A special kind of variation function widely used in the study of molecules is the linear variation function. A linear variation function is a linear combination of $n$ linearly independent functions $f{1}, f{2}, \ldots, f_{n}$ :
\(
\begin{equation}
\phi=c{1} f{1}+c{2} f{2}+\cdots+c{n} f{n}=\sum{j=1}^{n} c{j} f_{j} \tag{8.40}
\end{equation}
\)
where $\phi$ is the trial variation function and the coefficients $c{j}$ are parameters to be determined by minimizing the variational integral. The functions $f{j}$ (which are called basis functions) must satisfy the boundary conditions of the problem. We shall restrict ourselves to real $\phi$ so that the $c{j}$ 's and $f{j}$ 's are all real. In (8.40), the functions $f_{j}$ are known functions.
We now apply the variation theorem (8.9). For the real linear variation function, we have
$\int \phi^{*} \phi d \tau=\int \sum{j=1}^{n} c{j} f{j} \sum{k=1}^{n} c{k} f{k} d \tau=\sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} \int f{j} f{k} d \tau \equiv \sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} S{j k}$
where we defined the overlap integral $S{j k}$ as
\(
\begin{equation}
S{j k} \equiv \int f{j}^{} f_{k} d \tau \tag{8.42}
\end{}
\)
Note that $S{j k}$ is not necessarily equal to $\delta{j k}$, since there is no reason to suppose that the functions $f_{j}$ are mutually orthogonal. They are not necessarily the eigenfunctions of any operator. The numerator in (8.9) is
\(
\begin{aligned}
\int \phi^{*} \hat{H} \phi d \tau & =\int \sum{j=1}^{n} c{j} f{j} \hat{H} \sum{k=1}^{n} c{k} f{k} d \tau \
& =\sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} \int f{j} \hat{H} f{k} d \tau \equiv \sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} H_{j k}
\end{aligned}
\)
where we defined $H_{j k}$ as
\(
\begin{equation}
H{j k} \equiv \int f{j}^{} \hat{H} f_{k} d \tau \tag{8.43}
\end{}
\)
The variational integral $W$ is
\(
\begin{align}
W \equiv \frac{\int \phi^{} \hat{H} \phi d \tau}{\int \phi^{} \phi d \tau} & =\frac{\sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} H{j k}}{\sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} S{j k}} \tag{8.44}\
W \sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} S{j k} & =\sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} H{j k} \tag{8.45}
\end{align}
\)
We now minimize $W$ so as to approach as closely as we can to $E{1}\left(W \geq E{1}\right)$. The variational integral $W$ is a function of the $n$ independent variables $c{1}, c{2}, \ldots, c_{n}$ :
\(
W=W\left(c{1}, c{2}, \ldots, c_{n}\right)
\)
A necessary condition for a minimum in a function $W$ of several variables is that its partial derivatives with respect to each of the variables must be zero at the minimum:
\(
\begin{equation}
\frac{\partial W}{\partial c_{i}}=0, \quad i=1,2, \ldots, n \tag{8.46}
\end{equation}
\)
We now differentiate (8.45) partially with respect to each $c_{i}$ to obtain $n$ equations:
\(
\begin{equation}
\frac{\partial W}{\partial c{i}} \sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} S{j k}+W \frac{\partial}{\partial c{i}} \sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} S{j k}=\frac{\partial}{\partial c{i}} \sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} H{j k}, \quad i=1,2, \ldots, n \tag{8.47}
\end{equation}
\)
Now
\(
\frac{\partial}{\partial c{i}} \sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} S{j k}=\sum{j=1}^{n} \sum{k=1}^{n}\left[\frac{\partial}{\partial c{i}}\left(c{j} c{k}\right)\right] S{j k}=\sum{j=1}^{n} \sum{k=1}^{n}\left(c{k} \frac{\partial c{j}}{\partial c{i}}+c{j} \frac{\partial c{k}}{\partial c{i}}\right) S_{j k}
\)
The $c_{j}$ 's are independent variables, and therefore
\(
\begin{gather}
\frac{\partial c{j}}{\partial c{i}}=0 \quad \text { if } j \neq i, \quad \frac{\partial c{j}}{\partial c{i}}=1 \quad \text { if } j=i \
\frac{\partial c{j}}{\partial c{i}}=\delta_{i j} \tag{8.48}
\end{gather}
\)
We then have
\(
\frac{\partial}{\partial c{i}} \sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} S{j k}=\sum{k=1}^{n} \sum{j=1}^{n} c{k} \delta{i j} S{j k}+\sum{j=1}^{n} \sum{k=1}^{n} c{j} \delta{i k} S{j k}=\sum{k=1}^{n} c{k} S{i k}+\sum{j=1}^{n} c{j} S{j i}
\)
where we evaluated one of the sums in each double summation using Eq. (7.32). Use of (7.4) gives
\(
\begin{equation}
S{j i}=S{i j}^{}=S_{i j} \tag{8.49}
\end{}
\)
where the last equality follows because we are dealing with real functions. Hence,
\(
\begin{equation}
\frac{\partial}{\partial c{i}} \sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} S{j k}=\sum{k=1}^{n} c{k} S{i k}+\sum{j=1}^{n} c{j} S{i j}=\sum{k=1}^{n} c{k} S{i k}+\sum{k=1}^{n} c{k} S{i k}=2 \sum{k=1}^{n} c{k} S_{i k} \tag{8.50}
\end{equation}
\)
where the fact that $j$ is a dummy variable was used.
By replacing $S{j k}$ by $H{j k}$ in each of these manipulations, we get
\(
\begin{equation}
\frac{\partial}{\partial c{i}} \sum{j=1}^{n} \sum{k=1}^{n} c{j} c{k} H{j k}=2 \sum{k=1}^{n} c{k} H_{i k} \tag{8.51}
\end{equation}
\)
This result depends on the fact that
\(
\begin{equation}
H{j i}=H{i j}^{}=H_{i j} \tag{8.52}
\end{}
\)
which is true because $\hat{H}$ is Hermitian, and $f{i}, f{j}$, and $\hat{H}$ are real.
Substitution of Eqs. (8.46), (8.50), and (8.51) into (8.47) gives
\(
\begin{array}{ll}
2 W \sum{k=1}^{n} c{k} S{i k}=2 \sum{k=1}^{n} c{k} H{i k}, & i=1,2, \ldots, n \
\sum{k=1}^{n}\left[\left(H{i k}-S{i k} W\right) c{k}\right]=0, & i=1,2, \ldots, n \tag{8.53}
\end{array}
\)
Equation (8.53) is a set of $n$ simultaneous, linear, homogeneous equations in the $n$ unknowns $c{1}, c{2}, \ldots, c_{n}$ [the coefficients in the linear variation function (8.40)]. For example, for $n=2$, (8.53) gives
\(
\begin{align}
& \left(H{11}-S{11} W\right) c{1}+\left(H{12}-S{12} W\right) c{2}=0 \
& \left(H{21}-S{21} W\right) c{1}+\left(H{22}-S{22} W\right) c{2}=0 \tag{8.54}
\end{align}
\)
For the general case of $n$ functions $f{1}, \ldots, f{n}$, (8.53) is
\(
\begin{align}
& \left(H{11}-S{11} W\right) c{1}+\left(H{12}-S{12} W\right) c{2}+\cdots+\left(H{1 n}-S{1 n} W\right) c{n}=0 \
& \left(H{21}-S{21} W\right) c{1}+\left(H{22}-S{22} W\right) c{2}+\cdots+\left(H{2 n}-S{2 n} W\right) c{n}=0 \tag{8.55}\
& \left(H{n 1}-S{n 1} W\right) c{1}+\left(H{n 2}-S{n 2} W\right) c{2}+\cdots+\left(H{n n}-S{n n} W\right) c_{n}=0
\end{align}
\)
From the theorem of Section 8.4, for there to be a solution to the linear homogeneous equations (8.55) besides the trivial solution $0=c{1}=c{2}=\cdots=c_{n}$ (which would make the variation function $\phi$ zero), the determinant of the coefficients must vanish. For $n=2$ we have
\(
\left|\begin{array}{ll}
H{11}-S{11} W & H{12}-S{12} W \tag{8.56}\
H{21}-S{21} W & H{22}-S{22} W
