Application of the Variational Method to the Particle-in-the-Box Problem David Keeports Mills College, Oakland, CA 94613 In a well-known problem illustrating the use of the variational method,'.2 a trial wave function of the form Ji = N(Lx - x2) is shown to yield a calculated energy slightly in excess of the true ground state energy for a particle in a onedimensional box of length L when the trial wave function is substituted into the energy expression
loL
*R*dx
E=
(1)
In a recent paper in this J ~ u r n a ltrial , ~ wave functions of the form $" = (-l)"A,
n
(z- Lkln)
k-0
were considered as approximate solutions to the particle-inthe-box prohlem. I t is the purpose of the present paper t o consider a more general application of the variational method to the particle-in-the-box problem with polynomial trial functions, i.e., trial functions of the form
with corresponding energies of
where n is a positive integer. According to the variational method, approximate polynomial wave functions can be constructed by substitution of trial functions of the form given by eq 3 into eq 1,followed by the minimization of E in eq 1with respect to each of the ak,,'s in eq 3. This procedure, however, leads to approximate wave functions that in general do not obey the boundary conditions given by eq 5. Despite this difficulty, it is still possible to construct approximate polynomial solutions that do satisfy boundary conditions a t the ends of the box from starting functions individually obeying the boundary conditions following a convenient shift of the box's location along the x axis. First consider a one-dimensional box of length L lying along the x axis with the center of the box a t the origin so that the ends of the box areat x = -L/2 and a t x =LIZ. Exact wave functions for a particle of mass m in such a box are given by
m=o
,bk = (zlL)o-5COB
with the ah,, coefficients determined by the variational method. This problem illustrates the mathematical steps involved in applying the variational method t o the construction of molecular orbitals as linear combinations of atomic orbitals, hut with the advantage that all steps, including the evaluation of all required integrals, are mathematically simple. Additionally, the approximate results obtained when the variational method is applied to the particle-in-the-box problem are readily compared to well-known exact results.
+
(k 1)rx L
k = 0,2,4,
.. .
(8)
and
while energies are given by
Method
In the standard problem of a particle of mass m with zero potential energy confined to a one-dimensional box extending from the origin to the point x = L, a Hamiltonian operator of the form
Assume for mathematical simplicity that the mass of the particle and the box length aresuch that the quantities h2/ 2m and L/2 are of unit magnitude. The Hamilronian uperator and boundary conditions of eqs 4 and 5 then become
and boundary conditions
and *(0) = 0
and *(L) = 0
(5)
lead to wave functions of the form
Additionally, the wave function expressions in eqs 8 and 9 simplify to
'
Eggers, D. F.; Gregory, N. W.; Halsey, G. D.; Rabinovitch. 6 . S. PhysicalChemishy,Wiley: New York, 1964; p 84. Barrow. G. M.; Physical Chemistry, 4th ed.; McGraw-Hill: New York, 1979; p 381. Fucaloro, A. F. J. Chem. Educ. 1986, 63, 579. 314
Journal of Chemical Education
and
Ick = sin ax = ax - (&.~)~/3! + (axI5/5! - ...
.
k = 1,3,5,
. ..
Sample Calculation
(14)
As an example of the applic&ion of the variational method to the oarticle-in-the-box nroblem. the calculation of approximaie energies and w a v ~ f u n c t i ofor ~ s the three lowest energy states characterized by odd wave functions will be considered. The functious h,66,and +7 will be used as the set of functions for the construction of the aooroximate wave functious $I, $3, and G5 representing states of increasing energy. As in all applications of the variational method to construct functions as linear combinations of other functions, the secular equation
with 0 defined by a = (k
+l)d2
(15)
while the expression for energy reduces to
+
Ek = n2(k 1)'/4
(16)
Due to the fact that the V(x) = 0 potential inside the box is an even function, i.e., a function with the property V(-x) = V(x), the probability density function $2 associated with any solution to the Schr6dinger equation must also he an even function. The fact that $2(-x) = @(x) implies that either $(-x) = $(x) or $(-x) = -$(x), i.e., that any function $ satisfying the Schrodinger equation must be either an even function or an odd function. Consequently, the problem of finding solutions to the particle-in-the-box problem can be treated as two separate problems-one of constructing even approximate wave functions from a set of even functions, and one of constructing odd approximate wave functions from a set of odd functions. Consider the sets of functions ( 1 - ~ ~ , 1 - ~ ,... ~ , )1= -( +~2 ,~+ 4 , 6+, . . .
