量子化学与群论基础 4.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 Wavefunctionψ： 1 The state description 2ψ*ψ ? 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： 3.4 Vibration motion 3.4.1 The Harmonic Oscillator (1) Schr?dinger 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=ra+rb x y z ? ?? ra rb 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) Schr?dinger 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 = 2J + 1 Rotational energy levels Further Reading and Homework Identify which of the following functions of the operator d/dx:(a) eikx,,(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)x3-kx(b)coskx,(c) x2+3x-1. State the eigenvalue of i when relevent. 3. An electron in a one-dimensiona