PHY 1371Dr. Jie Zou1 Chapter 41 Quantum Mechanics (cont.)

PHY 1371Dr. Jie Zou2 Outline Probability density Expectation value A particle in a box: an infinite square potential well

PHY 1371Dr. Jie Zou3 Probability density Probability density p: The probability per unit volume that the particle will be found within an infinitesimal volume containing the point (x, y, z). p = |  | 2. In one-dimensional systems, the probability that a particle will be found in the infinitesimal interval dx around the point x is P=|  | 2 dx. The probability of finding the particle in the interval a  x  b is Normalization condition:

PHY 1371Dr. Jie Zou4 Example: Problem #49 A particle is described by the wave function (a) Find and sketch the probability density. (b) Find the probability that the particle will be at any point where x < 0. (c) Show that  is normalized, and then find the probability that the particle will be found between x = 0 and x =a.

PHY 1371Dr. Jie Zou5 Expectation value of a physical quantity All measurable quantities of a particle can be derived from a knowledge of the wave function . For example, given the wave function , it is possible to calculate the average position x of a particle, after many experimental trials: Expectation value of a physical quantity x: Expectation value of a function f(x):

PHY 1371Dr. Jie Zou6 A particle in a box A particle confined in an infinite potential well: Wave function is subject to boundary conditions:  (0) = 0 and  (L) = 0. sin (kL) = 0, k = n  /L, n = 1, 2, 3,.. = 2L/n, and p =nh/2L, n = 1,2,3,.. E n = (h 2 /8mL 2 )n 2, n = 1,2,3…. Quantization of energy  n (x) = A n sin(n  x/L), n = 1,2,3…. Stationary states Normalization: A n = (2/L) 1/2. 0 L V(x)  

PHY 1371Dr. Jie Zou7

PHY 1371Dr. Jie Zou8 Example 41.2 A bound electron: An electron is confined between two impenetrable walls nm apart. Determine the energy levels for the states n = 1,2,3.

PHY 1371Dr. Jie Zou9 Homework Ch. 41, P. 1346, Problems: # 17, 18, 48.