chapter 1

nullnull物理化学nullChapter One Introduction to Quantum Mechanics 1020 meters~ 10-20meters 1010years ~ 10-18seconds 1023stars ~ 102atomsCan we describe the universe with an uniform theory,especially at atomic scale?The WORLDnullClassical (Newtonian) MechanicsMaxwell’s Theory of Electricity and MagnetismIn the late of nineteenth century, some physicists believed that the theoretical structure of physics was complete, is it true?1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum Mechanics1900 1905 1911 1913Max Planck explains blackbody radiation in the context of quantized energy emission: Quantum theory is born. Albert Einstein proposes that light, which has wavelike properties, also consists of discrete, quantized bundles of energy, which are later called photons. Ernest Rutherford proposes the nuclear model of the atom. Niels Bohr proposes his planetary model of the atom, along with the concept of stationary energy states, and accounts for the spectrum of hydrogen.1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsArthur Compton observes that x-rays behave like miniature billiard balls in their interactions with electrons, thereby providing further evidence for the particle nature of light. Louis de Broglie generalizes wave-particle duality by suggesting that particles of matter are also wavelike. Wolfgang Pauli enunciates the exclusion principle. Werner Heisenberg, Max Born, and Pascual Jordan develop matrix mechanics, the first version of quantum mechanics, and make an initial step toward quantum field theory.1923 1923 1925 19251-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum Mechanics1926 1927 1928Erwin Schrodinger develops a second description of quantum physics, called wave mechanics. It includes what becomes one of the most famous formulae of science, which is later known as the Schrodinger equation. Heisenberg states his Uncertainty Principle, that it is impossible to exactly measure the position and momentum of a particle at the same time. Dirac presents a relativistic theory of the electron that includes the prediction of antimatter. null布鲁塞尔的女巫盛宴 1911年10月19日第一届索尔维会议1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsA picture of some famous physicists in 1927A.EinsteinCurieM.PlanckLorenzoLangevinDebyeBraggDiracPauliHeisenbergBoreDe BroglieComptonBorn1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsQuantum Mechanics is one of the most profound and beautiful creations of the human mind. … Their beauty is transcended only by those things that are beyond the reach of man, such as the smile of a child.1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsBlackbody is a substance that would absorb all frequencies of radiation falling on it ; it would reflect no radiation and appear black.nullWien’s equationUltraviolet catastropheRauleigh-Jeans Equation1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum Mechanics1. The radiating atoms or molecules of the blackbody could emit and absorb electromagnetic energy as coming in small packets. The packets of energy are called quanta and are only in multiples of . 2. The packets are of energy . M.Planck1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsLower FrequencyPlanck’s EquationRauleigh-Jeans EquationHigher FrequencyWien’s equation1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsProblem: a) Use the fact that to show that the radiant energy emitted per second by unit area of a blackbody is . b) The sun’s diameter is m and its effective surface temperature is 5800 K. Assume the sun is a blackbody and estimate the rate of energy loss by radiation from the sun. c) Use to calculate the relativistic mass of the photons lose by radiation from the sun in a year.1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum Mechanics金属Photoelectric Effect1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum Mechanics. The frequency of the light and not its intensity controls whether or not electrons are emitted。The frequency which is just sufficient to remove the electron, is known as threshold frequencyP.Lenard (1905 Nobel Prize）1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsClassical Physics：The energy of the radiation should depend only on its intensity. Whether or not electrons are emitted should therefore depend on the energy of the radiation, and even low intensities, regardless of frequency, should be effective if long periods of time were allowed for their absorption. 