# Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

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two or more identical nuclei. 2.8 Particle in a three-dimensional box A simple example of a three-dimensional system is a particle confined to a rectangular container with sides of lengths a, b, and c. Within the box there is no force acting on the particle, so that the potential V (r) is given by V (r) 0, 0 < x < a, 0 < y < b, 0 < z < c 1, x , 0, x . a; y , 0, y . b; z , 0, z . c The wave function \u142(r) outside the box vanishes because the potential is infinite there. Inside the box, the wave function obeys the Schro¨dinger equation (2.70) with the potential energy set equal to zero ÿ"2 2m @2\u142(r) @x2 @ 2\u142(r) @ y2 @ 2\u142(r) @z2 ! E\u142(r) (2:75) The standard procedure for solving a partial differential equation of this type is to assume that the function \u142(r) may be written as the product of three functions, one for each of the three variables \u142(r) \u142(x, y, z) X (x)Y (y)Z(z) (2:76) Thus, X (x) is a function only of the variable x, Y (y) only of y, and Z(z) only of z. Substitution of equation (2.76) into (2.75) and division by the product XYZ give ÿ"2 2mX d2 X dx2 ÿ" 2 2mY d2Y dy2 ÿ" 2 2mZ d2 Z dz2 E (2:77) The first term on the left-hand side of equation (2.77) depends only on the variable x, the second only on y, and the third only on z. No matter what the values of x, or y, or z, the sum of these three terms is always equal to the same constant E. The only way that this condition can be met is for each of the three terms to equal some constant, say Ex, Ey, and Ez, respectively. The partial differential equation (2.77) can then be separated into three equations, one for each variable d2 X dx2 2m "2 ExX 0, d 2Y dy2 2m "2 EyY 0, d 2 Z dz2 2m "2 EzZ 0 (2:78) where Ex Ey Ez E (2:79) 2.8 Particle in a three-dimensional box 61 Thus, the three-dimensional problem has been reduced to three one-dimen- sional problems. The differential equations (2.78) are identical in form to equation (2.34) and the boundary conditions are the same as before. Consequently, the solutions inside the box are given by equation (2.40) as X (x) 2 a r sin nxðx a , nx 1, 2, 3, . . . Y (y) 2 b r sin n yðy b , ny 1, 2, 3, . . . (2:80) Z(z) 2 c r sin nzðz c , nz 1, 2, 3, . . . and the constants Ex, Ey, Ez are given by equation (2.39) Ex n 2 x h 2 8ma2 , nx 1, 2, 3, . . . Ey n2y h 2 8mb2 , ny 1, 2, 3, . . . (2:81) Ez n 2 z h 2 8mc2 , nz 1, 2, 3, . . . The quantum numbers nx, ny, nz take on positive integer values independently of each other. Combining equations (2.76) and (2.80) gives the wave functions inside the three-dimensional box \u142nx,n y,nz(r) 8 v r sin nxðx a sin nyðy b sin nzðz c (2:82) where v abc is the volume of the box. The energy levels for the particle are obtained by substitution of equations (2.81) into (2.79) Enx,ny,nz h2 8m n2x a2 n 2 y b2 n 2 z c2 \ufffd \ufffd (2:83) Degeneracy of energy levels If the box is cubic, we have a b c and the energy levels become Enx,ny,nz h2 8ma2 (n2x n2y n2z) (2:84) The lowest or zero-point energy is E1,1,1 3h2=8ma2, which is three times the zero-point energy for a particle in a one-dimensional box of the same length. The second or next-highest value for the energy is obtained by setting one of 62 Schro¨dinger wave mechanics the integers nx, ny, nz equal to 2 and the remaining ones equal to unity. Thus, there are three ways of obtaining the value 6h2=8ma2, namely, E2,1,1, E1,2,1, and E1,1,2. Each of these three possibilities corresponds to a different wave function, respectively, \u1422,1,1(r), \u1421,2,1(r), and \u1421,1,2(r). An energy level that corresponds to more than one wave function is said to be degenerate. The second energy level in this case is