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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,Mathematical Background,and Postulates of Quantum Mechanics,2.1,Operators,Operator,An,operator,is a symbol that tells you to do something with whatever follows the symbol.,e.g., , , ,ln, sin, d/,d,x, ,An,operator,is a rule that transforms a given function or vector into another function or vector.,e.g.,2.1.1,Basic Properties of Operators,Two operators are,equal,if,The,sum,and,difference,of two operators,The,product,of two operators is defined by,The,identity operator,does nothing (or multiplies by 1),A common mathematical trick is to write this operator as a sum over a complete set of states (more on this later).,The,associative law,holds for operators,The,commutative law,does,not,generally hold for operators. In general,It is convenient to define the quantity,which is called the,commutator,of and . Note that the order matters, If and happen to commute, then,The,n-,th,power,of an operator,is defined as,n,successive applications of the operator, e.g.,The,exponential,of an operator,is defined via the power series,2.1.2,Linear Operators,Almost all operators encountered in quantum mechanics are,linear operators,. A linear operator is an operator which satisfies the following two conditions:,where,c,is a constant and,f,and,g,are functions.,As an example, consider the operators,d,/,d,x,and (),2,. We can see that,d,/,d,x,is a linear operator because,However, (),2,is not a linear operator because,The only other category of operators relevant to quantum mechanics is the set of,antilinear,operators, for which,Time-reversal operators are antilinear.,2.1.3,Eigenfunctions and Eigenvalues,An,eigenfunction,of an operator is a function,u,such that the application of on,u,gives,u,again, times a constant,Matrix description of an eigenvalue equation,2.1.4,Operator Expression of the Time-Independent Schrdinger Equation,Definite,Lapacian,then,Definite,Hamiltonian,then,2.2,Postulates of Quantum Mechanics,Postulate 1,The state of a quantum mechanical system is completely specified by a function,(,r,t,),that depends on the coordinates of the particle(s) and on time. This function, called the wave function or state function, has the important property that,*,(,r,t,),(,r,t,)d,is the probability that the particle lies in the volume element,d,located at,r,at time,t,.,The wavefunction must be single-valued, continuous, and finite.,Postulate 2,In any measurement of the observable associated with operator, the only values that will ever be observed are the eigenvalues,a, which satisfy the eigenvalue equation,Postulate 3,. If a system is in a state described by a wave function, then the average value of the observable corresponding to,is given by,Postulate 4,. To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics.,Table 1:,Physical observables and their corresponding quantum operators (single particle),Observable,Observable,Operator,Operator,Name,Symbol,Symbol,Operation,Position,r,Multiply by,r,Momentum,P,i,Kinetic energy,T,Potential energy,V,(,r,),Multiply by,V,(,r,),Total energy,E,Angular momentum,l,x,l,y,l,z,Postulate 4,. An arbitrary state can be expanded in the complete set of eigenvectors of,as,where,n,may go to infinity. In this case we only know that the measurement of,A,will yield,one,of the values,a,i, but we dont know which one. However, we do know the,probability,that eigenvalue,a,i,will occur-it is the absolute value squared of the coefficient, |,c,i,|,2,2.3,Hermitian Operators and Unitary Operators,2.3.1,Hermitian Operators,As mentioned previously, the expectation value of an operator,is given by,and all physical observables are represented by such expectation values. Obviously, the value of a physical observable such as energy or density must be real, so we require to be real. This means that we must have = ,*, or,Operators,which satisfy this condition are called,Hermitian,.,2.3.2,Unitary Operators,A linear operator whose inverse is its adjoint is called,unitary,. These operators can be thought of as generalizations of complex numbers whose absolute value is 1.,U,-1,=,U,UU,=,U,U =I,A unitary operator preserves the lengths and angles between vectors, and it can be considered as a type of rotation operator in abstract vector space. Like Hermitian operators, the eigenvectors of a unitary matrix are orthogonal. However, its eigenvalues are not necessarily real.,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：,3.1 The Free Particle,A free particle is one which moves through space without experiencing any forces. Hence it travels in a straight line. Its potential energy is everywhere constant, and so can be assigned to be 0.,The energy states are NOT quantized, but any value is allowed.,3.2 The Particle in a Box,3.2.1 The 1-Dimensional Particle-in-a-Box,(1),Schrdinger Equation,The particle of mass,m,is confined between two walls:,V,(,x,) = 0 (0,x,l,),V,(,x,) (,x,0 and,x,0),let,Boundary conditions,x,=0,(0),A,sin0,B,cos0 =0;,B,=0,(,x,),A,sin,kx,x,=,l,(,l,),A,sin,kl,= 0; sin,kl,= 0,kl,=,n,square,n,= 1,2,3 quantum number,The general solutions are,(,x,),A,sin,kx,B,cos,kx,n,= 1,2,3 .,(2) Properties of the solutions,Therefore, the complete solution to the problem is,(,i) The quantization of energy,n,= 1,2,3 .,quantum number,This lowest, irremovable energy is called the zero-point energy.,E,=,T,+,V,The 1-Dimensional Particle-in-a-Box,V,= 0,E,=,T,（a）Zero-point energy,(,b),E,l,or,m,，,E,Classical or free particle，,E,0.,(,ii) Wavefunction,and quantum number,n,Ground state and excitated state,(iii)Probability distributions,(iv) Applications,1,3-butadiene,b,-carotene,l,=,21,0.140,nm = 3.08 nm.,And the lowest 11 energy levels will be filled.,Carrots are orange because the absorption of the short wavelength (blue) light leaves only the red-orange to reflect.,(v) Orthogonality and the bracket notation,Two wavefuctions are orthogonal if their product vanishes.,e.g.,The integral is often written,=0 (,n,n,),Dirac,bracket notation, ket,Normalized wavefuctions =1,These two expressions can be combined into a single expression:,Kronecker delta,3.3 The Two and Three-Dimensional Particle-in-a-Box,3.3.1,Motion in two dimensions,(1) Schrdinger Equation,In box，,V,0,(2) Separation of variables,X,(,x,),Y,(,y,),E,E,x,E,y,(3) The solution,(4) Degeneracy,Consider the case,n,x,=1,n,y,=2 and,n,x,=2,n,y,=1,When,a,=,b,We say that the states |1,2 and |2,1 are degenerate.,3.3.2,Motion in three dimensions,(1) Schrdinger Equation,In box，,V,0,Separation of variables,X,(,x,),Y,(,y,),Z,(,z,),E,E,x,E,y,E,z,(2),Solution,(3) Degeneracy,Cubic，,a,b,c,112,121,211,E,112,=,E,121,=,E,211,3.4,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.,
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