Physical Chemistry 2nd Edition Chapter 15 Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition Thomas Engel, Philip Reid

Objectives Using the postulates to understand the particle in the box (1-D, 2-D and 3-D)

Outline The Free Particle The Particle in a One-Dimensional Box Two- and Three-Dimensional Boxes Using the Postulates to Understand the Particle in the Box and Vice Versa

15.1 The Free Particle For free particle in a one-dimensional space on which no forces are acting, the Schrödinger equation is is a function that can be differentiated twice to return to the same function where

15.1 The Free Particle If x is restricted to the interval then the probability of finding the particle in an interval of length dx can be calculated.

15.2 The Particle in a One-Dimensional Box 15.1 The Classical Particle in a Box When consider particle confined to a box in 1-D, the potential is

15.2 The Particle in a One-Dimensional Box Consider the boundary condition satisfying 1-D, The acceptable wave functions must have the form of Thus the normalized eigenfunctions are

15.2 The Particle in a One-Dimensional Box 15.2 Energy Levels for the Particle in a Box 15.3 Probability of Finding the Particle in a Given Interval

Example 15.1 From the formula given for the energy levels for the particle in the box, for n = 1, 2, 3, 4… , we can see that the spacing between adjacent levels increases with n. This appears to indicate that the energy spectrum does not become continuous for large n, which must be the case for the quantum mechanical result to be identical to the classical result in the high-energy limit.

Example 15.1 A better way to look at the spacing between levels is to form the ratio . By forming this ratio, we see that becomes a smaller fraction of the energy as . This shows that the energy spectrum becomes continuous for large n.

Solution We have, which approaches zero as . Both the level spacing and the energy increase with n, but the energy increases faster (as n2), making the energy spectrum appear to be continuous as n→∞

15.3 Two- and Three-Dimensional Boxes 1-D box is useful model system as it allows focus to be on quantum mechanics instead of mathematics. For 3-D box, the potential energy is Inside the box, the Schrödinger equation can be written as

15.3 Two- and Three-Dimensional Boxes The total energy eigenfunctions have the form And the total energy has the form 15.4 Eigenfunctions for the Two- Dimensional Box

15.4 Using the Postulates to Understand the Particle in the Box and Vice Versa The state of a quantum mechanical system is completely specified by a wave function . The probability that a particle will be found at time t in a spatial interval of width dx centered at x0 is given by . This postulate states that all information obtained about the system is contained in the wave function.

Example 15.2 Consider the function a. Is an acceptable wave function for the particle in the box? b. Is an eigenfunction of the total energy operator, ? c. Is normalized?

Solution a. If is to be an acceptable wave function, it must satisfy the boundary conditions =0 at x=0 and x=a. The first and second derivatives of must also be well-behaved functions between x=0 and x=a. This is the case for . We conclude that is an acceptable wave function for the particle in the box.

Solution b. Although may be an acceptable wave function, it need not be an eigenfunction of a given operator. To see if is an eigenfunction of the total energy operator, the operator is applied to the function: The result of this operation is not multiplied by a constant. Therefore, is not an eigenfunction of the total energy operator.

Solution c. To see if is normalized, the following integral is evaluated:

Solution Using the standard integral and recognizing that the third

Solution Therefore, is not normalized, but the function is normalized for the condition that Note that a superposition wave function has a more complicated dependence on time than does an eigenfunction of the total energy operator.

Solution For instance, for the wave function under consideration is given by This wave function cannot be written as a product of a function of x and a function of t. Therefore, it is not a standing wave and does not describe a state whose properties are, in general, independent of time.

15.5 Acceptable Wave Functions for the Particle in a Box 15.4 Using the Postulates to Understand the Particle in the Box and Vice Versa 15.5 Acceptable Wave Functions for the Particle in a Box

Example 15.3 What is the probability, P, of finding the particle in the central third of the box if it is in its ground state?

Solution For the ground state, . From the postulate, P is the sum of all the probabilities of finding the particle in intervals of width dx within the central third of the box. This probability is given by the integral

Solution Solving this integral, Although we cannot predict the outcome of a single measurement, we can predict that for 60.9% of a large number of individual measurements, the particle is found in the central third of the box.

15.4 Using the Postulates to Understand the Particle in the Box and Vice Versa In any single measurement of the observable that corresponds to the operator , the only values that will ever be measured are the eigenvalues of that operator.

If the system is in a state described by the wave 15.4 Using the Postulates to Understand the Particle in the Box and Vice Versa Postulate 4 If the system is in a state described by the wave function , and the value of the observable a is measured once each on many identically prepared systems, the average value of all of these measurements is given by

15.4 Using the Postulates to Understand the Particle in the Box and Vice Versa 15.6 Expectation Values for E, p, and x for a Superposition Wave Function

Example 15.4 Assume that a particle is confined to a box of length a, and that the system wave function is a. Is this state an eigenfunction of the position operator? b. Calculate the average value of the position that would be obtained for a large number of measurements. Explain your result.

Example 15.4 a. The position operator . Because , where c is a constant, the wave function is not an eigenfunction of the position operator.

Example 15.4 b. The expectation value is calculated using the fourth postulate: Using the standard integral

Example 15.4 We have The average position is midway in the box. This is exactly what we would expect, because the particle is equally likely to be in each half of the box.