\end{array}\right|=0
\)
and for the general case
\(
\begin{align}
& \operatorname{det}\left(H{i j}-S{i j} W\right)=0 \tag{8.57}\
& \left|\begin{array}{cccc}
H{11}-S{11} W & H{12}-S{12} W & \cdots & H{1 n}-S{1 n} W \
H{21}-S{21} W & H{22}-S{22} W & \cdots & H{2 n}-S{2 n} W \
\cdot & \cdot & \cdots & \cdot \
\cdot & \cdot & \cdots & \cdot \
\cdot & \cdot & \cdots & \cdot \
H{n 1}-S{n 1} W & H{n 2}-S{n 2} W & \cdots & H{n n}-S{n n} W
\end{array}\right|=0 \tag{8.58}
\end{align}
\)
Expansion of the determinant in (8.58) gives an algebraic equation of degree $n$ in the unknown $W$. This algebraic equation has $n$ roots, which can be shown to be real. Arranging these roots in order of increasing value, we denote them as
\(
\begin{equation}
W{1} \leq W{2} \leq \cdots \leq W_{n} \tag{8.59}
\end{equation}
\)
If we number the bound states of the system in order of increasing energy, we have
\(
\begin{equation}
E{1} \leq E{2} \leq \cdots \leq E{n} \leq E{n+1} \leq \cdots \tag{8.60}
\end{equation}
\)
where the $E$ 's denote the true energies of various states. From the variation theorem, we know that $E{1} \leq W{1}$. Moreover, it can be proved that [J. K. L. MacDonald, Phys. Rev., 43, 830 (1933); R. H. Young, Int. J. Quantum Chem., 6, 596 (1972); see Prob. 8.40]
\(
\begin{equation}
E{1} \leq W{1}, \quad E{2} \leq W{2}, \quad E{3} \leq W{3}, \ldots, \quad E{n} \leq W{n} \tag{8.61}
\end{equation}
\)
Thus, the linear variation method provides upper bounds to the energies of the lowest $n$ bound states of the system. We use the roots $W{1}, W{2}, \ldots, W{n}$ as approximations to the energies of the lowest states. If approximations to the energies of more states are wanted, we add more functions $f{k}$ to the trial function $\phi$. The addition of more functions $f{k}$ can be shown to increase (or cause no change in) the accuracy of the previously calculated energies. If the functions $f{k}$ in $\phi=\sum{k} c{k} f_{k}$ form a complete set, then we will obtain the
true wave functions of the system. Unfortunately, we usually need an infinite number of functions to have a complete set.
Quantum chemists may use dozens, hundreds, thousands, or even millions of terms in linear variation functions so as to get accurate results for molecules. Obviously, a computer is essential for this work. The most efficient way to solve (8.58) (which is called the secular equation) and the associated linear equations (8.55) is by matrix methods (Section 8.6).
To obtain an approximation to the wave function of the ground state, we take the lowest root $W{1}$ of the secular equation and substitute it in the set of equations (8.55); we then solve this set of equations for the coefficients $c{1}^{(1)}, c{2}^{(1)}, \ldots, c{n}^{(1)}$, where the superscript ${ }^{(1)}$ was added to indicate that these coefficients correspond to $W{1}$. [As noted in the previous section, we can determine only the ratios of the coefficients. We solve for $c{2}^{(1)}, \ldots, c{n}^{(1)}$ in terms of $c{1}^{(1)}$, and then determine $c{1}^{(1)}$ by normalization.] Having found the $c{k}^{(1)}$ 's, we take $\phi{1}=\sum{k} c{k}^{(1)} f{k}$ as an approximate ground-state wave function. Use of higher roots of (8.58) in (8.55) gives approximations to excited-state wave functions. These approximate wave functions can be shown to be orthogonal (Prob. 8.40).
