量子力學導論 Chap 1 - The Wave Function Chap 2 - The Time-independent Schrödinger Equation Chap 3 - Formalism in Hilbert Space Chap 4 - 表象理論.

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量子力學導論 Chap 1 - The Wave Function Chap 2 - The Time-independent Schrödinger Equation Chap 3 - Formalism in Hilbert Space Chap 4 - 表象理論

Quantum Mechanics Chap 2 - The Time-independent Schrödinger Equation ► 2.1 Stationary state Assume V is independent of t, use separation of variables Deduce from equation (2.1), then

Quantum Mechanics...(2.2) time-independent Schrödinger equation

Quantum Mechanics ■ properties of (i) Stationary state for every expectation value is constant in time (ii) Definite total energy Classical mechanics : total energy is Hamiltonian Quantum mechanics : corresponding Hamiltonian operator

Quantum Mechanics thus equation (2.2)

Quantum Mechanics (iii) Linear combination of separable solution ► 2.2 Infinite square well( one dimensional box) and boundary conditions:

Quantum Mechanics deduce: outside the potential well, probability is zero for finding the particle inside the well V = 0 thus

Quantum Mechanics normalize

Quantum Mechanics ■ first three states and probability density of infinite square well

Quantum Mechanics ► 2.3 Harmonic oscillator Classical treatment : solution potential energy V is related to F : Quantum treatment :

Quantum Mechanics solve equation (2.3) by use ladder operator rewrite equation (2.3) by ladder operator :

Quantum Mechanics compare equation(2.3) similarly discussion (i)

Quantum Mechanics and (ii) and

Quantum Mechanics and (iii) there must exist a min state with and from

Quantum Mechanics and so the ladder of stationary states can illustrate :

Quantum Mechanics and ► 2.4 Delta-function potential Energy E Consider potential then

Quantum Mechanics (i) bound state : E < 0 similarly use boundary condition : find k : from

Quantum Mechanics and soand normalize

Quantum Mechanics (ii) scattering state : E > 0 and

Quantum Mechanics for wave coming in from left ( D = 0 ), equation (2.5),(2.6) rewrite is incident wave is reflected wave is transmitted wave R is reflection coefficientT is transmission coefficient

Quantum Mechanics ► 2.5 Free particle

Quantum Mechanics Fourier transform wave packet moves along at group velocity and phase velocity so wave packet

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