2. The number of electrons emitted is proportional to the intensity, and the electron emission occurs the instant the radiation strikes the metal surface if the frequency is above threshold.1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum Mechanics2. Photons obey the laws of conservation of energy and momentum when the photons interact with the material. Einstein’s Theory1. Light itself is a beam of particles each of which has energy equal to and momentum to .3. The intensity of light is proportional to the number of photons per unit volume in the beam.1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsThe Explain of Photoelectric effect kinetic energy K = E (radiation)- work function W1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsCompton’s effect, Nobel Prize in 1927 ——investigated the scattering of monochromatic x-ray and found that the scattered beam consisted of radiation of two different wavelengths, one wavelength was the same as for original beam, whereas the other was slightly longer.1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsProblem: The work function of K is 2.2eV and that of Ni is 5.0eV. a) Calculate the threshold frequencies and wavelengths for these two metals. b) Will violet light of wavelength 4000A cause the photoelectric effect in K? In Ni? c) Calculate the maximum kinetic energy of the electron emitted in b).1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsBohr’s Atomic TheoryWhat is the regularities in atomic spectra1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsJ.J.Balmer’s seriesHow to explain these rules？ --------The Model of Atomic StructureSome empirical relationships:1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsJ.J.TomsonJ.J.Thomson’s Model1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsE.Rutherford’s model1.The nucleus of an atom is very much smaller than the atom itself;2. The electrons revolved around this dense positively charged nucleus;3. The size of the atom is determined by the size of the electronic orbits1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsRutherford’s model presented a serious problem in terms of classical physics. 1. An orbitng charged particle must continuously lose energy. The electron would therefore fall into the nucleus, and the atom would not survive.2.The kinds of atom is limited, why?Bohr’s atomic theory= Planck’s Theory + Einstein’s theory of radiation + Rutherford’s concept of atom1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum Mechanicsis a positive integer1. There are certain allowed energy, known as stationary states in the atom.2. These states are characterized by discrete values on the angular momentum3. When an electronic transition occurs between two states of energies, the frequency is given by1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsApplication to hydrogen atomBohr radius109737.31cm-11-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsBohr & Copenhagen SchoolCorrection of Bohr’s Theory109677.60cm-1109737.31cm-1Sommerfeld1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsProblem: On the basis of the Bohr theory, calculate the ionization energy of the hydrogen atom and the linear velocity of an electron in the ground state of the hydrogen atom.1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsDual Theory of RadiationNewtonHuygensYoungFresnel1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsA.Einstein’s RelationshipInteraction PropagationWave-particle dualityNewtonMaxwellEinstein1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsElementary particles such as electrons can also have wave properties, just as radiation has particle properties. ——de BroglieINTEGER? Dual Theory of Particle Louis de Broglie generalizes wave-particle duality by suggesting that particles of matter are also wavelike1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum Mechanics Show the de Broglie wavelengths of a 10-g bullet moving at 10m/s and a electron moving at 108cm/s.The extremely small size of wavelength indicates that quantum effects are unobservable for the motion of macroscopic objects. 1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsDe Broglie’s prediction was confirmed experimentally in 1927C.J.Davtsson & L.H.GermerG.P.Thomson 1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsG.P.Thomson’s resultC.J.Davtsson & L.H.Germer’s resultThe Rule of Silver Coin1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsParticleWavenull1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum Mechanicsthe Scattering of Electrons1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsBorn’s Interpretation 对应于空间的一个状态，就有一个伴随这状态的德布罗依波的几率。若与电子对应的波函数在空间某点为零，这就意味着在这点发现电子的几率小到零。 ——M.BornThe intensity of a particle at some point is proportional to the probability density at that point. 