Solution of (8.58) and (8.55) is simplified by having as many of the integrals equal to zero as possible. We can make some of the off-diagonal $H{i j}$ 's vanish by choosing the functions $f{k}$ as eigenfunctions of some operator $\hat{A}$ that commutes with $\hat{H}$. If $f{i}$ and $f{j}$ correspond to different eigenvalues of $\hat{A}$, then $H{i j}$ vanishes (Theorem 6 of Section 7.4). If the functions $f{k}$ are orthonormal, the off-diagonal $S{i j}$ 's vanish $\left(S{i j}=\delta{i j}\right)$. If the initially chosen $f{k}$ 's are not orthogonal, we can use the Schmidt (or some other) procedure to find $n$ linear combinations of these $f_{k}$ 's that are orthogonal and then use the orthogonalized functions.
Equations (8.55) and (8.58) are also valid when the restriction that the variation function be real is removed (Prob. 8.39).
EXAMPLE
Add functions to the function $x(l-x)$ of the first example of Section 8.1 to form a linear variation function for the particle in a one-dimensional box of length $l$ and find approximate energies and wave functions for the lowest four states.
In the trial function $\phi=\sum{k=1}^{n} c{k} f{k}$, we take $f{1}=x(l-x)$. Since we want approximations to the lowest four states, $n$ must be at least 4. There are an infinite number of possible well-behaved functions that could be used for $f{2}, f{3}$, and $f{4}$. The function $x^{2}(l-x)^{2}$ obeys the boundary conditions of vanishing at $x=0$ and $x=l$ and leads to simple integrals, so we take $f{2}=x^{2}(l-x)^{2}$.
If the origin is placed at the center of the box, the potential energy (Fig. 2.1) is an even function, and, as noted in Section 8.2, the wave functions alternate between being even and odd functions (see also Fig. 2.3). (Throughout this example, the terms even and odd will refer to having the origin at the box's center.) The functions $f{1}=x(l-x)$ and $f{2}=x^{2}(l-x)^{2}$ are both even functions (see Prob. 8.34). If we were to take $\phi=c{1} x(l-x)+c{2} x^{2}(l-x)^{2}$, we would end up with upper bounds to the energies of the lowest two states with even wave functions (the $n=1$ and $n=3$ states) and would get approximate wave functions for these two states. Since we also want to approximate the $n=2$ and $n=4$ states, we shall add in two functions that are odd. An odd function must vanish at the origin [as noted after Eq. (4.50)], so we need functions that vanish at the box midpoint $x=\frac{1}{2} l$, as well as at $x=0$ and $l$. A simple function with these properties is $f{3}=x(l-x)\left(\frac{1}{2} l-x\right)$. To get $f{4}$, we shall multiply $f{2}$ by $\left(\frac{1}{2} l-x\right)$. Thus we take $\phi=\sum{k=1}^{4} c{k} f{k}$, with
\(
\begin{equation}
f{1}=x(l-x), f{2}=x^{2}(l-x)^{2}, f{3}=x(l-x)\left(\frac{1}{2} l-x\right), f{4}=x^{2}(l-x)^{2}\left(\frac{1}{2} l-x\right) \tag{8.62}
\end{equation}
\)
Note that $f{1}, f{2}, f{3}$, and $f{4}$ are linearly independent, as assumed in (8.40).
Because $f{1}$ and $f{2}$ are even, while $f{3}$ and $f{4}$ are odd, many integrals will vanish.
