize 1922Explanation of Bohr Hydrogen Model (continue 4)Energy Levels and SpectraWe obtainedand Energy Levels: The Energy Levels are all negative signifying that the electron does not have enough energy to escape from the atom. The lowest energy level E1 is called the Ground State. The higher levels E2, E3, E4,, are called Excited States.In the limit n , E=0 and the electron is no longer bound to the nucleus to form an atom.QUANTUM MECHANICS*


*1913SOLONobel Prize 1922Explanation of Bohr Hydrogen Model (continue 5)According to the Bohr Hydrogen Model when the electron is excited he drops to a lower state, and a single photon of light is emittedInitial Energy Final Energy = Photon Energy where v is the photon frequency.If is the Wavelength of the photon we haveWe recovered the Rydberg Formula (1887) QUANTUM MECHANICS

*Beiser, Perspectives of Modern Physics, McGraw Hill-Kogakusha, International Student Edition, 1961, Ch.6

*1913SOLONobel Prize 1922Explanation of Bohr Hydrogen Model (continue 6)The Bohr model treats the electron as if it were a miniature planet, with definite radius

and momentum. This is in direct violation of the uncertainty principle (formulated by Werner Heisenberg in 1927) which dictates that position and momentum cannot be simultaneously determined. It fails to provide any understanding of why certain spectral lines are brighter than others.

There is no mechanism for the calculation of transition probabilities. While the Bohr model was a major step toward understanding the quantum theory of the atom, it is not in fact a correct description of the nature of electron orbits. Some of the shortcomings of the model are: The electrons in free atoms can will be found in only certain discrete energy states. These sharp energy states are associated with the orbits or shells of electrons in an atom, e.g., a hydrogen atom. One of the implications of these quantized energy states is that only certain photon energies are allowed when electrons jump down from higher levels to lower levels, producing the hydrogen spectrum. The electron must jump instantaneously because if he moves gradually it will radiate and lose energy in the process. The Bohr model successfully predicted the energies for the hydrogen atom, but had significant failures.Quantized Energy StatesQUANTUM MECHANICS

*http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html

1915Einsteins General Theory of RelativityThe General Theory of Relativity takes in consideration the action of Gravity and does not assume Unaccelerated Observer like Special Theory of Relativity. Principle of Equivalence The Inertial Mass and the Gravitational Mass of the same body are always equal. (checked by experiments first performed by Etvos in 1890) Principle of Covariance -- The General Laws of Physics can be expressed in a form that is independent of the choise of the coordinate system. Principle of Mach -- The Gravitation Field and Metric (Space Curvature) depend on the distribution of Matter and Energy.SOLOGENERAL RELATIVITY Dissatisfied with the Nonlocality (Action at a Distance) of Newtons Law of GravityEinstein developed the General Theory of Gravity.*

GENERAL RELATIVITYEinsteins General Theory Equation The Matter Energy Distribution produces the Bending (Curvature) of the Space-Time.All Masses are moving on the Shortest Path (Geodesic) of the Curved Space-Time. In the limit (Weak Gravitation Fields) this Equation reduce to thePoissons Equation of Newtons Gravitation LawSOLO*

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SOLOGENERAL RELATIVITY*

GENERAL RELATIVITYEinsteins General Theory of Relativity (Summary)Gravity is Geometry

Mass Curves Space Time

Free Mass moves on the

Shortest Path in Curved Space Time.SOLONewtons Gravity The Earth travels around the Sun because it is pulled by the Gravitational Force exerted by the Mass of the Sun. Mass (somehow) causes a Gravitational Force which propagates instantaneously (Action at a Distance) and causes True Acceleration.Einsteins Gravity The Earth travels around the Sun because is the Shortest Path in the Curved Space Time produced by the Mass of the Sun. Mass (somehow) causes a Warping, which propagates with the Speed of Light, and results in Apparent Acceleration.*

*Photons EmissionSOLOTheory of Light Emission. Concept of Stimulated Emission 1916 Albert Einstein1879 - 1955Nobel Prize 1921http://members.aol.com/WSRNet/tut/ut4.htmSpontaneous Emission& AbsorptionStimulated Emission& AbsorptionOn the Quantum Mechanics of Radiation Einsteins work laid the foundation of the Theory of LASER (Light Amplification by Stimulated Emission)

2602.bin

*W.M. Steen, Laser Material Processing, 2nd Ed., 1998http://www.laser.org.uk/laser_welding/briefhistory.htmhttp://sol.sci.uop.edu/~jfalward/natureofatom/natureofatom.html

QUANTUM MECHANICS1919: ERNEST RUTHERFORD FINDS THE FIRST EVIDENCE OF PROTONS. HE SUGGESTED IN 1914 THAT THE POSITIVELY CHARGED ATOMIC NUCLEUS CONTAINS PROTONS. 1922: OTTO STERN AND WALTER GERLACH SHOW SPACE QUANTIZATION1923: ARTHUR COMPTON DISCOVERS THE QUANTUM NATURE OF X RAYS, THUS CONFIRMS PHOTONS AS PARTICLES.1924: LOUIS DE BROGLIE PROPOSES THAT MATTER HAS WAVE PROPERTIES.1924: WOLFGANG PAULI STATES THE QUANTUM EXCLUSION PRINCIPLE.*

E. RUTHERFORD

OTTO STERN

W. GERLACH

A. COMPTON

L. de BROGLIE

W. PAULI

QUANTUM MECHANICS1925: WERNER HEISENBERG, MAX BORN, AND PASCAL JORDAN FORMULATE QUANTUM MATRIX MECHANICS.1925: SAMUEL A. GOUDSMITH AND GEORGE E. UHLENBECK POSTULATE THE ELECTRON SPIN*

W. HEISENBERG

MAX BORN

P. JORDAN

QUANTUM MECHANICS1926: MAX BORN GIVES A PROBABILISTIC INTERPRATATION OF THE WAVE FUNCTION.1927: WERNER HEISENBERG STATES THE QUANTUM UNCERTAINTY PRINCIPLE.1928: PAUL DIRAC STATES HIS RELATIVISTIC QUANTUM WAVE EQUATION. HE PREDICTS THE EXISTENCE OF THE POSITRON.1932: JHON von NEUMANN WROTE THE FOUNDATION OF QUANTUM MECHANICS*