1
arises as a necessary condition for a nontrivial solution to exist for the system of simultaneous equations arising when E in eq 1 is minimized with respect to all of the constant coefficients used to construct linear combinations of the starting function^.^ In eq 23, Hij and S;, are integrals defined by
H, = J+;R+,dr
(24)
Sij = J+T+,d,dr
(25)
and
(17)
For the simple real functions given in eqn 17 and 18, the evaluation of all integrals is straightforward. For example,
and These function sets respectively consist of only even and onlv odd functions. Furthermore. everv function in both sets obeys the boundary conditions give; by eq 12. Thus, any approximate wave function constructed as a linear combination of functions from these sets must also obey the required boundary conditions. In the following discussion, the construction of approximate particle-in-the-box wavefunctions of the forms *k =
C
ck,~m+2m
m-1
and
will be considered. Such construction of approximate wave functions is equivalent to the construction of simple even and odd polynomial wave functions of the forms
and
It is readily shown by integration that HG equals Hjj for any pair of functions given in eqs 17 and 18, a result anticipated from the fact that the Hamiltonian operator fi is Hermitian. Substitution of aII integral values into eq 23 leads to the equation 1.MXXX)- 0.15238E 2.28571 - 0.20317E 2.66667 - 0.22626E 2.28571 - 0.20317E 3.55556 - 0.271WE 4.36364 - 0.31258E = 0 2.66667 - 0.22626E 4.36364 - 0.312583 5.53846 - 0.35556E (28)
Expansion of the secular determinant in eq 28 (followed by multidvina the resultina equation bv 109 results in the
and
subject to the constraints that
and
The roots of eq 29 are easily found by trial and error with the use of a programmable calculator. Values of 9.8696,39.998, and 142.63, respectively, are found for E1,E3, and Es, whereas eq 16, the expression for the exact energies, gives values of 9.8696, 39.478, and 88.826. Equation 16 in general provides good first estimates of secular equation roots of lower energy; however, the secular equation roots of largest magnitude are considerably underestimated by eq 16. Equation 23 arises in the solution of the system of equations (H - SE)(c) = 0
and with and
(30)
In eq 30, H and S are r X r matrices of the integrals Hij and Stj, c is a column matrix of the r coefficients used to construct wave functions, and 0 is a column matrix of r zeroes. Successive substitution of E's found as roots of eq 29 into eq Moore, W. J. Physical Chemishy, 4th ed.; Prentlce-Hall: Englewood Cliffs. NJ, 1972; pp 718-22. ,,
Volume 66 Number 4
April 1989
315
30 provides relations among the c's in each approximate wave function. For example, substitution of El = 9.8696 into eq 30 leads to the matrix equation 0.09606 0.28046 0.28046 0.82112 1.27863 Clr 2.02926 c , , ~ 0.43354 1.27863 0.43354]r]
I:[
=
(31)
= - 0 . 4 7 1 3 3 ~ ~and , ~ that c1.7 = from which it is found that 0.08334~1~3. Application of the normalization condition
leads to a c l r value of 5.1278. Similar substitutions of E values of 39.998 and 142.63 into eq 30 lead to the set of approximate wave functions
Addltlonal Results In principle, the procedure of the above example can be extended to the construction of approximate wave functions from sets of functions containing arbitrarily large numbers of functions. However, certain steps of the calculation, particularly the expansion of the secular determinant and the determination of relationships among the coefficients used to form approximate wave functions as linear combinations of starting functions, become quite laborious as the number of starting functions is increased. Direct substitution of even and odd functions of the forms
and into eqs 24 and 25 yields the following general results for the integrals Hij and Sjj: for even i and j
By the use of Stjvalues from eq 28, the orthogonality of the approximate wave functions can be readily verified. This orthogonality arises as a mathematical consequence of the fact that both H and S in eq 30 are symmetric matrices. Through the use of eqs 20-22, the approximate wave functions given by eq 33 may be expressed in the alternate form
These forms can he compared term by term to the series expansion given in eq 14. For example, from eq 14, Graphs of & vs. x are presented in Figure 1 for the three approximate wave functions found by the application of the variational method along with plots of the exact wave functions as given by eq 14. In summary of this example, very good agreement is found between approximate and exact energies and between approximate and exact wave functions for the two states of lower energy characterized by odd wave functions, while poor agreement is found hetween approximate and exact energies and between approximate and exact wave functions for the highest energy state,
Zij Hjj=-2 L + J - ~ 1 1 1 r + l ~ + 1& + ] + I sjj=2P----+-)
-)
1 -1 + 1 S - -= 2(-1 - 3 i+2 j+2 &+]+I
"
for odd i and j
(37)
for even i and j for odd i and j
(38)
To facilitate calculations, FORTICON4 the extended Hiickel program available through the Quantum Chemistry Program E x ~ h a n g ewas , ~ modified to calculate approximate energies and wave functions from secular equations constructed from the integrals given by eqs 37 and 38. Tables 17 summarize results from the 10 problems in which approximate particle-in-the-box energies are calculated from function sets consisting of the first r functions from the setsgiven in eqs 17 and 18. Values of the integrals Hi, and Sjj are presented in Tables 1-4.Substitution of values from these tables into eq 23 yields secular equations whose expanded secular determinants are given in Table 5. Tables 6 and 7 compare the approximate energies found as secular equation roots to exact energies. As seen in Tables 6 and 7, approximate particle-in-thebox energies found from the useof the variational principle show excellent agreement with exact energies except in the cases of the highest energies predicted by the use of a given set of starting functions. Poorly calculated energies are Howell. J.: Rossi. A.: Wallace, D.; Haraki, K.: Hoffmann R. oCPE 1977, 19, 344.