Thus
\(
\begin{equation}
S{13}=S{31}=0, \quad S{14}=S{41}=0, \quad S{23}=S{32}=0, \quad S{24}=S{42}=0 \tag{8.63}
\end{equation}
\)
because the integrand in each of these overlap integrals is an odd function with respect to the origin at the box center. The functions $f{1}, f{2}, f{3}, f{4}$ are eigenfunctions of the parity operator $\hat{\Pi}$ (Section 7.5) with the even functions $f{1}$ and $f{2}$ having parity eigenvalue +1 and $f{3}$ and $f{4}$ having eigenvalue -1 . The operator $\hat{\Pi}$ commutes with $\hat{H}$ (since $V$ is an even function), so by Theorem 6 of Section 7.4, $H{i j}$ vanishes if $f{i}$ is an odd function and $f_{j}$ is even, or vice versa. Thus
\(
\begin{equation}
H{13}=H{31}=0, \quad H{14}=H{41}=0, \quad H{23}=H{32}=0, \quad H{24}=H{42}=0 \tag{8.64}
\end{equation}
\)
From (8.63) and (8.64), the $n=4$ secular equation (8.58) becomes
\(
\left|\begin{array}{cccc}
H{11}-S{11} W & H{12}-S{12} W & 0 & 0 \tag{8.65}\
H{21}-S{21} W & H{22}-S{22} W & 0 & 0 \
0 & 0 & H{33}-S{33} W & H{34}-S{34} W \
0 & 0 & H{43}-S{43} W & H{44}-S{44} W
\end{array}\right|=0
\)
The secular determinant is in block-diagonal form and so is equal to the product of its blocks [Eq. (8.34)]:
\(
\left|\begin{array}{ll}
H{11}-S{11} W & H{12}-S{12} W \
H{21}-S{21} W & H{22}-S{22} W
\end{array}\right| \times\left|\begin{array}{ll}
H{33}-S{33} W & H{34}-S{34} W \
H{43}-S{43} W & H{44}-S{44} W
\end{array}\right|=0
\)
The four roots of this equation are found from the equations
\(
\begin{align}
& \left|\begin{array}{ll}
H{11}-S{11} W & H{12}-S{12} W \
H{21}-S{21} W & H{22}-S{22} W
\end{array}\right|=0 \tag{8.66}\
& \left|\begin{array}{ll}
H{33}-S{33} W & H{34}-S{34} W \
H{43}-S{43} W & H{44}-S{44} W
\end{array}\right|=0 \tag{8.67}
\end{align}
\)
Let the roots of (8.66) (which are approximations to the $n=1$ and $n=3$ energies) be $W{1}$ and $W{3}$ and let the roots of (8.67) be $W{2}$ and $W{4}$. After solving the secular equation for the $W$ 's, we substitute them one at a time into the set of equations (8.55) to find the coefficients $c{k}$ in the variation function. From the secular equation (8.65), the set of equations (8.55) with the root $W{1}$ is
\(
\begin{align}
\left.\begin{array}{c}
\left(H{11}-S{11} W{1}\right) c{1}^{(1)}+\left(H{12}-S{12} W{1}\right) c{2}^{(1)} \
\left(H{21}-S{21} W{1}\right) c{1}^{(1)}+\left(H{22}-S{22} W{1}\right) c{2}^{(1)} \
\
\left(H{33}-S{33} W{1}\right) c{3}^{(1)}+\left(H{34}-S{34} W{1}\right) c{4}^{(1)}=0
\end{array}\right} \tag{8.68a}\
\left.\begin{array}{r}
\left(H{43}-S{43} W{1}\right) c{3}^{(1)}+\left(H{44}-S{44} W{1}\right) c{4}^{(1)}=0
\end{array}\right}
\end{align}
\)
Because $W{1}$ is a root of (8.66), the set of equations (8.68a) has the determinant of its coefficients [which is the determinant in (8.66)] equal to zero. Hence (8.68a) has a nontrivial solution for $c{1}^{(1)}$ and $c{2}^{(1)}$. However, $W{1}$ is not a root of (8.67), so the determinant of the coefficients of the set of equations (8.68b) is nonzero. Hence, (8.68b) has only the trivial solution $c{3}^{(1)}=c{4}^{(1)}=0$. The trial function $\phi{1}$ corresponding to the root $W{1}$ thus has the form $\phi{1}=\sum{k=1}^{4} c{k}^{(1)} f{k}=c{1}^{(1)} f{1}+c{2}^{(1)} f{2}$. The same reasoning
shows that $\phi{3}$ is a linear combination of $f{1}$ and $f{2}$, while $\phi{2}$ and $\phi{4}$ are each linear combinations of $f{3}$ and $f_{4}$ :
\(
\begin{array}{ll}
\phi{1}=c{1}^{(1)} f{1}+c{2}^{(1)} f{2}, & \phi{3}=c{1}^{(3)} f{1}+c{2}^{(3)} f{2} \tag{8.69}\
\phi{2}=c{3}^{(2)} f{3}+c{4}^{(2)} f{4}, & \phi{4}=c{3}^{(4)} f{3}+c{4}^{(4)} f{4}
\end{array}
\)
The even wave functions $\psi{1}$ and $\psi{3}$ are approximated by linear combinations of the even functions $f{1}$ and $f{2}$; the odd functions $\psi{2}$ and $\psi{4}$ are approximated by linear combinations of the odd functions $f{3}$ and $f{4}$.