*SOLO1922 Otto Stern and Walter Gerlach show Space Quantization They designed the Stern-GerlachExperiment that determine if a particle has angular momentum.http://en.wikipedia.org/wiki/Stern-Gerlach_experiment They directed a beam of neutral silver atoms from an oven trough a set of collimating slits into an inhomogeneous magnetic field. A photographic plate recorded the configuration of the beam. They found that the beam split into two parts, corresponding to the two opposite spin orientations, that are permitted by space quantization.QUANTUM MECHANICS

*Beiser, Perspectives of Modern Physics, McGraw Hill-Kogakusha, International Student Edition, 1961, pg. 227-228

*SOLO1923 Arthur Compton discovers the quantum nature of X rays, thus confirms photonsas particles. Compton effect consists of a X ray (incident photons) colliding with rest electronsis scattered in the direction (detected by an X-ray spectrometer) and the electrons in the direction.QUANTUM MECHANICS

*Beiser, Perspectives of Modern Physics, McGraw Hill-Kogakusha, International Student Edition, 1961, pp. 68-73http://www.regentsprep.org/Regents/physics/phys05/bcompton/default.htm

*SOLO1924Louis de Broglie proposes that matter has wave properties and using the relation between Wavelength and Photon mass:He postulate that any Particle of mass m and velocity v has an associate Wave with a Wavelength .QUANTUM MECHANICS

*

SOLOExplanation of Bohr Model using de Broglie Relation To understand Bohr novelty let look at an Elastic Wire that vibrates transversally. At Steady State the Wavelengths always fit an integral number of times into the Wire Length. This is true if we bend the Wire and even if we obtain a Closed Loop Wire. If the Wire is perfectly elastic the vibration will continue indefinitely.This is Resonance.We found the Electron Orbital VelocityReturn to Bohr Hydrogen Model using de Broglie RelationUsing de Broglie RelationAt Steady State the Wavelengths always fit an integral number of times into the Wire Length.We obtain the same relation as Bohr for the Orbit radius:QUANTUM MECHANICS*


*SOLO1924Wolfgang Pauli states the Quantum Exclusion PrincipleWolfgang Pauli1900 - 1958Nobel Prize 1945QUANTUM MECHANICS

*

QUANTUM THEORIES Werner Heisenberg, Max Born, and Pascal Jordan formulate Quantum Matrix Mechanics.QUANTUM MATRIX MECHANICS.http://en.wikipedia.org/wiki/Matrix_mechanics1925 Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps occur. It did so by interpreting the physical properties of particles as matrices that evolve in time. It is equivalent to the Schrdinger wave formulation of quantum mechanics, and is the basis of Dirac's bra-ket notation for the wave function.SOLOIn 1928, Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics, but Heisenberg alone won the 1932 Prize "for the creation of quantum mechanics, the application of which has led to the discovery of the allotropic forms of hydrogen",[47] while Schrdinger and Dirac shared the 1933 Prize "for the discovery of new productive forms of atomic theory".[47] On 25 November 1933, Born received a letter from Heisenberg in which he said he had been delayed in writing due to a "bad conscience" that he alone had received the Prize "for work done in Gottingen in collaboration you, Jordan and I."[48] Heisenberg went on to say that Born and Jordan's contribution to quantum mechanics cannot be changed by "a wrong decision from the outside."*

http://en.wikipedia.org/wiki/Matrix_mechanicshttp://en.wikipedia.org/wiki/Max_Born*

1925SAMUEL A. GOUDSMITH AND GEORGE E. UHLENBECK POSTULATE THE ELECTRON SPINTwo types of experimental evidence which arose in the 1920s suggested an additional property of the electron. One was the closely spaced splitting of the hydrogen spectral lines, called fine structure.

The other was the Stern-Gerlach experiment which showed in 1922 that a beam of silver atoms directed through an inhomogeneous magnetic field would be forced into two beams. Both of these experimental situations were consistent with the possession of an intrinsic angular momentum and a magnetic moment by individual electrons. Classically this could occur if the electron were a spinning ball of charge, and this property was called electron spin.In 1925, the Dutch Physicists S.A. Goudsmith and G.E. Uhlenbeckrealized that the experiments can be explained if the electron has anmagnetic property of Rotation or Spin. They work actually showed that the electron has a quantum-mechanical notion of spin that is similarto the mechanical rotation of particles.http://hyperphysics.phy-astr.gsu.edu/hbase/spin.htmlZeemans EffectQUANTUM MECHANICS*

http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html*

Spin In quantum mechanics and particle physics, Spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei. Spin is a solely quantum-mechanical phenomenon; it does not have a counterpart in classical mechanics (despite the term spin being reminiscent of classical phenomena such as a planet spinning on its axis).[ Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. Orbital angular momentum is the quantum-mechanical counterpart to the classical notion of angular momentum: it arises when a particle executes a rotating or twisting trajectory (such as when an electron orbits a nucleus).The existence of spin angular momentum is inferred from experiments, such as the SternGerlach experiment, in which particles are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone.[http://en.wikipedia.org/wiki/Spin_(physics)In some ways, spin is like a vector quantity; it has a definite magnitude; and it has a "direction" (but quantization makes this "direction" different from the direction of an ordinary vector). All elementary particles of a given kind have the same magnitude of spin angular momentum, which is indicated by assigning the particle a spin quantum number.[2] However, in a technical sense, spins are not strictly vectors, and they are instead described as a related quantity: a Spinor. In particular, unlike a Euclidean vector, a spin when rotated by 360 degrees can have its sign reversedQUANTUM MECHANICS*

http://hyperphysics.phy-astr.gsu.edu/hbase/spin.htmlSin-itiro Tomonaga, The Story of Spin, University of Chicago Press, 1997*