Figure 1. Approximate and exact parllcle-ln-the-box wave functions. Approxi. mate wave functions calculated from a set of three odd functlons.
316
Journal of Chemical Education
Figure 2. Approximate and exact particl~ln-the-box wave functions. Approxifunctions constructed hom sets consisting of three. four, and five mate J., even funotions.
Table 1.
HyValues lor a Set of Flve Even Functions
Table 3.
S,,Values for e Set of Flve Even Functlons
Table 2.
HllValues for a Set of Flve Odd Functlons
Table 4.
S,,Values for a Set of Five Odd Functlons
I
7
5
11
9
Coenlclenis In Expanded Secular Determinantsa
Table 5. &ng
functions
coetflcients
n.
in
2
v,E" = 0
n=o
number
NPB
n=O
n = l
1 2 3 4 5 1 2 3 4 5
even even
26666.67 19504.76 20586.35 24435.62 28523.68 16000.00 46439.91 16795.39 63325.49 22416.48
-10666.67 -8668.78 -9501.39 -11499.11 -13582.70 -1523.81 -5629.08 -2239.39 -8887.79 -3248.77
even men
even
odd odd odd odd
odd
n=5
n=2
n=3
n=4
309.5994 479.8683 670.7817 857.8549
-4.26550 -9.62830 -15.69931
0.03191007 0.09344825
93.6160 57.4201 274.4758 111.3863
-0.298286 -2.489577 -1.302763
0.005319606 0.005321745
-0.0001369205
-0.000005529086
' M l c i e n t a lor a given squstlon rmnipliedby an appropriate power 01 10 to facilitate display.
Table 6.
Approxlmate and Exact Energles lor States wlth Even Wave Functions number r of starting functions
wavefiinctlon
number k
1
2
3
4
5
m
0 2 4 6
2.50000
2.46744 25.53256
2.46740 22.29341 87.73919
2.46740 22.20737 63.60446 219.72077
2.46740 22.20661 61.76068 132.91796 463.14734
2.46740 22.20661 61.68503 120.90265 199.85949
n
Table 7.
Approxlrnate and Exact Energies lor States wlth Odd Wave Functlons number r of starting funnions
~a~efunctlon
number k
1
2
3
4
5
m
1 3 5 7
10.50000
9.87539 50.12461
9.86962 39.99796 142.63240
9.86960 39.48924 94.11862 324.52254
9,86960 39.47850 89.16438 181.68712 642.30039
9.86960 39.47842 88.82644 157.91367 246.74011
0
meatlv - .imnroved when the number of starting functions is increased.'For example, in the problem discussed above, the true EI of 88.826 is ~oorlvan~roximatedas 142.63when only starting functions. 4% h; and 6, are used successive inclusion of the functions 49,411,613,and 41s provides respective values of 94.119, 89.164, 88.836, and 88.827 as upper limits for Es. In general, as the number of functions used in the construction of approximate wave functions is increased, calcu-
ow ever,
lated wave functions are seen to resemble more closely the exact wave functions given by eqs 13 and 14. Figure 2 shows the improvement in the calculated 11.4 when $4 is constructed from subsets containing the first three, four, and five elements of the function set {Q*, 91, gC,. . .I. Table 6 shows how the calculated El values converge to the exact El value as the size of the set ofstarting functibns is increased.~ In summary, the variational method treatment of the particle-in-the-box problem discussed here provides a set of Volume 66 Number 4
April 1989
317
mathematically tractable problems illustrating the formation of approximate trial wave functions as linear combinations of other functions. Besides the problems summarized above in Tables 1-7, problems involving the formation of wave functions from subsets of the starting functions used in the above examples are readily constructed. As an example, two approximate even wave functions can be constructed from only the functions & and 46. The mathematical sim-
318
Journal of Chemical Education
plicity of approximate wave functions of the forms given by eqs 19 and 20 make such functions convenient for demonstrating the postulates of quantum mechanics. For example, expectation values for position and linear momentum are easily calculated, and the orthonormality of wave functions is easily shown. Approximate wave functions of the forms given by eqs 19 and 20 are conveniently plotted by the use of a programmable calculator with graphics capabilities.
