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Transcript

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) Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Outline 1. 2. 3. 4. 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 Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 15.1 The Free Particle • For free particle in a one-dimensional space on which no forces are acting, the Schrödinger equation is d 2 x E x 2 dx • x is a function that can be differentiated twice to return to the same function x A e 2 mE / h x A e ikx 2 where k x A e i 2 mE / h x A e ikx i 2 2 Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 2mE h2 15.1 The Free Particle • If x is restricted to the interval L x L then the probability of finding the particle in an interval of length dx can be calculated. Px dx * x x dx L * x x dx L Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd A A e ikxe ikx dx L A A e ikxe ikx dx L dx 2L 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 1D, the potential is V x 0, for a x 0 V x , for x a, x 0 Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 15.2 The Particle in a One-Dimensional Box • Consider the boundary condition satisfying 1-D, 0 a 0 • The acceptable wave functions must have the form of nx , for n 1,2,3,4... a n A sin • Thus the normalized eigenfunctions are n x Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 2 nx sin a a 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 Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Example 15.1 From the formula given for the energy levels for the particle in the box, En h 2 n 2 / 8ma2 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. Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Example 15.1 A better way to look at the spacing between levels is to form the ratio En1 En / En . By forming this ratio, we see that E / E becomes a smaller fraction of the energy as n . This shows that the energy spectrum becomes continuous for large n. Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Solution We have, En 1 En En h 2 n 12 n 2 / 8 ma 2 h 2 n 2 / 8 ma 2 2n 1 n2 which approaches zero as n . 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→∞ Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 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 V x, y, z 0 for 0 x a; 0 y b; 0 z c otherwise • Inside the box, the Schrödinger equation can be written as h2 2 2 2 2 2 2 x, y, z E x, y, z 2m x y z Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 15.3 Two- and Three-Dimensional Boxes • The total energy eigenfunctions have the form n yy nxx n z nx n y nz x, y, z N sin sin sin z a b c • And the total energy has the form 2 h 2 nx2 n y nz2 E 2 2 2 8m a b c • 15.4 Eigenfunctions for the TwoDimensional Box Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 15.4 Using the Postulates to Understand the Particle in the Box and Vice Versa Postulate 1 The state of a quantum mechanical system is completely specified by a wave function ( x, t ) . The probability that a particle will be found at time t in a spatial interval of width dx centered at x0 is given by * ( x0 , t ) x0 , t dx . • This postulate states that all information obtained about the system is contained in the wave function. Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Example 15.2 Consider the function x c sin x / a d sin 2x / a a. Is (x) an acceptable wave function for the particle in the box? b. Is (x) an eigenfunction of the total energy operator, ? c. Is (x) normalized? Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Solution a. If (x) is to be an acceptable wave function, it must satisfy the boundary conditions (x) =0 at x=0 and x=a. The first and second derivatives of (x) be well-behaved functions between x=0 must also and x=a. This is the case for (x) . We conclude that (x) x c sin x / a d sin 2x / a is an acceptable wave function for the particle in the box. Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Solution b. Although x c sin x / a d sin 2x / a may be an acceptable wave function, it need not be an eigenfunction of a given operator. To see if (x) is (x) an eigenfunction of the total energy operator, the operator is applied to the function: 2 (x ) 2 2 h2 d 2 x h x 2x x c sin a d sin c sin 4d sin 2 2 2m dx a 2ma a a The result of this operation is not (x) multiplied by a constant. Therefore, (x) is not an eigenfunction of the total energy operator. Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Solution c. To see if (x) is normalized, the following integral is evaluated: 2 x (x ) 2x 0 c sin a d sin a dx a 2(x ) x d * d sin 2 2x cd * c * d sin x sin 2x dx c * c sin a a a a 0 a 2 2 x 2x x 2x c sin 2 dx d sin 2 dx cd * c * d sin sin dx a a a a 0 0 0 a a Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd a Solution Using the standard integral sin 2bydy y / 2 1 / 4b sin by and recognizing that the third 2 2 2(x 2 )x 2 2x c sin d sin dx 0 a a asin 2 sin 0 asin 4 sin 0 a 2 2 a 2 a 2 c (x) d c d 2 2 4 8 2 a Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Solution Therefore, (x) is not normalized, but the function (x) 2 x 2 c sin d sin a a a is normalized for the condition that c 2 d 2 1 (x) Note that a superposition wave function has a more complicated dependence on time than does an eigenfunction of the total energy operator. Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Solution For instance, (x) for the wave function under consideration is given by 2 iE / h x iE (x ) x, t ce sin ce a a 1 (x) 2 /h 2x sin x f t a 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. Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 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 Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 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? Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Solution For the ground state, 1 x 2 / a sin x / a . 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 2 P a x a / 3sin a dx 2a / 3 Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 2 Solution Solving this integral, 2 a a P a 6 4 4 2 sin sin 0.609 3 3 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. Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 15.4 Using the Postulates to Understand the Particle in the Box and Vice Versa Postulate 3 In any single measurement of the observable that corresponds to the operator Â , the only values that will ever be measured are the eigenvalues of that operator. Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 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 x, t , 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 a ˆ x, t dx * x , t A * x, t x, t dx Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 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 Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Example 15.4 Assume that a particle is confined to a box of length a, and that the system wave function is x 2 / a sin x / a 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. Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Example 15.4 a. The position operator xˆ x . Because , x x 2 / a sin x / a c x where c is a constant, the wave function is not an eigenfunction of the position operator. Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Example 15.4 b. The expectation value is calculated using the fourth postulate: x 2 2 x 2 x x sin x sin dx x sin dx a 0 a a 0 a a a a Using the standard integral a 2 x 2 cos 2bx x sin 2bx 0 x(sin bx) dx 4 8b 2 4b Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Example 15.4 We have a 2x 2x x sin 2 cos a2 a 2 x 2 a 2 a 2 a a x 2 0 2 2 a 4 a 4 8 8 2 4 8 a a 0 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. Chapter 15: Using Quantum Mechanics on Simple Systems Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd