Section 2-7
Section 2-7 The Particle in a Three-Dimensional Box: Separation of Variables
2-7 The Particle in a Three-DimensionalBox: Separation of Variables
Let us now consider the three-dimensional analog of the square well of Section 2-1.
This would be a three-dimensional box with zero potential inside and infinite potential outside. As before, the particle has no probability for penetrating beyond the box.
Therefore, the Schr¨odinger equation is just
−h2
∂2 + ∂2 + ∂2 ψ =Eψ
(2-53)
8π 2m
∂x2
∂y2
∂z2
and ψ vanishes at the box edges.
The hamiltonian operator on the left side of Eq. (2-53) can be written as a sum of operators, one in each variable (e.g., Hx = (−h2/8π2m)∂2/∂x2). Let us assume for the moment that ψ can be written as a product of three functions, each one being a function of a different variable, x, y, or z (i.e., ψ = X(x)Y (y)Z(z)). If we can show that such a ψ satisfies Eq. (2-53), we will have a much simpler problem to solve. Using this assumption, Eq. (2-53) becomes (Hx + Hy + Hz)XY Z = EXY Z
(2-54)
This can be expanded and then divided through by XY Z to obtain HxXY Z
H + yXY Z + HzXY Z = E (a constant)
(2-55)
XY Z
XY Z
XY Z Now, since Hx, for example, operates only on functions of x, but not y or z, we can carry out some limited cancellation. Those functions that are not operated on in a numerator can be canceled against the denominator. Those that are operated on cannot be canceled since these are differential operators [e.g., in (1/x)dx2/dx it is not permissible to cancel 1/x against x2 before differentiating: (1/x)dx2/dx = dx/dx].
Such cancellation gives
HxX
H + yY + HzZ = E (a constant)
(2-56)
X
Y
Z
Now, suppose the particle is moving in the box parallel to the x axis so that the variables y and z are not changing. Then, of course, the functions Y and Z are also not changing, so HyY/Y and HzZ/Z are both constant. Only HxX/X can vary—but does it vary?
Not according to Eq. (2-56), which reduces under these conditions to HxX +constant+constant=E (a constant)
(2-57)
X
Therefore, even though the particle is moving in the x direction, HxX/X must also be a constant, which we shall call Ex. Similar reasoning leads to analogous constants Ey and Ez. Furthermore, the behavior of HxX/X must really be independent of whether the particle is moving parallel to the y and z axes. Even if y and z do change, they do not appear in the quantity HxX/X anyway. Thus we may write, without restriction,
HxX
H
=
y Y
HzZ
Ex, = Ey, = Ez
(2-58)
X
Y
Z
Chapter 2 Quantum Mechanics of Some Simple Systems and, from Eq. (2-56), Ex + Ey + Ez = E
(2-59)
Our original equation in three variables has been separated into three equations, one in each variable. The first of these equations may be rewritten HxX = ExX
(2-60)
which is just the Schr¨odinger equation for the particle in the one-dimensional square well, which we have already solved. For a rectangular box with Lx = Ly = Lz we have the general solution
ψ = XY Z = 2/Lx sin (nxπx/Lx) 2/Ly sin nyπy/Ly 2/Lz sin (nzπz/Lz)
(2-61)
and
E = Ex + Ey + Ez = h2/8m
n2
+
+ x /L2
x
n2y/L2y
n2z/L2z
(2-62)
For a cubical box, Lx = Ly = Lz = L, and the energy expression simplifies to
E = h2/8mL2 n2 + +
x
n2y
n2z
(2-63)
The lowest energy occurs when nx = ny = nz = 1, and so E1(1) = 3h2/8mL2
(2-64)
Thus, the cubical box has three times the zero point energy of the corresponding onedimensional well, one-third coming from each independent coordinate for motion (i.e., “degree of freedom”). The one in parentheses indicates that this level is nondegenerate.
The next level is produced when one of the quantum numbers n has a value of two while the others have values of one. There are three independent ways of doing this; therefore, the second level is triply degenerate, and E2(3) = 6h2/8mL2. Proceeding, E3(3) = 9h2/8mL2, E4(3) = 11h2/8mL2, E5(1) = 12h2/8mL2, E6(6) = 14h2/8mL2, etc. Apparently, the energy level scheme and degeneracies of these levels do not proceed in the regular manner which is found in the one-dimensional cases we have studied.