threefold or triply degenerate. The zero- point energy level is non-degenerate. The energies and degeneracies for the first six energy levels are listed in Table 2.1. The degeneracies of the energy levels in this example are the result of symmetry in the lengths of the sides of the box. If, instead of the box being cubic, the lengths of b and c in terms of a were b a=2, c a=3, then the values of the energy levels and their degeneracies are different, as shown in Table 2.2 for the lowest eight levels. Degeneracy is discussed in more detail in Chapter 3. Table 2.1. Energy levels for a particle in a three- dimensional box with a b c Energy Degeneracy Values of nx, ny, nz 3(h2/8ma2) 1 1,1,1 6(h2/8ma2) 3 2,1,1 1,2,1 1,1,2 9(h2/8ma2) 3 2,2,1 2,1,2 1,2,2 11(h2/8ma2) 3 3,1,1 1,3,1 1,1,3 12(h2/8ma2) 1 2,2,2 14(h2/8ma2) 6 3,2,1 3,1,2 2,3,1 2,1,3 1,3,2 1,2,3 Table 2.2. Energy levels for a particle in a three- dimensional box with b a=2, c a=3 Energy Degeneracy Values of nx, ny, nz 14(h2/8ma2) 1 1,1,1 17(h2/8ma2) 1 2,1,1 22(h2/8ma2) 1 3,1,1 26(h2/8ma2) 1 1,2,1 29(h2/8ma2) 2 2,2,1 4,1,1 34(h2/8ma2) 1 3,2,1 38(h2/8ma2) 1 5,1,1 41(h2/8ma2) 2 1,1,2 4,2,1 2.8 Particle in a three-dimensional box 63 Problems 2.1 Consider a particle in a one-dimensional box of length a and in quantum state n. What is the probability that the particle is in the left quarter of the box (0 < x < a=4)? For which state n is the probability a maximum? What is the probability that the particle is in the left half of the box (0 < x < a=2)? 2.2 Consider a particle of mass m in a one-dimensional potential such that V (x) 0, ÿa=2 < x < a=2 1, x ,ÿa=2, x . a=2 Solve the time-independent Schro¨dinger equation for this particle to obtain the energy levels and the normalized wave functions. (Note that the boundary conditions are different from those in Section 2.5.) 2.3 Consider a particle of mass m confined to move on a circle of radius a. Express the Hamiltonian operator in plane polar coordinates and then determine the energy levels and wave functions. 2.4 Consider a particle of mass m and energy E approaching from the left a potential barrier of height V0, as shown in Figure 2.3 and discussed in Section 2.6. However, suppose now that E is greater than V0 (E . V0). Obtain expressions for the reflection and transmission coefficients for this case. Show that T equals unity when E ÿ V0 n2ð2"2=2ma2 for n 1, 2, . . . Show that between these periodic maxima T has minima which lie progressively closer to unity as E increases. 2.5 Find the expression for the transmission coefficient T for Problem 2.4 when the energy E of the particle is equal to the potential barrier height V0. 64 Schro¨dinger wave mechanics 3 General principles of quantum theory 3.1 Linear operators The wave mechanics discussed in Chapter 2 is a linear theory. In order to develop the theory in a more formal manner, we need to discuss the properties of linear operators. An operator A^ is a mathematical entity that transforms a function \u142 into another function ö ö A^\u142 (3:1) Throughout this book a circumflex is used to denote operators. For example, multiplying the function \u142(x) by the variable x to give a new function ö(x) may be regarded as operating on the function \u142(x) with the operator x^, where x^ means multiply by x: ö(x) x^\u142(x) x\u142(x). Generally, when the operation is simple multiplication, the circumflex on the operator is omitted. The operator D^x, defined as d=dx, acting on \u142(x) gives the first derivative of \u142(x) with respect to x, so that in this case ö D^x\u142 d\u142 dx The operator A^ may involve a more complex procedure, such as taking the integral of \u142 with respect to x either implicitly or between a pair of limits. The operator A^ is linear if