ERWIN SCHRDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT1926In January 1926, Schrdinger published in Annalen der Physik the paper "Quantisierung als Eigenwertproblem" [Quantization as an Eigenvalue Problem] on wave mechanics and presented what is now known as the Schrdinger equation. In this paper, he gave a "derivation" of the wave equation for time-independent systems and showed that it gave the correct energy eigenvalues for a hydrogen-like atom. This paper has been universally celebrated as one of the most important achievements of the twentieth century and created a revolution in quantum mechanics and indeed of all physics and chemistry. A second paper was submitted just four weeks later that solved the quantum harmonic oscillator, rigid rotor, and diatomic molecule problems and gave a new derivation of the Schrdinger equation. A third paper in May showed the equivalence of his approach to that of Heisenberg.http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dingerQUANTUM MECHANICSSOLO*

*

MAX BORN GIVES A PROBABILISTIC INTERPRATATION OF THE WAVE FUNCTION.1926 Max Born wrote in 1926 a short paper on collisions between particles, about the same time as Schrdinger paper Quantization as an Eigenvalue Problem. Born rejected the Schrdinger Wave Field approach. He had been influenced by a suggestion made by Einstein that, for photons, the Wave Field acts as strange kind of phantom Field guiding the photon particles on paths which could therefore be determined by Wave Interference Effects. Max Born reasoned that the Square of the Amplitude of the Waveform in some specific region of configuration space is related to the Probability of finding the associated quantum particle in that region of configuration space. Since Probability is a real number, and the integral of all Probabilities over all regions of configuration space, the Wave Function must satisfy Condition of Normalization of the Wave Function Therefore the probability of finding the particle between a and b is given byEinstein rejected this interpretation. In a 1926 letter to Max Born, Einstein wrote: "I, at any rate, am convinced that He [God] does not throw dice."[QUANTUM MECHANICSSOLO*

Jim Baggott, The Meaning of Quantum Theory, Oxford University Press, 1992*

QUANTUM MECHANICSIn December 1926 Einstein wrote a letter to Bohr which contains a phrase that has since become symbolic of Einsteins lasting dislike of the element of chance implied by the quantum theory:J. Baggott, The Meaning of Quantum Theory, Oxford University Press, 1992, pp. 28-29SOLO1926http://en.wikipedia.org/wiki/Max_Born Quantum mechanics is very impressive. But an inner voice tells me that it is not the real thing. The theory produce a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice.*

*SOLOWavelike Behavior for Electrons In 1927, the wavelike behavior of the electrons was demonstrated by Davisson and Germer in USA and by G.P. Thomson in Scotland.Quantum 1927

*

*SOLOWavelike Behavior for Electrons Quantum 1927 G.P. Thomson carried a series of experiments using an apparatus called an electron diffraction camera. With it he bombarded very thin metal and celluloid foils with a narrow electron beam. The beam then was scattered into a series of rings. Using these results G.P. Thomson proved mathematically that the electron particles acted like waves, for which he received the Nobel Prize in 1937. J.J. Thomson the father of G.P. proved that the electron is a particle in 1897, for which he received the Nobel Prize in 1906.Results of a double-slit-experiment performed by Dr. Tonomura showing the build-up of an interference pattern of single electrons. Numbers of electrons are 11 (a), 200 (b), 6000 (c), 40000 (d), 140000(e).

*SOLOOptics History Raman Effect1928http://en.wikipedia.org/wiki/Raman_scatteringhttp://en.wikipedia.org/wiki/Chandrasekhara_Venkata_Raman Raman Effect was discovered in 1928 by C.V. Raman incollaboration with K.S. Krishnan and independently byGrigory Landsberg and Leonid Mandelstam. Monochromatic light is scattered when hitting molecules. The spectral analysis of the scattered light shows an intense spectral line matching the wavelength of the light source (Rayleigh or elastic scattering). Additional, weaker lines are observed at wavelength which are shifted compared to the wavelength of the light source. These are the Raman lines.

*

*SOLOStimulated Emission and Negative Absorption1928Rudolph W. Landenburg confirmed existence of stimulated emission and Negative AbsorptionLasers History Rudolf Walter Ladenburg (June 6, 1882 April 6, 1952) was a German atomic physicist. He emigrated from Germany as early as 1932 and became a Brackett Research Professor at Princeton University. When the wave of German emigration began in 1933, he was the principal coordinator for job placement of exiled physicist in the United States.

*W.M. Steen, Laser Material Processing, 2nd Ed., 1998http://www.laser.org.uk/laser_welding/briefhistory.htmhttp://en.wikipedia.org/wiki/Rudolf_Ladenburghttp://www.aip.org/history/newsletter/fall2000/pic_einstein_lg.htm

QUANTUM MECHANICSSOLOWave Packet A wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere Depending on the evolution equation, the wave packet's envelope may remain constant (no dispersion, see figure) or it may change (dispersion) while propagating. As an example of propagation without dispersion, consider wave solutions to the following wave equation:where v is the speed of the wave's propagation in a given medium.*

http://en.wikipedia.org/wiki/Wave_packet*

QUANTUM MECHANICSSOLOWave Packetwhere v is the velocity , v is the frequency, is the Wavelength of the Wave Packet.*


QUANTUM MECHANICSSOLOWave PacketA wave packet is a localized disturbance that results from the sum of many different wave forms. If the packet is strongly localized, more frequencies are needed to allow the constructive superposition in the region of localization and destructive superposition outside the region. From the basic solutions in one dimension, a general form of a wave packet can be expressed aswhereDefineWave Function in Momentum Space*


ERWIN SCHRDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT1926 Following Max Planck's quantization of light (see black body radiation), Albert Einstein interpreted Planck's quanta to be photons, particles of light, and proposed that the energy of a photon is proportional to its frequency, one of the first signs of waveparticle duality. Since energy and momentum are related in the same way as frequency and wavenumber in special relativity, it followed that the momentum p of a photon is proportional to its wavenumber k.Louis de Broglie hypothesized that this is true for all particles, even particles such as electrons. He showed that, assuming that the matter waves propagate along with their particle counterparts, electrons form standing waves, meaning that only certain discrete rotational frequencies about the nucleus of an atom are allowed.[7] These quantized orbits correspond to discrete energy levels, and de Broglie reproduced the Bohr model formula for the energy levels. The Bohr model was based on the assumed quantization of angular momentum:According to de Broglie the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit: n = 2 rhttp://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationHistorical Background and DevelopmentQUANTUM MECHANICSSOLO*