EXAMPLE 2-8 Verify that E6 is six-fold degenerate.
SOLUTION E6 = 14h2/8mL2, so n2x + n2y + n2z = 14. There is only one combination of integers that satisfies this relation, namely 1, 2, and 3. So we simply need to deduce how many unique ways we can assign these integers to nx , ny , and nz. There are three ways to assign 1. For each of these three choices, there remain but two ways to assign 2, and then there is only one way to assign 3. So the number of unique ways is 3 × 2 × 1 = 6. (Or one can simply write down all of the possibilities and observe that there are six of them.)
We shall now briefly consider what probability distributions for the particle are predicted by these solutions. The lowest-energy solution has its largest value at the box center where all three sine functions are simultaneously largest. The particle
Section 2-7 The Particle in a Three-Dimensional Box: Separation of VariablesFigure 2-16
Sketches of particle probability distributions for a particle in a cubical box.
(a) nx = ny = nz = 1. (b) nx = 2, ny = nz = 1. (c) nx = ny = nz = 2.
distribution is sketched in Fig. 2-16a. The second level may be produced by setting nx = 2, and ny = nz = 1. Then there will be a nodal plane running through the box perpendicular to the x axis, producing the split distribution shown in Fig. 2-16b. Since there are three ways this node can be oriented to produce distinct but energetically equal distributions, this energy level is triply degenerate. The particle distribution for the state where nx = ny = nz = 2 is sketched in Fig. 2-16c. It is apparent that, in the high energy limit, the particle distribution becomes spread out uniformly throughout the box in accord with the classical prediction.
The separation of variables technique which we have used to convert our three dimensional problem into three independent one-dimensional problems will recur in other quantum-chemical applications. Reviewing the procedure makes it apparent that this technique will work whenever the hamiltonian operator can be cleanly broken into parts dependent on completely different coordinates. This is always possible for the kinetic energy operator in cartesian coordinates. However, the potential energy operator often prevents separation of variables in physical systems of interest.
It is useful to state the general results of separation of variables. Suppose we have a hamiltonian operator, with associated eigenfunctions and eigenvalues: H ψi = Eiψi
(2-65)
Suppose this hamiltonian can be separated, for example, H (α, β) = Hα (α) + Hβ (β)
(2-66)
2-7 The Particle in a Three-DimensionalBox: Separation of Variables
Let us now consider the three-dimensional analog of the square well of Section 2-1.
This would be a three-dimensional box with zero potential inside and infinite potential outside. As before, the particle has no probability for penetrating beyond the box.
Therefore, the Schr¨odinger equation is just
−h2
∂2 + ∂2 + ∂2 ψ =Eψ
(2-53)
8π 2m
∂x2
∂y2
∂z2
and ψ vanishes at the box edges.
The hamiltonian operator on the left side of Eq. (2-53) can be written as a sum of operators, one in each variable (e.g., Hx = (−h2/8π2m)∂2/∂x2). Let us assume for the moment that ψ can be written as a product of three functions, each one being a function of a different variable, x, y, or z (i.e., ψ = X(x)Y (y)Z(z)). If we can show that such a ψ satisfies Eq. (2-53), we will have a much simpler problem to solve. Using this assumption, Eq. (2-53) becomes (Hx + Hy + Hz)XY Z = EXY Z
(2-54)
This can be expanded and then divided through by XY Z to obtain HxXY Z
H + yXY Z + HzXY Z = E (a constant)
(2-55)
XY Z
XY Z
XY Z Now, since Hx, for example, operates only on functions of x, but not y or z, we can carry out some limited cancellation. Those functions that are not operated on in a numerator can be canceled against the denominator. Those that are operated on cannot be canceled since these are differential operators [e.g., in (1/x)dx2/dx it is not permissible to cancel 1/x against x2 before differentiating: (1/x)dx2/dx = dx/dx].
Such cancellation gives
HxX
H + yY + HzZ = E (a constant)
(2-56)
X
Y
Z
Now, suppose the particle is moving in the box parallel to the x axis so that the variables y and z are not changing. Then, of course, the functions Y and Z are also not changing, so HyY/Y and HzZ/Z are both constant. Only HxX/X can vary—but does it vary?
