Presentation on theme: "Physical Chemistry 2nd Edition"— Presentation transcript:
1 Physical Chemistry 2nd Edition Chapter 15Using Quantum Mechanics on Simple SystemsPhysical Chemistry 2nd EditionThomas Engel, Philip Reid
2 ObjectivesUsing the postulates to understand the particle in the box (1-D, 2-D and 3-D)
3 Outline The Free Particle The Particle in a One-Dimensional Box Two- and Three-Dimensional BoxesUsing the Postulates to Understand the Particle in the Box and Vice Versa
4 15.1 The Free ParticleFor free particle in a one-dimensional space on which no forces are acting, the Schrödinger equation isis a function that can be differentiated twice to return to the same functionwhere
5 15.1 The Free ParticleIf x is restricted to the interval then the probability of finding the particle in an interval of length dx can be calculated.
6 15.2 The Particle in a One-Dimensional Box 15.1 The Classical Particle in a BoxWhen consider particle confined to a box in 1-D, the potential is
7 15.2 The Particle in a One-Dimensional Box Consider the boundary condition satisfying 1-D,The acceptable wave functions must have the form ofThus the normalized eigenfunctions are
8 15.2 The Particle in a One-Dimensional Box 15.2 Energy Levels for the Particle in a Box15.3 Probability of Finding the Particle in a Given Interval
9 Example 15.1From 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.
10 Example 15.1A 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 asThis shows that the energy spectrum becomes continuous for large n.
11 SolutionWe 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→∞
12 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 isInside the box, the Schrödinger equation can be written as
13 15.3 Two- and Three-Dimensional Boxes The total energy eigenfunctions have the formAnd the total energy has the form15.4 Eigenfunctions for the Two- Dimensional Box
14 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 byThis postulate states that all information obtained about the system is contained in the wave function.
15 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 energyoperator, ?c. Is normalized?
16 Solutiona. 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 thatis an acceptable wave function for the particle in the box.
17 Solutionb. 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.
18 Solutionc. To see if is normalized, the following integral is evaluated:
19 SolutionUsing the standard integral and recognizing that the third
20 Solution Therefore, is not normalized, but the function is normalized for the condition thatNote that a superposition wave function has a morecomplicated dependence on time than does aneigenfunction of the total energy operator.
21 Solution For instance, for the wave function under consideration is given byThis wave function cannot be written as a product of afunction of x and a function of t. Therefore, it is not astanding wave and does not describe a statewhose properties are, in general, independent of time.
22 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 Versa15.5 Acceptable Wave Functions for the Particle in a Box
23 Example 15.3What is the probability, P, of finding the particle in the central third of the box if it is in its ground state?
24 SolutionFor 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
25 Solution Solving this integral, Although we cannot predict the outcome of a singlemeasurement, 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.
26 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.
27 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 VersaPostulate 4If the system is in a state described by the wavefunction , and the value of the observable a ismeasured once each on many identically preparedsystems, the average value of all of thesemeasurements is given by
28 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
29 Example 15.4Assume that a particle is confined to a box of length a, and that the system wave function isa. 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.
30 Example 15.4 a. The position operator . Because , where c is a constant, the wave function is not aneigenfunction of the position operator.
31 Example 15.4b. The expectation value is calculated using the fourth postulate:Using the standard integral
32 Example 15.4We haveThe average position is midway in the box. This isexactly what we would expect, because the particle is equally likely to be in each half of the box.