ERWIN SCHRDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT1926http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationHistorical Background and Development (continue 1)Following up on de Broglie's ideas, physicist Peter Debye made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, Schrdinger decided to find a proper 3-dimensional wave equation for the electron. He was guided by William R. Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system the trajectories of light rays become sharp tracks that obey Fermat's principle, an analog of the principle of least action.For a general form of a Progressive Wave Function in + x direction with velocity v and frequency v:ThereforeQUANTUM MECHANICSSOLO*

Arthur Beiser, Perspectives of Modern Physics, McGraw-Hill, International Student Edition, 1969, Schrdingers Equation: Time-dependent Form, pp. 153-156*

ERWIN SCHRDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT1926Historical Background and Development (continue 2)We want to find the Differential Equation yielding the Wave Function .We haveWave Function:At particle speeds small compared to speed of light c, the Total Energy E is the sum of theKinetic Energy p2/2m and the Potential Energy V (function of position and time):

QUANTUM MECHANICSSOLO*


ERWIN SCHRDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT1926Historical Background and Development (continue 3)Non-RelativisticOne-DimensionalTime DependentSchrdinger EquationIn the same wayNon-RelativisticThree-DimensionalTime DependentSchrdinger EquationQUANTUM MECHANICSSOLOThis is a Linear Partial Differential Equation. It is also a Diffusion Equation (with an Imaginary Diffusion Coefficient), but unlike the Heat Equation, this one is also a Wave Equation given the imaginary unit present in the transient term.*


1926Schrdinger EquationTime-dependent Schrdinger equation (single non-relativistic particle)A wave function that satisfies the non-relativistic Schrdinger equation with V=0. In other words, this corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted hereEach of these three rows is a wave function which satisfies the time-dependent Schrdinger equation for a harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The probability distribution of finding the particle with this wave function at a given position. The top two rows are examples of stationary states, which correspond to standing waves. The bottom row an example of a state which is not a stationary state. The right column illustrates why stationary states are called "stationary".QUANTUM MECHANICSSOLO*

Schrdinger Equation: Steady State FormUsing and the Time-dependent Schrdinger equationsNon-RelativisticOne-DimensionalTime DependentSchrdinger EquationNon-RelativisticThree-DimensionalTime DependentSchrdinger Equationwe can write Non-RelativisticOne-DimensionalSteady-StateSchrdinger EquationNon-RelativisticThree-DimensionalSteady-StateSchrdinger EquationQUANTUM MECHANICS1926SOLO*

Operators in Quantum MechanicsSince, according to Born, * represents Probability of finding the associated quantum particle in a region we can compute the Expectation (Mean) Value of the Total Energy E and of the Momentum p in that region using Rearranging we obtainQUANTUM MECHANICSSOLO*

Arthur Beiser, Perspectives of Modern Physics, McGraw-Hill, International Student Edition, 1969*

Operators in Quantum Mechanics (continue 1)We obtainedMoment OperatorTotal Energy Operator Although we derived those operators for free particles, they are entire general results, equivalent to Schrdinger Equation. To see this let write the Operator EquationWe have

Applying this Operator on Wave Function we recover the Schrdinger EquationThe two descriptions (Operator and Schrdingers) are equivalent.QUANTUM MECHANICSSOLO*

Arthur Beiser, Perspectives of Modern Physics, McGraw-Hill, International Student Edition, 1969*

QUANTUM MECHANICSOperators in Quantum Mechanics (continue 3)We obtainedMoment OperatorTotal Energy Operator Because p and E can be replaced by their Operators in an equation, we can use those Operators to obtain Expectation Values for p and E.Let define the Hamiltonian OperatorSchrdinger Equation in Operator form isSOLO*

Arthur Beiser, Perspectives of Modern Physics, McGraw-Hill, International Student Edition, 1969J. Baggott, The Meaning of Quantum Theory, Oxford University Press,1992*

Dirac bracket notationA elegant shorthand notation for the integrals used to define Operators was introduced by Dirac in 1939Instead of dealing with Wavefunctions n, we defined a related Quantum State,denoted | which is called a ket, ket vector, state or state vector.The complex conjugate of | is called the bra and is denoted by |.When a bra is combined with a ket we obtain a bracket.The following integrals are represented by bra and ket Operators in Quantum Mechanics (continue 5)QUANTUM MECHANICSSOLO*

J. Baggott, The Meaning of Quantum Theory, Oxford University Press,1992C. Cohen-Tannoudji, B. Diu, F. Lalo, Quantum Mechanics, Volume I, Wiley Interscience, 1977, Ch. II, The Mathematical Tools of Quantum MechanicsArthur Beiser, Perspectives of Modern Physics, McGraw-Hill, International Student Edition, 1969*

QUANTUM THEORIESHILBERT SPACE AND QUANTUM MECHANICS.http://en.wikipedia.org/wiki/Matrix_mechanicsBorn had also learned Hilberts theory of integral equations and quadratic forms for an infinite number of variables as was apparent from a citation by Born of Hilberts work Grundzge einer allgemeinen Theorie der Linearen Integralgleichungen published in 1912. Jordan, too was well equipped for the task. For a number of years, he had been an assistant to Richard Courant at Gttingen in the preparation of Courant and David Hilberts book Methoden der mathematischen Physik I, which was published in 1924. This book, fortuitously, contained a great many of the mathematical tools necessary for the continued development of quantum mechanics.In 1926, John von Neumann became assistant to David Hilbert, and he would coin the term Hilbert Space to describe the algebra and analysis which were used in the development of quantum mechanicsSOLO*

*Functional Analysis SOLOVector (Linear) Space: E is a Linear Space if Addition and Scalar Multiplication are Defined Linear Independence, Dimensionality and BasesA set of vectors |i (i=1,,n) that satisfy the relation Dimension of a Vector Space E , is the maximum number N of Linear Independent Vectors in this space. Thus, between any set of more that N Vectors |i (i=1,2,,n>N), there exist a relation of Linear Dependency. Any set of N Linearly Independent Vectors |i (i=1,2,,N), form a Basis of the Vector Space E ,of Dimension N, meaning that any vector | E can be written as a Linear Combination of those Vectors. In the case of an Infinite Dimensional Space (N), the space will be defined by a Complete Set of Basis Vectors. This is a Set of Linearly Independent Vectors of the Space, such that if any other Vector of the Space is added to the set, there will exist a relation of Linear Dependency to the Basis Vectors.

