Dr. Jie ZouPHY Chapter 41 Quantum Mechanics (Cont.)

Dr. Jie ZouPHY Outline The Schrödinger equation The particle in a box revisited

Dr. Jie ZouPHY The Schrödinger equation The time-independent Schrödinger equation for a particle of mass m confined to moving along the x axis is E: a constant = the total energy of the system = K + U. U = U(x): the potential energy of the system (the particle and its environment). Approach of quantum mechanics: Given U(x), solve the Schrödinger equation for  Apply the appropriate boundary conditions to  and find the allowed wave functions  and energies E Restrictions on  (x): (1)  (x) must be continuous; (2)  (x)  0 when x . (3)  (x) must be single-valued, and (4) d  /dx must be continuous for all finite values of U(x).

Dr. Jie ZouPHY The particle in a box revisited System under consideration: Particle in a one- dimensional box of length L. Potential energy function U(x) is given: Schrödinger equation for 0<x<L: Solution to the above equation:  (x) = A sin(kx) + B cos(kx) Apply boundary conditions:  (0) = 0 and  (L) = 0 and normalization conditions: Find allowed wave functions and energies:  n (x) = (2/L) 1/2 sin(n  x/L) and E n = (h 2 /8mL 2 ) n 2, n = 1,2,3… Potential-energy diagram for an infinite square potential well

Dr. Jie ZouPHY (a)and (b): The first three allowed states for a particle confined to a one- dimensional box. (c): Energy-level diagram. Ground state: n = 1 and E 1 = h 2 /8mL 2. Excited states: E n = n 2 E 1, n = 2,3,4… (c)