量子力学教程《一维量子力学》.pdf

Lecture 2 Quantum mechanics in one dimension Quantum mechanics in one dimension Schr¨odinger equation for non-relativistic quantum particle: ˆ i (r, t) = H(r, t) t 2 2 ˆ where H = + V(r) denotes quantum Hamiltonian. 2m To acquire intuition into general properties, we will review some simple and familiar(?) applications to one-dimensional systems. Divide consideration between potentials, V(x ), which leave particle free (i.e. unbound), and those that bind particle. Quantum mechanics in 1d: Outline 1 Unbound states Free particle Potential step Potential barrier Rectangular potential well 2 Bound states Rectangular potential well (continued) -function potential 3 Beyond local potentials Kronig-Penney model of a crystal Anderson localization Unbound particles: free particle 2 2 i (x , t) = x (x , t) t 2m For V = 0 Schr¨odinger equation describes travelling waves. 2 2 k (x , t) = A e i (kx t) , E (k) = (k) = 2m where k = 2 with the wavelength; momentum p = k = h .