Not according to Eq. (2-56), which reduces under these conditions to HxX +constant+constant=E (a constant)
(2-57)
X
Therefore, even though the particle is moving in the x direction, HxX/X must also be a constant, which we shall call Ex. Similar reasoning leads to analogous constants Ey and Ez. Furthermore, the behavior of HxX/X must really be independent of whether the particle is moving parallel to the y and z axes. Even if y and z do change, they do not appear in the quantity HxX/X anyway. Thus we may write, without restriction,
HxX
H
=
y Y
HzZ
Ex, = Ey, = Ez
(2-58)
X
Y
Z
Chapter 2 Quantum Mechanics of Some Simple Systems and, from Eq. (2-56), Ex + Ey + Ez = E
(2-59)
Our original equation in three variables has been separated into three equations, one in each variable. The first of these equations may be rewritten HxX = ExX
(2-60)
which is just the Schr¨odinger equation for the particle in the one-dimensional square well, which we have already solved. For a rectangular box with Lx = Ly = Lz we have the general solution
ψ = XY Z = 2/Lx sin (nxπx/Lx) 2/Ly sin nyπy/Ly 2/Lz sin (nzπz/Lz)
(2-61)
and
E = Ex + Ey + Ez = h2/8m
n2
+
+ x /L2
x
n2y/L2y
n2z/L2z
(2-62)
For a cubical box, Lx = Ly = Lz = L, and the energy expression simplifies to
E = h2/8mL2 n2 + +
x
n2y
n2z
(2-63)
The lowest energy occurs when nx = ny = nz = 1, and so E1(1) = 3h2/8mL2
(2-64)
Thus, the cubical box has three times the zero point energy of the corresponding onedimensional well, one-third coming from each independent coordinate for motion (i.e., “degree of freedom”). The one in parentheses indicates that this level is nondegenerate.
The next level is produced when one of the quantum numbers n has a value of two while the others have values of one. There are three independent ways of doing this; therefore, the second level is triply degenerate, and E2(3) = 6h2/8mL2. Proceeding, E3(3) = 9h2/8mL2, E4(3) = 11h2/8mL2, E5(1) = 12h2/8mL2, E6(6) = 14h2/8mL2, etc. Apparently, the energy level scheme and degeneracies of these levels do not proceed in the regular manner which is found in the one-dimensional cases we have studied.
EXAMPLE 2-8 Verify that E6 is six-fold degenerate.
SOLUTION E6 = 14h2/8mL2, so n2x + n2y + n2z = 14. There is only one combination of integers that satisfies this relation, namely 1, 2, and 3. So we simply need to deduce how many unique ways we can assign these integers to nx , ny , and nz. There are three ways to assign 1. For each of these three choices, there remain but two ways to assign 2, and then there is only one way to assign 3. So the number of unique ways is 3 × 2 × 1 = 6. (Or one can simply write down all of the possibilities and observe that there are six of them.)
We shall now briefly consider what probability distributions for the particle are predicted by these solutions. The lowest-energy solution has its largest value at the box center where all three sine functions are simultaneously largest. The particle
Section 2-7 The Particle in a Three-Dimensional Box: Separation of VariablesFigure 2-16
Sketches of particle probability distributions for a particle in a cubical box.
(a) nx = ny = nz = 1. (b) nx = 2, ny = nz = 1. (c) nx = ny = nz = 2.
distribution is sketched in Fig. 2-16a. The second level may be produced by setting nx = 2, and ny = nz = 1. Then there will be a nodal plane running through the box perpendicular to the x axis, producing the split distribution shown in Fig. 2-16b. Since there are three ways this node can be oriented to produce distinct but energetically equal distributions, this energy level is triply degenerate. The particle distribution for the state where nx = ny = nz = 2 is sketched in Fig. 2-16c. It is apparent that, in the high energy limit, the particle distribution becomes spread out uniformly throughout the box in accord with the classical prediction.
The separation of variables technique which we have used to convert our three dimensional problem into three independent one-dimensional problems will recur in other quantum-chemical applications. Reviewing the procedure makes it apparent that this technique will work whenever the hamiltonian operator can be cleanly broken into parts dependent on completely different coordinates. This is always possible for the kinetic energy operator in cartesian coordinates. However, the potential energy operator often prevents separation of variables in physical systems of interest.
It is useful to state the general results of separation of variables. Suppose we have a hamiltonian operator, with associated eigenfunctions and eigenvalues: H ψi = Eiψi
(2-65)
Suppose this hamiltonian can be separated, for example, H (α, β) = Hα (α) + Hβ (β)
(2-66)