1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsQuantum mechanics…says a lot, but does not really bring us any closer to the secret of the Old One. I, at any rate, am convinced that He does not throw dice. ------ A. Einstein1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsThe Uncertainty PrincipleProblem：Can we make accurate measurements of both two properties, such as the momentum and position, at same time? It is impossible, because any technique for measuring one of them will necessarily disturb the system and will cause the measurement of other one to be imprecise.1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum Mechanics Problem: The electron does not fall into the nucleus, why?1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsA.H.Zewail & FemtochemistryNobel Prize in 1999Application of Uncertainty Principle ------ lifetime of excited state1-1 Naissance of the Quantum Mechanics1-1 Naissance of the Quantum MechanicsProblem: What is the de Broglie wavelength of an oxygen molecule at room temperature? Compare this to the average distance between oxygen molecules in a gas at 1 bar at room temperature. What is the de Broglie wavelength of an electron that has been accelerated through a potential difference of 100V. What is the width in energy domain of a 4fs pulse?1-2 Wavefunction1-2 WavefunctionMatrix MechanicsW.HeisenbergWave MechanicsE.SchrodingerHow do we describe the behavior of micro-particles ?1-2 Wavefunction1-2 WavefunctionWavefunction: Matter is described by a wave, called the wavefunction , that depends upon the coordinates and time. In general, wavefunction is finite,single valued, and continuous at all point.Stationary state wavefunction & Time-dependent wavefunction gives the probability density for finding the particles at given locations in space.1-2 Wavefunction1-2 WavefunctionLinear Combination of WavefunctionSchrodinger Cat ?1-2 Wavefunction1-2 WavefunctionCoherent control of reaction1-2 Wavefunction1-2 WavefunctionProblem: 1. A possible eigenfunction for the system is: Show that , the probability, is independent of time. 2. Prove that m must be the integral in order for the function to be an acceptable wave function.1-3 The Solution of Wavefunctions1-3 The Solution of WavefunctionsFunctionsFunction1-3 The Solution of Wavefunctions1-3 The Solution of WavefunctionsOperatorAn operator is a mathematical operation that is applied to a function. 1-3 The Solution of Wavefunctions1-3 The Solution of WavefunctionsMultiplication & CommutationMultiplicationCommutationThe multiplication of operators is in general not commutative.1-3 The Solution of Wavefunctions1-3 The Solution of WavefunctionsLinear operatorHermitian operatorEigenequation Eigenfunction EigenvalueDegenerate & DegeneracynullSome properties of Hermitian operatorsTheorem 1: The eigenvalue of a Hermitian operator must be real.Theorem 2: The eigenfunctions for the Hermitian operators corresponding to different eigenvalues must be orthogonal.nullProblem: Prove that momentum operator corresponding to is a Hermitian operator. nullPostulates of Quantum MechanicsThe first Postulate: A state of a quantum mechanical system is completely specified by a wavefunction that is a function of the coordinates of the particle and time.nullThe process of converting a function for a classical system into the corresponding operator for the quantum mechanical system is formalized by the following rules: 1) Each Cartesian coordinate in the Hamiltonian function is replaced by the coordinate itself. 2) Each Cartesian component of linear momentum in the Hamiltonian function is replaced by the operator of momentum. The second Postulate: Each classical observable is associated with a quantum mechanical operator.nullnullThe third Postulate: The possible measured value of the physical observable corresponding with an operator is given by:Where is the complex conjugate of . nullSchrödinger Equationis called HamiltoniannullOne-dimensional BoxnullnullnullnullTwo-dimensional BoxnullHarmonic OscillatornullnullZero-point EnergynullnullWhen a particle passes through a region that is classically prohibited, the process is called tunneling.nullTunneling Effect of Stationary WavenullnullA particle in one dimension box with nullnullScanning Tunneling MicroscopeG.Binnig, H.Rohrer, E.Ruska Nobel Prize in 1986nullProblem: What is the degree of the degeneracy if the three quantum numbers of a three-dimensional box have the values 1, 2 and 3? Calculate the lowest possible energy for an electron confined in a cube of sides equal to a) 10pm and b)10-15m. The latter cube is the order of the magnitude of an atomic nucleus; what do you conclude from the energy you calculate about the probability of a free electron being present in a nucleus?nullProblem: Prove that any two wave functions for a particle in a one-dimensional box are orthogonal to each other. The vibration frequency of the N2 molecule corresponds a wave number of 2360cm-1. Calculate the zero-point energy and the energy corresponding to v=1.nullThe Rigid RotornullSeparate variablenullNormalizenullnullLegendre equationnullLegendre polynomialnullor EnergySpherical functionWavefunctionAngular momentumnull