SOLOFunctional Analysis Use of bra-ket notation of Dirac for Vectors. *The Inner Product of the Vectors f and e is defined asInner Product Using Dirac NotationTo every ket corresponds a bra.

http://en.wikipedia.org/wiki/Paul_Dirachttp://en.wikipedia.org/wiki/Spectral_theory*

*Functional AnalysisSOLOInner Product Using Dirac NotationIf E is an Inner Product Space, than we can induce the Norm:

*C. Cohen-Tannoudji, B. Diu, F. Lalo, Quantum Mechanics, Volume I, Wiley Interscience, 1977, pg.111

*Functional AnalysisSOLOInner ProductCauchy, Bunyakovsky, Schwarz Inequality known as Schwarz Inequality Let |, | be the elements of an Inner Product space E, than :we have:which reduce to:or:q.e.d.Proof:

*http://en.wikipedia.org/wiki/Cauchy-Schwarz_inequality

*Functional AnalysisSOLOHilbert Space A Linear Space E is called a Hilbert Space if E is an Inner Product Space that is complete with respect to the Norm induced by the Inner Product ||1||=[< 1, 1>]1/2. Equivalently, a Hilbert Space is a Banach Space (Complete Metric Space) whoseNorm is induced by the Inner Product ||1||=[< 1, 1>]1/2 .Orthogonal Vectors in a Hilbert Space: Theorem: Given a Set of Linearly Independent Vectors in a Hilbert Space |i (i=1,,n) and any Vector |m Orthogonal to all |i, than it is also Linearly Independent.Proof: Suppose that the Vector |m is Linearly Dependent on |i (i=1,,n) We obtain a inconsistency, therefore |m is Linearly Independent on |i (i=1,,n) Therefore in a Hilbert Space, of Finite or Infinite Dimension, by finding the MaximumSet of Orthogonal Vectors we find a Basis that Complete covers the Space. q.e.d.

*

*Functional AnalysisSOLOHilbert SpaceOrthonormal Sets Let |1, |2, ,, |n, denote a set of elements in the Hilbert Space H. Define the Gram Matrix of the set: Theorem: A set of functions |1, |2, ,, |n of the Hilbert Space H is linearly dependent if and only if the Gram determinant of the set is zero. Proof: Linearly Dependent Set:Multiplying (inner product) this equation consecutively by |1, |2, ,, |n, we obtain:q.e.d.

*

*Functional AnalysisSOLOHilbert SpaceOrthonormal Sets (continue 2) Theorem: A set of functions |1, |2, ,, |n, of the Hilbert space H is linearly dependent if and only if the Gram Determinant of the Set is zero. Proof: The Gram Matrix of an Orthogonal Set has only nonzero diagonal; thereforeDeterminant G (|1, |2, ,, |n ).=| 1 2 | 2 2 | n 2 0, and the Set is LinearlyIndependent. q.e.d. Corollary: The rank of the Gram Matrix equals the dimension of the Linear ManifoldL (|1, |2, ,, |n ). If Determinant G (1, |2, ,, |n) is nonzero, the Gram Determinant of any other Subset is also nonzero. Definition 1: Two elements |,| of a Hilbert Space H are said to be orthogonal if =0. Definition 2: Let S be a nonempty Subset of a Hilbert Space H. S is called an OrthogonalSet if || for every pair |,| S and | |. If in addition |=1 for every | S, then S is called an Orthonormal Set. Lemma: Every Orthogonal Set is Linearly Independent. If | is Orthogonal to every element of the Set (|1, |2, ,, |n ), then | is Orthogonal to Manifold L (|1, |2, ,, |n ).

*

*Functional AnalysisSOLOHilbert SpaceOrthonormal Sets (continue 3) Gram-Schmidt Orthogonalization Process Let =(|1, |2, ,, |n ) any finite Set of Linearly Independent Vectorsand L (|1, |2, ,, |n ) the Manifold spanned by the Set . The Gram-Schmidt Orthogonalization Process derive a Set (|e1, |e2, ,, |en ) of Orthonormal Elements from the Set .

*

*Functional AnalysisSOLOHilbert SpaceOrthonormal Sets (continue 4) Gram-Schmidt Orthogonalization Process (continue)

OrthogonalizationNormalization

*

*Functional AnalysisSOLOHilbert SpaceDiscrete |ei and Continuous |w Orthonormal BasesFrom those equations we obtainThe Orthonormalization RelationA Vector | will be represented by ThereforeThe Closure RelationsI the Identity Operator (its action on any state leaves it unchanged). - a real number or vector, not complex-valuedThe Vectors in Hilbert Space can be Countable (Discrete) or Uncountable (Continuous).

*C. Cohen-Tannoudji, B. Diu, F. Lalo, Quantum Mechanics, Volume I, Wiley Interscience, 1977, pg.122

*Functional AnalysisSOLOHilbert SpaceSeries Expansions of Arbitrary FunctionsDefinitions: Approximation in the Mean by an Orthonormal Series |1, |2, ,, |n :Let | be any function. The numbers:

are called the Expansions Coefficients or Components of | with respect to the given Orthonormal System From the relationwe obtainor

*Courant, R., Hilbert, D., Methods of Mathematical Physics, Vol. I, Interscience Publishers, 1953

*Functional AnalysisSOLOHilbert SpaceSeries Expansions of Arbitrary Functions Approximation in the Mean by an Orthonormal Series |1, |2, |3, :Since the sum on the right is independent on n, is true also for n , we haveBessels Inequality Bessels Inequality is true for every Orthonormal System. It proves that the sum of the square of the Expansion Coefficients always converges.

