量子化学与群论基础 2.ppt

1、1.4 The structure of science,Law a summary of experience,Hypothesis a tentative explanation that accounts for a set of facts and can be tested by further investigation.,Theory a system of assumptions, accepted principles, and rules of procedure devised to analyze, predict, or explain the nature or b

2、ehavior of phenomenon.,Model a simplified version of the system that focuses on the essentials of the problem.,1.5 Wavefunction,The Uncertainty Principle xPxh/ 4,The state description Thermodynamics: Bulk properties(p,V,T,) Classical mechanics:,de Broglie Hypothesis 1. Matter could also behave as wa

3、ves. 2. Matter would obey the same equation as light Eh P h 3. The states of matter can be described by wavefunction. The 1-dimensional free particle:,1.5.1 Interpretation of the wavefunction,What is this wavefunction? What does it mean?,Born(1926) suggested that wavefunction was that the square at

4、a given point in space was proportional to the probability of finding the particle at that point in space.,The square is called the probability density while we can call the wavefunction is probability amplitude.,A single electron xPxh/ 4,A swarm of electrons,Probability density,Probability,Statisti

5、cal definition:,Calculation of probability,Properties of probability,Probability deals with measuring or determining the likelihood that an event will have a particular outcome.,P(A)=0 impossible event P(A)=1 inevitable event,e.g. , throw a coin , dice,ANormalization constant,For the case of a singl

6、e particle,1.6 The dynamics of microscopic systems,Wave Mechanics Schrdinger (1925) developed Broglies idea into the differential equations capable of dealing with a number of physical phenomena and with problems that could not be handled by classical physics.,Matrix Mechanics Heisenberg (1925)chose

7、 to pursue the Matrix pathway and started to associate matrices with the properties of matter.,Relativistic Quantum Theory A synthesis of quantum mechanics and relativity was made in 1928 by the British mathematical physicist Dirac, leading to the prediction of the existence of the positron and brin

8、ging the development of quantum mechanics to a culmination.,Hilbert was a well-known mathematician during this time. Hilbert spaces,1.6.1 The Schrdinger Equation,The Time-Dependent Schrdinger Equation,The Time-Independent Schrdinger Equation,1.6.2 The acceptability of wavefunction,(1) must be contin

9、uous, have a continuous slop;,(3)and be finite everywhere.,(2) be single-valued;,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/dx An operator is a rul

10、e 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 mathem

11、atical 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

12、 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 operat

13、ors. 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/dx and ()2. We can see that d/dx is a linear operator because,However, ()2 is not a linear operator because,2.1.3 Eigenfunction

14、s 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,DefiniteLapacian,then,Definite Hamiltonian,then,2.2

15、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

16、, 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

17、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 mecha

18、nics.,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 ai, but we dont know which one. However, we do know the probability that eigenvalue ai will occ

19、ur-it is the absolute value squared of the coefficient, |ci|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 valu

20、e 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= UU =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, t