量子化学与群论基础 3.ppt

* * Brief Review The most important properties of particle 1 The quantization e.g quantization of energy ? energy levels 2 Particle - Wave Duality Ε＝ hν P＝h /λ Planck-Eistain- de Broglie relations Particle Wave Interference and Diffraction Δx ΔPx≥ h/4? impossible to specify simultaneously the precise position and momentum. state— wavefunction Dynamic equation—wave equation amplitudeψ*ψ ? 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/dx … … 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/dx and ()2. We can see that d/dx is a linear operator because However, ()2 is not a linear operator because The only other category of operators relevant to quantum me