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单击此处编辑母版标题样式,*,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,Brief Review,The most important properties of particle,1,The quantization,e.g quantization of energy, energy levels,2,Particle - Wave Duality,h,P,h,/,Planck-Eistain- de,Broglie,relations,Particle,Wave,Interference and Diffraction,x P,x,h/4,impossible to specify simultaneously the precise position and 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 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：,3.4,Vibration motion,3.4.1,The Harmonic Oscillator,(1),Schrdinger Equation,Consider a particle subject to a restoring,force,F,= -,kx, the potential is then,Zero-point：,(2),The solutions,(,i)The energy levels,v,= 0, 1, 2, 3,(,ii)The wavefunctions,3.5,Rotational Motion,R=r,a,+r,b,x,y,z,r,a,r,b,B,A,O,The rigid rotor is a simple model of a rotating diatomic molecule. We consider the diatomic to consist of two point masses at a fixed internuclear distance.,(1),Schrdinger Equation,For a rigid rotor,so,(2),The solutions,After a little effort, the eigenfunctions can be shown to be the spherical harmonics,(,) =,Y,(,),J,0、1、2、3，,J,Rotational quantum number,degeneracy,g =,2,J,+ 1,Rotational energy levels,Further Reading and Homework,Identify which of the following functions of the operator d/,dx,:(a),e,i,kx,(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,):(a),x,3,-,kx,(b),cos,kx,(c),x,2,+3,x,-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 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,the hydrogen atom and hydrogen-like ions, He,+,Li,2+, 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 atoms,so the Schrdinger equation as,(2),The solutions,or,The spherical coordinates used for discussing systems with spherical symmetry.,since,Separation of variables,X,(,x,),Y,(,y,),Z,(,z,) ?,x,=,r,sin,cos, y,=,r,sin,sin, z,=,r,cos,so we write out the Schrdinger equation in spherical polar coordinates as,R,equation,Radial wave equation,equation,equation,Separation of variables,=R(,r,)(,)(,),The radial solutions,The radial part,R,(,r,) then can be shown to obey the equation,eV,n =,1,2,3,principal quantum number,.,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,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 continuous and,single-valued,，,(,)=,(,+2),m,= 0,1,2,l,the,magnetic,momentum quantum number,Total wavefunction of single-electron atoms,R,nl,(,r,) is called,radial wavefunction.,Taking,Y,lm,(,) yields spherical harmonics.,4.2,Atomic orbitals and their energyies,The,atomic orbital and electron cloud,An atomic orbital,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,r,R,nl,(,r,),r,radial distribution functions of,Y,lm,(,),angular,distribution functions of,radial distribution functions,The probability of finding an electron in a unit volume,d,V,is given by,Probability 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,(,)|,orbital,spherical harmonic,s,Y,0,0,p,x,Y,1,0,p,y,Y,1,1,p,z,Y,1,-1,Y,2,0,d,xz,Y,2,1,d,yz,Y,2,-1,Y,2,2,d,xy,Y,2,-2,d,xz,d,xz,d,yz,|6,4,1,|3,2,1 and |3,1,-1,|3,2,2 and |3,1,-1,|4,3,3 and |4,1,0,|3,2,0,Properties of the solutions,The quantization of energy,eV,RRydberg 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 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.,m,s,is called the,spin quantum number,.,l,= 0,L,= 0 .,As nicely explained in this angular momentum is,intrinsic,to the electron.,Overall wavefunctions of atom,-,orbital,wavefunction,-,spin,wavefunction,Quantum number,Integer values,Quantized quantity,n,l,m,m,s,n, 1,0 ,l,n,-l,m,l, 1/2,Energy,Magnitude of orbital angular momentum,z-component of orbital angular momentum,z-component of spin angular momentum,n =,1,2,3,l,= 0,1,2,n,-1,m =,0, 1,2,l,m,s=, 1/2,The states of the single-electron atoms,6.3,Many-electron atoms,1,The Schr,dinger,equation of many-electron atoms,（,Born-Oppenheimer,Approximation,）,Unfortunately, precise solutions are not available through the Schr,d,inger equation, even for the simplest many-electron, helium, because,Independent particle model,The Schr,dinger equation,Separation of variables,Mean field model,An electron at a distance r from the nucleus experiences a Coulombic repulsion from all the electrons within a sphere of radius r and which is equivalent to a point negative charge located on the nucleus.,，,n,=1,2,3,Symmetric,Bosons,Antisymmetric,Fermions,The Pauli principle,All electronic wavefunctions must be antisymmetric under the interchange of any two electrons.,2,Identical particles and the Pauli principle,Identical particles,Identical particles cannot be distinguished by means of any intrinsic properties.,Slater,determinant,Normalization constant,(,i),(,ii) No two electrons in an atom can have the same values for all four quantum numbers.,4,Electron configurations,The Pauli exclusion principle,No two electrons in an atom can have the same values for all four quantum numbers.,Ground state electron configuration Aufbau principle,Hund,s rule,Electrons occupy the orbitals of a subshell singly until each orbital has one electron.,p,6, d,10, f,14,p,3, d,5, f,7,p,0, d,0, f,0,Further Reading,http:/tesla.phys.unm.edu/phy537/1/node1.html,http:/blakey.sci.ccny.cuny.edu_themislecture_ch8_htmlindex.html,http:/www.pha.jhu.edu_rt19hydrohydro.html,http:/library.thinkquest.orgC0110925htmlindex.html,http:/rugth30.phys.rug.nl/quantummechanics/,
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