*Courant, R., Hilbert, D., Methods of Mathematical Physics, Vol. I, Interscience Publishers, 1953

*Functional AnalysisSOLOHilbert SpaceSeries Expansions of Arbitrary Functions Approximation in the Mean by an Orthonormal Series |1, |2, |3, :If for a given Orthonormal System |1, |2, |3, any piecewise continuous function | can be approximated in the mean to any desired degree of accuracy by choosing a n large enough ( n>N () ), i.e.: then the Orthonormal System |1, |2, |3, is said to be Complete.For a Complete Orthonormal System |1, |2, |3, the Bessels Inequality becomes an Equality:Parsevals Equality applies forComplete Orthonormal SystemsThis relation is known as the Completeness Relation.

Marc-Antoine Parseval des Chnes1755 - 1836

*B.H. Bransden, C.J. Joachain, Quantum Mechanics, 2nd Ed., Prentice Hall, 1989,2000, pp.96, 213Curant, R., Hilbert, D., Methods of Mathematical Physics, Vol. I, Interscience Publishers, 1953

Functional AnalysisSOLOHilbert SpaceLinear Operators in Hilbert SpaceAn Operator L in Hilbert Space acting on a Vector |, produces a Vector |.The arrow over L means that the Operator is acting on the Vector on the Left, |.Eigenvalues and Eigenfunction of a Linear Operator are defined byThe Eigenfunction | is transformed by the Operator L into multiple of itself, by theEigenvalue . The conjugate equation is*

*Courant, R., Hilbert, D., Methods of Mathematical Physics, Vol. I, Interscience Publishers, 1953B.H. Bransden, C.J. Joachain, Quantum Mechanics, 2nd Ed., Prentice Hall, 1989,2000, pp.96, 213

Functional AnalysisSOLOHilbert SpaceAdjoint or Hermitian Conjugates OperatorsAn Operator L1 in Hilbert Space acting on a Vector |, produces a Vector |.Let have another Operator in Hilbert Space acting on the Vector |, and produce a Vector |.ThereforeThe Adjoint of a Product of Operators is obtained by Reversing the order of theProduct of Adjoint of Operators.*


Functional AnalysisSOLOHilbert SpaceGivenTherefore

In the same way

Not all Operators have an Inverse.*


Functional AnalysisSOLOHilbert SpaceProperties of Hermitian OperatorsFrom the definition we can see that the direction of the arrow is not important and we can write1*


Functional AnalysisSOLOHilbert SpaceProperties of Hermitian OperatorsHermitian Operator*


Functional AnalysisSOLOHilbert SpaceProperties of Hermitian OperatorsIf |n and |m are two Eigenfunctions of the Hermitian Operator L, with eigenvalues n and m, respectivelyHermitian Operator: If m n this equality is possible only if n and m are Orthogonal If m = n we can use the Gram-Schmidt Procedure to obtain a new Eigenfunction Orthogonal to |n.

The Hermitian Operators have Real Eigenvalues and Orthogonal Eigenfunctions. *


Functional AnalysisSOLOHilbert SpaceProperties of Unitary OperatorsA Unitary Matrix is such that its Adjoint is equal to its Inverse.All Eigenvalues of a Unitary Matrix have absolute values equal to 1.Suppose |i is an Eigenfunction and i is the corresponding Eigenvalue of a Unitary Operator.12For all

Let | and | be two State Vectors and A be a Linear Hermitian Operator, such thatLet apply the Unitary U, so thatWe define a new Operator A such thatSince this is true for any |, we obtainFunctional AnalysisSOLOHilbert SpaceProperties of Unitary Operators4*

J. Baggott, The Meaning of Quantum Theory, Oxford University Press,1992C. Cohen-Tannoudji, B. Diu, F. Lalo, Quantum Mechanics, Volume I, Wiley Interscience, 1977, Ch. II, The Mathematical Tools of Quantum MechanicsArthur Beiser, Perspectives of Modern Physics, McGraw-Hill, International Student Edition, 1969B.H. Bransden, C.J. Joachain, Quantum Mechanics, 2nd Ed., Prentice Hall, 1989,2000, pp.96, 213*

Properties of Unitary Operators If A is Hermitian, then A is also Hermitian Operator Equations remain unchanged in form under Unitary TransformationsConsider the Operatoror The Eigenvalues of A are the same as those of A Functional AnalysisSOLOHilbert Space4a4b4c*

4J. Baggott, The Meaning of Quantum Theory, Oxford University Press,1992C. Cohen-Tannoudji, B. Diu, F. Lalo, Quantum Mechanics, Volume I, Wiley Interscience, 1977, Ch. II, The Mathematical Tools of Quantum MechanicsArthur Beiser, Perspectives of Modern Physics, McGraw-Hill, International Student Edition, 1969B.H. Bransden, C.J. Joachain, Quantum Mechanics, 2nd Ed., Prentice Hall, 1989,2000, pp.96, 213*

QUANTUM THEORIESQUANTUM MECHANICSVon NEUMANN WROTE THE FOUNDATION OF QUANTUM MECHANICS1932Von Neumann was the first to rigorously establish a mathematical framework for quantum mechanics, known as the Diracvon Neumann Axioms, with his 1932 work Mathematische Grundlagen der Quantenmechanik. After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of quantum mechanics. He immediately realized, in 1926, that a quantum system could be considered as a point in a so-called Hilbert space, analogous to the 6N dimension (N is the number of particles, 3 general coordinate and 3 canonical momentum for each) phase space of classical mechanics but with infinitely many dimensions (corresponding to the infinitely many possible states of the system) instead: the traditional physical quantities (e.g., position and momentum) could therefore be represented as particular linear operators operating in these spaces. The physics of quantum mechanics was thereby reduced to the mathematics of the linear Hermitian Operators on Hilbert Spaces. For example, the Uncertainty Principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into the non-commutativity of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrdinger.http://en.wikipedia.org/wiki/John_von_NeumannSOLO*

