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1、Brief Review,The most important properties of particle,1 The quantization e.g quantization of energy energy levels,2 Particle - Wave Duality, h Ph /,Planck-Eistain- de Broglie relations,Particle,Wave,Interference and Diffraction,x Px h/4 impossible to specify simultaneously the precise position and
2、momentum.,state wavefunction,Dynamic equationwave equation,amplitude* the probability of finding the particle,Probability wave,Wavefunction：,1 The state description,2* Probability density,3 The value of observable,4 The average value of the observable,The problem is How to get Wavefunction?,The only
3、 way is,3 Some Analytically Soluble Problems,The motions of particle Translational motion Rotational motion Vibrational motion Electronic motion Nuclear motion,The Energy of the particle：,Zero-point：,(ii)The wavefunctions,(1) Schrdinger Equation,Rotational energy levels,Further Reading and Homework,
4、Identify which of the following functions of the operator d/dx:(a) eikx,(b)cos x,(c)k,(d)kx,(e)e-x. Gave the corresponding eigenvalue where appropriate. Determine which of the following functions are aigenfunctions of the inversion operator i(which has the effect of making the replacement x to -x):(
5、a)x3-kx(b)coskx,(c) x2+3x-1. State the eigenvalue of i when relevent.,3. An electron in a one-dimensional box undergoes a transition from the n=3 level to the n=6 level by absorbing a photon of wavelength 500 nm. What is the width of the box? 4. What is the average location of a particle in a box of
6、 length l in the n=3 quantum state? 5. Calculate the lowest energy transition in the butadiene molecule.,GREEK VIEWS word for atom means not divisible.,4 Atoms,Rutherford (1901) proposed that electrons orbit about the nucleus of an atom.,1. The Schrdinger equation of single-electron atoms,Consider t
7、he hydrogen atom and hydrogen-like ions, He+,Li2+ as a proton fixed at the origin, orbited by an electron of reduced mass .,The Born-Oppenheimer Approximation The approximation that the nuclei remain stationary on the time scale of electron movement.,(1) Schrdinger equation,4. 1 The single-electron
8、atoms,so the Schrdinger equation as,(2) The solutions,or,The spherical coordinates used for discussing systems with spherical symmetry.,x=rsincos y=rsinsin z=rcos,so we write out the Schrdinger equation in spherical polar coordinates as,R equation Radial wave equation, equation, equation,Separation
9、of variables =R(r)()( ),The radial solutions,The radial part R(r) then can be shown to obey the equation,which is called the radial equation (associated Laguerre equation).,Its (messy) solutions are (associated Laguerre functions),The single-electron atoms eigenvalues are, The solutions of equation,
10、This is simply the associated Legendre differential equation with solutions given by,l = 0, 1, 2, 3 (n-1) s, p, d, f l - angular momentum quantum number.,With the correct normalization constant when l =0,1,2(n-1), the solution is, The solutions of equation,solutions of equation,or,() must be continu
11、ous and single-valued， ()=(+2),m = 0,1,2,l,the magnetic momentum quantum number, Total wavefunction of single-electron atoms,Rnl(r) is called radial wavefunction.,Taking Ylm(,) yields spherical harmonics.,4.2 Atomic orbitals and their energyies, The atomic orbital and electron cloud,An atomic orbita
12、l is one-electron wavefunction for an electron in an atom. Probability of finding electron in a atom or molecule is called electron cloud., Representations of atomic orbitals and electron clouds, radial distribution functions,The probability of finding an electron in a unit volume dV is given by,Pro
13、bability of finding electron in a spherical shell of radius r?,Shells and subshells n = 1, 2, 3, 4 K, L, M, N,l = 0, 1, 2, 3 (n-1) s, p, d, f ,Spherical harmonics Y(,),draw a line from origin: the direction - (,) the length - |Y(,)|, Properties of the solutions, The quantization of energy,eV,RRydber
14、g energy, n =1，2，3，,n is called the principle quantum number., The quantization of orbital angular momentum of the electron,l = 0, 1, 2, n-1 s, p, d , ,m is called the magnetic momentum quantum number., The quantization of orbital magnetic momentum of the electron,m= 0, 1,2,l,l is called the angular
15、 momentum quantum number,3 The states of the single-electron atoms, Spin,The Stern-Gerlach experiment performed in 1925 showed that the electron itself also carries angular momentum which has only 2 possible orientations.,ms is called the spin quantum number.,l = 0 , L = 0 .,As nicely explained in t
16、his angular momentum is intrinsic to the electron., Overall wavefunctions of atom,- orbital wavefunction,- spin wavefunction, The states of the single-electron atoms,6.3 Many-electron atoms,1 The Schrdinger equation of many-electron atoms,（Born-Oppenheimer Approximation）,Unfortunately, precise solutions are not available through the Schrdinger equation, even for the simplest many-electron, helium, because, Independent particle model,The Schrdinger equation,Separation of varia