When the secular equation is in block-diagonal form, it factors into two or more smaller secular equations, and the set of simultaneous equations (8.55) breaks up into two or more smaller sets of equations.
We now must evaluate the $H{i j}$ and $S{i j}$ integrals so as to solve (8.66) and (8.67) for $W{1}, W{2}, W{3}$, and $W{4}$. We have
\(
\begin{aligned}
H{11}=\left\langle f{1}\right| \hat{H}\left|f{1}\right\rangle & =\int{0}^{l} x(l-x)\left(\frac{-\hbar^{2}}{2 m}\right) \frac{d^{2}}{d x^{2}}[x(l-x)] d x=\frac{\hbar^{2} l^{3}}{6 m} \
S{11} & =\left\langle f{1} \mid f{1}\right\rangle=\int{0}^{l} x^{2}(l-x)^{2} d x=\frac{l^{5}}{30}
\end{aligned}
\)
where Eqs. (8.12) and (8.13) were used. Evaluation of the remaining integrals using (8.62), (8.49), and (8.52) gives (Prob. 8.35)
\(
\begin{gathered}
H{12}=H{21}=\left\langle f{2}\right| \hat{H}\left|f{1}\right\rangle=\hbar^{2} l^{5} / 30 m, \quad H{22}=\hbar^{2} l^{7} / 105 m \
H{33}=\hbar^{2} l^{5} / 40 m, \quad H{44}=\hbar^{2} l^{9} / 1260 m, \quad H{34}=H{43}=\hbar^{2} l^{7} / 280 m \
S{12}=S{21}=\left\langle f{1} \mid f{2}\right\rangle=l^{7} / 140, \quad S{22}=l^{9} / 630 \
S{33}=l^{7} / 840, \quad S{44}=l^{11} / 27720, \quad S{34}=S{43}=l^{9} / 5040
\end{gathered}
\)
Equation (8.66) becomes
\(
\left|\begin{array}{ll}
\frac{\hbar^{2} l^{3}}{6 m}-\frac{l^{5}}{30} W & \frac{\hbar^{2} l^{5}}{30 m}-\frac{l^{7}}{140} W \tag{8.70}\
\frac{\hbar^{2} l^{5}}{30 m}-\frac{l^{7}}{140} W & \frac{\hbar^{2} l^{7}}{105 m}-\frac{l^{9}}{630} W
\end{array}\right|=0
\)
Using Theorem IV of Section 8.3, we eliminate the fractions by multiplying row one of the determinant by $420 \mathrm{~m} / \mathrm{l}^{3}$, row two by $1260 \mathrm{~m} / \mathrm{l}^{5}$, and the right side of (8.70) by both factors, to get
\(
\begin{gather}
\left|\begin{array}{cc}
70 \hbar^{2}-14 m l^{2} W & 14 \hbar^{2} l^{2}-3 m l^{4} W \
42 \hbar^{2}-9 m l^{2} W & 12 \hbar^{2} l^{2}-2 m l^{4} W
\end{array}\right|=0 \
m^{2} l^{4} W^{2}-56 m l^{2} \hbar^{2} W+252 \hbar^{4}=0 \
W=\left(\hbar^{2} / m l^{2}\right)(28 \pm \sqrt{532})=0.1250018 h^{2} / m l^{2}, \quad 1.293495 h^{2} / m l^{2} \tag{8.71}
\end{gather}
\)
Substitution of the integrals into (8.67) leads to the roots (Prob. 8.36)
\(
\begin{equation}
W=\left(\hbar^{2} / m l^{2}\right)(60 \pm \sqrt{1620})=0.5002930 h^{2} / m l^{2}, \quad 2.5393425 h^{2} / m l^{2} \tag{8.72}
\end{equation}
\)
The approximate values $\left(m l^{2} / h^{2}\right) W=0.1250018,0.5002930,1.293495$, and 2.5393425 may be compared with the exact values [Eq. (2.20)] $\left(\mathrm{ml}^{2} / h^{2}\right) E=0.125,0.5,1.125$, and 2 for the four lowest states. The percent errors are $0.0014 \%, 0.059 \%, 15.0 \%$, and $27.0 \%$ for $n=1,2,3$, and 4 , respectively. We did great for $n=1$ and 2 ; lousy for $n=3$ and 4 .
