Chapt15-量子力学初步-2015-中文版课件.ppt

* * * * * * * * * * * * * * * * * * * * * * n =10 系统能量 引入算子 定义其本征态 |n (n=0,1,2,3,…), 使得 对易关系 粒子数表示和 Dirac 符号 Hamiltonian 作用于 |n a 和 分别称为湮灭算子和产生算子。作用于 |n |0 为真空态, 使得 Quantum Flow §15.6 Quantum Mechanics Representations Mathematically, When r→∞, the integral of right hand side is 0. Dirac’s Notations, States Superposition Principle, State Vector, and Inner Product Base vectors : Pay your attention: Ground state Any state: where are the amplitudes, or projection coefficients on states Similar to where We can write a similar equation for any other state vector , say , with, of course, different coefficients: The Di are just the amplitudes . Define a “bra”, the conjugate state vector of as follows Then we have inner product of and which is the close analogy with the dot product Matrix Express of a Mechanical Quantity Operator If where are two states, and , they satisfy the relations under the basis Operators (representing observables) are linear transformations – they “transform ’’ one vector into another: or, Let left time above equation: So we have relation: where are matrix elements of operator . This form is equivalent to Example: Two states system: The general state is a normalized linear combination: The Hamiltonian is supposed to have a special form: where g and h are real constants. If the system starts out (at t=0) in state . What is the state at time t? Solution: The time-dependent Schr?dinger equation says The time-independent (stationary) (How to solve this equation?) The characteristic equation determines the eigenvalues: The eigenvectors are assumed to be the form Then, the stationary Schroedinger equation is written: The normalized eigenvectors of the stationary S-equation are So, The normalization condition requires that The initial state can be expanded as a linear combination of eigenvectors of the Hamiltonian: Finally, we take on the standard time-dependence Problem: The Hamiltoni