*

QUANTUM THEORIESQUANTUM MECHANICSVon NEUMANN WROTE THE FOUNDATION OF QUANTUM MECHANICS1932Postulates of Quantum MechanicsFirst Postulate:At a fixed time t0, the state of a physical system is completely defined by specifying a ket | (t0) belonging to the state space E.Second Postulate:Every measurable physical quantity A is described by an operator A acting in E; this operator is an observable.Third Postulate:The only possible result of the measurement of a physical quantity A is one of the eigenvalues of the corresponding observable A.In his book Von Neumann based the Mathematics of Quantum Mechanics on Five Postulates.SOLO*

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, John Wiley & Sons, 1977Jim Baggott, The Meaning of Quantum Theory, Oxford University Press, 1992, pp. 42-48*

QUANTUM THEORIESQUANTUM MECHANICSVon NEUMANN THE FOUNDATION OF QUANTUM MECHANICS1932Fourth Postulate (case of discrete non-degenerate spectrum):When the physical quantity A is measured on a system in the normalized state | the probability P (an) of obtaining a non-degenerate eigenvalue an of the corresponding observable A is

where |u is the normalized eigenvector of A associated with the eigenvalue an. SOLO*

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, John Wiley & Sons, 1977Jim Baggott, The Meaning of Quantum Theory, Oxford University Press, 1992, pp. 42-48*

QUANTUM MECHANICSVon NEUMANN THE FOUNDATION OF QUANTUM MECHANICS1932Fourth Postulate (case of continuous non-degenerate spectrum):When the physical quantity A is measured on a system in the normalized state | the probability dP (an) of obtaining a result included between and + dis equal to

where |v is the eigenvector corresponding to the eigenvalue of the observableA associated with A.Postulates of Quantum Mechanics (continue 1)SOLO*

C. Cohen-Tannoudji, B. Diu, F. Lalo, Quantum Mechanics, John Wiley & Sons, 1977*

QUANTUM MECHANICSConservation of ProbabilitySOLO that is related to the Wave Vector in Momentum Space by:*

C. Cohen-Tannoudji, B. Diu, F. Lalo, Quantum Mechanics, Volume I, Wiley Interscience, 1977B.H. Bransden, C.J. Joachain, Quantum Mechanics, 2nd Ed., Prentice Hall, 1989,2000*

QUANTUM MECHANICSSOLOUseFinally we obtainConservation of Probability*

http://en.wikipedia.org/wiki/Wave_packetC. Cohen-Tannoudji, B. Diu, F. Lalo, Quantum Mechanics, Volume I, Wiley Interscience, 1977B.H. Bransden, C.J. Joachain, Quantum Mechanics, 2nd Ed., Prentice Hall, 1989,2000*

QUANTUM MECHANICSConservation of ProbabilitySOLOSince at any time t, the Probability of finding the particle somewhere is unity, and theProbability of the particle being in the Momentum Space is unity we have Let consider a finite Region T in the Position Space:*


QUANTUM MECHANICSConservation of ProbabilitySOLOwhereSince in the Equation above we have any time independent Spatial Volume T we can write *


QUANTUM MECHANICSConservation of ProbabilitySOLO*Connection with classical mechanicsThe wave function can also be written in the complex exponential (polar) form

http://en.wikipedia.org/wiki/Probability_current*

QUANTUM MECHANICSExpectations Value and OperatorsSOLO

*


QUANTUM MECHANICSSOLOExpectations Value and Operators*


QUANTUM MECHANICSSOLOand

We foundWe observe thatwe obtainExpectations Value and Operators*


QUANTUM MECHANICSSOLOWe found

Expectations Value and Operators*


QUANTUM MECHANICSSOLOWe foundExpectations Value and Operators

andWe observe thatIntegration by partsthereforeusingwe obtain*


QUANTUM MECHANICSSOLOWe foundExpectations Value and Operators*


QUANTUM MECHANICSSOLOPhysical Quantities and Corresponding Operators in Configuration SpaceExpectations Value and Operators*


QUANTUM MECHANICSValue of Observables The Main Value of an Observable is equal to the Expectation Value of its corresponding Operator.This expression is due to the Probability property associated to n function. If n is Eigenfunction of the Operator , i.e.then

Since the Observable are Real the Expectation Value an must be Real, therefore theOperator has only Real Eigenvalues meaning that it is Hermitian.SOLO*

J. Baggott, The Meaning of Quantum Theory, Oxford University Press,1992C. Cohen-Tannoudji, B. Diu, F. Lalo, Quantum Mechanics, Volume I, Wiley Interscience, 1977, Ch. II, The Mathematical Tools of Quantum MechanicsArthur Beiser, Perspectives of Modern Physics, McGraw-Hill, International Student Edition, 1969*

QUANTUM MECHANICSThe Expansion Theorem or Superposition Principle An arbitrary, well behaved State Vector | can be expanded as a Linear Superposition of the Complete Set of Eigenstate |i (i=1,,n) of any Hermitian Operator .By Complete Set of Eigenstate we mean the full set of Eigenstate of the Hermitian Operator . whereThe Expectation Value of in terms of | is be given by If | is Normalized andSOLO*

B.H. Bransden, C.J. Joachain, Quantum Mechanics, 2nd Ed., Prentice Hall, 1989,2000, pp.96, 213C. Cohen-Tannoudji, B. Diu, F. Lalo, Quantum Mechanics, John Wiley & Sons, 1977*

QUANTUM MECHANICSThe Expansion Theorem or Superposition PrincipleTo find the coefficients ci let multiply by the bra