We now find the approximate wave functions corresponding to these $W$ 's. Substitution of $W_{1}=0.1250018 h^{2} / \mathrm{ml}^{2}$ into the set of equations (8.68a) corresponding to (8.71) gives (after division by $h^{2}$ )
\(
\begin{align}
0.023095 c{1}^{(1)}-0.020381 c{2}^{(1)} l^{2} & =0 \
-0.061144 c{1}^{(1)}+0.053960 c{2}^{(1)} l^{2} & =0 \tag{8.73}
\end{align}
\)
where, for example, the first coefficient is found from
\(
70 \hbar^{2}-14 m l^{2} W=70 h^{2} / 4 \pi^{2}-14(0.1250018) h^{2}=0.023095 h^{2}
\)
To solve the homogeneous equations (8.73), we follow the procedure given near the end of Section 8.4. We discard the second equation of (8.73), transfer the $c_{2}^{(1)}$ term to the right side, and solve for the coefficient ratio; we get
\(
c{1}^{(1)}=k, \quad c{2}^{(1)}=1.133 k / l^{2}
\)
where $k$ is a constant. We find $k$ from the normalization condition:
\(
\begin{aligned}
\left\langle\phi{1} \mid \phi{1}\right\rangle & =1=\left\langle k f{1}+1.133 k f{2} / l^{2} \mid k f{1}+1.133 k f{2} / l^{2}\right\rangle \
& =k^{2}\left(\left\langle f{1} \mid f{1}\right\rangle+2.266\left\langle f{1} \mid f{2}\right\rangle / l^{2}+1.284\left\langle f{2} \mid f{2}\right\rangle / l^{4}\right) \
& =k^{2}\left(S{11}+2.266 S{12} / l^{2}+1.284 S_{22} / l^{4}\right)=0.05156 k^{2} l^{5}
\end{aligned}
\)
where the previously found values of the overlap integrals were used. Therefore $k=4.404 / l^{5 / 2}$ and
\(
\begin{align}
& \phi{1}=c{1}^{(1)} f{1}+c{2}^{(1)} f{2}=4.404 f{1} / l^{5 / 2}+4.990 f{2} / l^{9 / 2} \
& \phi{1}=l^{-1 / 2}\left[4.404(x / l)(1-x / l)+4.990(x / l)^{2}(1-x / l)^{2}\right] \tag{8.74}
\end{align}
\)
where (8.62) was used.
Using $W{2}, W{3}$, and $W_{4}$ in turn in (8.55), we find the following normalized linear variation functions (Prob. 8.38), where $X \equiv x / l$ :
\(
\begin{align}
& \phi{2}=l^{-1 / 2}\left[16.78 X(1-X)\left(\frac{1}{2}-X\right)+71.85 X^{2}(1-X)^{2}\left(\frac{1}{2}-X\right)\right] \
& \phi{3}=l^{-1 / 2}\left[28.65 X(1-X)-132.7 X^{2}(1-X)^{2}\right] \tag{8.75}\
& \phi_{4}=l^{-1 / 2}\left[98.99 X(1-X)\left(\frac{1}{2}-X\right)-572.3 X^{2}(1-X)^{2}\left(\frac{1}{2}-X\right)\right]
\end{align}
\)