量子力学的几个基本假设Postulates.PDF

量子力學的幾個基本假設 (Postulates) • Postulate 1 The state of a quantum system is described by a function Ψ (x, y, z, t) or Ψ (r, t), of the configuration space variables and time. The function is referred to as the state function or the wavefunction. It contains all the information that can be determined about the system. Furthermore , we require that Ψ(x, y, z, t) be single valued, continuous, differentiable to all orders, and quadratically (or square) integrable. The wavefunction for a single particle may be interpreted as follows: Ψ* (r, t) Ψ (r, t) dx dy dz is the probability that the particle is in the volume dx dy dz located at r at time t. • 假設1: 一個系統的波函數包含系統所有的全部信息，系統的狀態可用波函數 Ψ充分完全描述。一個包含N個微觀粒子的系統，波函數其中的變數 可由各個粒子的座標變量與時間變量來確定。合格良質的波函數 必須滿足三個條件：單值，是連續，平方可積分。 2006/3/2 57 Unacceptable functions as wavefunctions • The wavefunction must satisfy stringent conditions for it to be acceptable. (a) Unacceptable because it is not continuous; (b) unacceptable because its slope is discontinuous; (c) unacceptable because it is not single-valued; (d) unacceptable because it is infinite over a finite region. 2006/3/2 58 Postulates of Quantum Mechanics • Postulate 2 For every observable in classical mechanics there is a linear quantum mechanical operator. The operator is obtained from the classical mechanical expression for the observable written in terms of Cartesian coordinates and conjugate momenta by replacing each coordinate q by itself and the conjugate momentum component by p̂ = –i ℏ ∂/∂q . q • 假設2: 一個系統的每一個可觀測的物理量，必存在一個對應的線性厄米算 符。量子力學物理量的算符是將古典物理量中直角座標系統的座 標，時間與動量三個變量以下式變換成相應算符表示: x ≡ x; t ≡ t; px ≡ -i ℏ ∂ / ∂x; 2006/3/2