QUANTUM MECHANICSMatrix Representation of Wave Functions and OperatorsThis ket Vector can be expressed as nx1 Matrix Given a Complete Set of Eigenstate |i (i=1,,n) (n can be finite or infinite) of any Hermitian Operator (orthonormal Eigenfunctions) we found that any Vector | can be expressed asIn the same way the bra Vectorcan be expressed as 1xn Matrixin the reciprocal basis i|, (i=1,,n)H is the Transpose, Complex ConjugateNow assume that an Operator acts on the Vector | to obtain The Vectors | and | can also be expressed in the bases|i and its reciprocal i|, (i=1,,n).SOLO*


QUANTUM MECHANICSMatrix Representation of Wave Functions and Operators (continue 1) Given a Complete Set of Eigenfunctions |i (i=1,,n) (n can be finite or infinite) of any Hermitian Operator (orthonormal Eigenfunctions) we found that any Vector | and | can be expressed as MatricesUsingand

We obtainSOLO*


QUANTUM MECHANICSMatrix Representation of Wave Functions and Operators (continue 2) In the same wayUsingand

Inner Product with |j:We obtainSOLO*


QUANTUM MECHANICSMatrix Representation of Wave Functions and Operators (continue 3) By tacking the Conjugate Complex, Transpose (H) of this equationWe obtainedwe see that (since is Hermitian)SOLOComparing with

*


QUANTUM MECHANICSMatrix Representation of Wave Functions and Operators (continue 4)Matrix Properties and DefinitionsSOLOIf A, B, C are Matrices of corresponding dimensions thenInverse A-1 of Square Matrix A, if exists, then A is Nonsingular (det A0) andTranspose AT of A, change the rows by the columnsAdjoint AH of A, is the Complex Conjugate of the TransposeHermitian Square MatrixSee also Matrix Presentation for a detailed description*


QUANTUM MECHANICSMatrix Representation of Wave Functions and Operators (continue 5)Matrix Properties and DefinitionsSOLOEigenvectors and Eigenvalues of Square MatricesSee also Matrix Presentation for a detailed descriptionwhere ui nx1 are Eigenvectors and i (i=1,,n) are Eigenvalues of A Unitary Square MatricesFor every Unitary Matrix U exists a Hermitian Matrix A such that*


Commutator of two Operators A and BGeneral Commutator PropertiesIf A B = B A we say that the two Operators A and B Commute, and then [A,B] = 0 .In general A B B A and we say that A and B dont commute.Theorem: A and B Commute if and only if they have the same Eigenfunctions iUsing Expansion Theorem any VectorProof: If A and B have the same Eigenfunctions i thenTherefore A and B Commute. QUANTUM MECHANICSAntisymmetryAssociativityJacobi IdentitySOLO*

B.H. Bransden, C.J. Joachain, Quantum Mechanics, 2nd Ed., Prentice Hall, 1989,2000, pp.96, 213*

Commutator of two Operators A and BTheorem: A and B Commute if and only if they have the same Eigenfunctions nSuppose that A have a complete set of EigenfunctionsProof (continue - 1): If A and B Commute, then they have the same Eigenfunctions n thenTherefore Bi and i are both Eigenfunctions of A, having the Eigenvalue ai, but this is possible only if they differ by a constant which will call biAssume first that A has non-degenerate Eigenvalues aiTherefore A and B have the same Eigenfunctions i , if ai are non-degenerate Eigenvalues. .Now assume that A has a degenerate Eigenvalue ai, of degree , with corresponding linearly independent Eigenfunctions ir (r=1,2,,). Since A B Commute (B i) is an Eigenfunction of A belonging to the degenerated Eigenvalue ai. It follows that (B i) can be expanded in terms of the linearly independent functions i1 , i2 ,, i QUANTUM MECHANICSSOLO*


Commutator of two Operators A and BTheorem: A and B Commute if and only if they have the same Eigenfunctions iProof (continue - 2): If A and B Commute, then they have the same Eigenfunctions i Therefore A and B have the same Eigenfunctions , even if ai are degenerate Eigenvalues. .Let form a linear combination of the functions ir with constants dr (r=1,2,,) to be defined If we can find constants bi (=1,..,) such thatWe find Eigenvectors (d1,,d)T and Eigenvalues bi of Matrix {cis}.QUANTUM MECHANICSSOLO*


Commutator of two Operators A and BExamplesIn the same way1Since this is true for all State Vectors QUANTUM MECHANICSSOLO

also*


Heisenberg Uncertainty RelationsConsider two Observable A and B, and a given Normalized State |.We define the Uncertainties A and B (Observable Variances) asDefineDefine the Linear but no Hermitian Operator1927The Expectations (Observables Main Values) of A and B are given by:QUANTUM MECHANICSSOLO*

B.H. Bransden, C.J. Joachain, Quantum Mechanics, 2nd Ed., Prentice Hall, 1989,2000, pp.96, 213P.R. Wallace, Mathematical Analysis of Phisical Problems, Dover Publications, 1972, 1984, Ch. 8, pp. 454-456*

Heisenberg Uncertainty Relations (continue 1)Define the Real and Nonnegative Function of Since f () is Real i [A,B] is Real [A,B] is Purely Imaginary [A,B] 20

QUANTUM MECHANICSSOLO*


Heisenberg Uncertainty Relations (continue 2)Since A and B are real and positive, we foundandTherefore

Heisenberg Uncertainty Relation for Simultaneous Position & MomentumMeasurements

Heisenberg Uncertainty Relation for Simultaneous Time & EnergyMeasurementsQUANTUM MECHANICSSOLO*


Heisenberg Uncertainty Relations1927QUANTUM MECHANICSSOLO*

Second way using Fourier Transform Properties

http://en.wikipedia.org/wiki/Uncertainty_principle*

Heisenberg Uncertainty Relations1927QUANTUM MECHANICSSOLO*We found

Second way using Fourier Transform Properties


Heisenberg Uncertainty Relations1927QUANTUM MECHANICSSOLO*andSecond way using Fourier Transform Properties


Heisenberg Uncertainty Relations1927QUANTUM MECHANICSSOLO*Let computeSecond way using Fourier Transform Properties

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