Exercise 17.7.6 opens up a relation between the calculus of variations and eigenfunction–
eigenvalue problems. We may rewrite the expression of Exercise 17.7.6 as F
y(x) = b
a(pyx2−qy2) dx b
a y2w dx , (17.128)
in which the constraint appears in the denominator as a normalizing condition. After the unconstrained minimum ofF has been found,y can be normalized without changing the stationary value ofF because stationary values ofJ correspond to stationary values ofF. Then from Exercise 17.7.6, wheny(x)is such thatJandF take on a stationary value, the optimum functiony(x)satisfies the Sturm–Liouville equation
d dx
pdy
dx
+qy+λwy=0, (17.129)
withλthe eigenvalue (not a Lagrangian multiplier). Integrating the first term in the numer- ator of Eq. (17.128) by parts and using the boundary condition,
pyxy b
a=0, (17.130)
we obtain F
y(x) = − b
a
y d
dx
pdy dx
+qy
dx
1 b
a
y2w dx. (17.131) Then substituting in Eq. (17.129), the stationary values ofF[y(x)]are given by
F
y(x) =λn, (17.132)
withλn the eigenvalue corresponding to the eigenfunctionyn. Equation (17.132) withF given by either Eq. (17.128) or Eq. (17.131) forms the basis of the Rayleigh–Ritz method for the computation of eigenfunctions and eigenvalues.
Ground State Eigenfunction
Suppose that we seek to compute the ground-state eigenfunctiony0and eigenvalue18 λ0
of some complicated atomic or nuclear system. The classical example, for which no exact solution exists, is the helium atom problem. The eigenfunctiony0is unknown, but we shall assume we can make a pretty good guess at an approximate functiony, so mathematically we may write19
y=y0+ ∞
i=1
ciyi. (17.133)
Theci are small quantities. (How small depends on how good our guess was.) Theyi are orthonormalized eigenfunctions (also unknown), and therefore our trial functiony is not normalized.
Substituting the approximate functionyinto Eq. (17.131) and noting that b
a
yi d
dx
pdyj
dx
+qyi
dx= −λiδij, (17.134)
F
y(x) =λ0+∞
i=1ci2λi 1+∞
i=1ci2 . (17.135)
Here we have taken the eigenfunctions to be orthonormal — since they are solutions of the Sturm–Liouville equation, Eq. (17.129). We also assume thaty0is nondegenerate. Now, if we replace
ic2iλi→
ic2iλ0+
ic2i(λi−λ0)we obtain F
y(x) =λ0+ ∞
i=1c2i(λi−λ0) 1+∞
i=1ci2 . (17.136)
Equation (17.136) contains two important results.
• Whereas the error in the eigenfunctionywasO(ci), the error inλis onlyO(ci2). Even a poor approximation of the eigenfunctions may yield an accurate calculation of the eigenvalue.
• Ifλ0is the lowest eigenvalue (ground state), then sinceλi−λ0>0, F
y(x) =λ≥λ0, (17.137)
or our approximation is always on the high side and becoming lower, converging on λ0 as our approximate eigenfunction y improves (ci →0). Note that Eq. (17.137) is a direct consequence of Eq. (17.135). More directly, F[y(x)] in Eq. (17.135) is a positively weighted average of the λi and, therefore, must be no smaller than the smallestλi, to wit,λ0. In practical problems in quantum mechanics,y often depends on parameters that may be varied to minimizeF and thereby improve the estimate of the ground-state energyλ0. This is the “variational method” discussed in quantum mechanics texts.
18This means thatλ0is the lowest eigenvalue. It is clear from Eq. (17.128) that ifp(x)≥0 andq(x)≤0 (compare Table 10.1), thenF[y(x)]has a lower bound and this lower bound is nonnegative. Recall from Section 10.1 thatw(x)≥0.
19We are guessing at the form of the function. The normalization is irrelevant.
Example 17.8.1 VIBRATINGSTRING
A vibrating string, clamped atx=0 and 1, satisfies the eigenvalue equation d2y
dx2+λy=0 (17.138)
and the boundary conditiony(0)=y(1)=0. For this simple example we recognize imme- diately thaty0(x)=sinπ x (unnormalized) andλ0=π2. But let us try out the Rayleigh–
Ritz technique.
With one eye on the boundary conditions, we try
y(x)=x(1−x). (17.139)
Then withp=1 andw=1, Eq. (17.128) yields F[y(x)] =
1
0(1−2x)2dx 1
0x2(1−x)2dx = 1/3
1/30=10. (17.140)
This result,λ=10, is a fairly good approximation (1.3% error)20 ofλ0=π2=9.8696.
You may have noted thaty(x), Eq. (17.139), is not normalized to unity. The denominator inF[y(x)]compensates for the lack of unit normalization.F may also be calculated from Eq. (17.131) since Eq. (17.130) is satisfied byy from Eq. (17.139).
In the usual scientific calculation the eigenfunction would be improved by introducing more terms and adjustable parameters, such as
y=x(1−x)+a2x2(1−x)2. (17.141) It is convenient to have the additional terms orthogonal, but it is not necessary. The parame- tera2is adjusted to minimizeF[y(x)]. In this case, choosinga2=1.1353 drivesF[y(x)] down to 9.8697, very close to the correct eigenvalue value.
Exercises
17.8.1 From Eq. (17.128) develop in detail the argument whenλ≥0 orλ <0. Explain the circumstances under whichλ=0, and illustrate with several examples.
17.8.2 An unknown function satisfies the differential equation y′′+
π 2
2
y=0 and the boundary conditions
y(0)=1, y(1)=0.
20The closeness of the fit may be checked by a Fourier sine expansion (compare Exercise 14.2.3 over the half-interval[0,1]or, equivalently, over the interval[−1,1], withy(x)taken to be odd). Because of the even symmetry relative tox=1/2, only odd nterms appear:
y(x)=x(1−x)= 8
π3
sinπ x+sin 3π x
33 +sin 5π x 53 + ã ã ã
.
(a) Calculate the approximation
λ=F[ytrial] for
ytrial=1−x2. (b) Compare with the exact eigenvalue.
ANS. (a)λ=2.5, (b)λ/λexact=1.013.
17.8.3 In Exercise 17.8.2 use a trial function
y=1−xn. (a) Find the value ofnthat will minimizeF[ytrial].
(b) Show that the optimum value ofndrives the ratioλ/λexactdown to 1.003.
ANS.(a) n=1.7247.
17.8.4 A quantum mechanical particle in a sphere (Example 11.7.1) satisfies
∇2ψ+k2ψ=0,
withk2=2mE/h¯2. The boundary condition is thatψ (r=a)=0, whereais the radius of the sphere. For the ground state [whereψ=ψ (r)] try an approximate wave function
ψa(r)=1− r
a 2
and calculate an approximate eigenvaluek2a.
Hint. To determinep(r)andw(r), put your equation in self-adjoint form (in spherical polar coordinates).
ANS. ka2=10.5
a2 , kexact2 =π2 a2. 17.8.5 The wave equation for the quantum mechanical oscillator may be written as
d2ψ (x) dx2 +
λ−x2
ψ (x)=0, withλ=1 for the ground state (Eq. (13.18)). Take
ψtrial= +
1−xa22, x2≤a2 0, x2> a2
for the ground-state wave function (witha2an adjustable parameter) and calculate the corresponding ground-state energy. How much error do you have?
Note. Your parabola is really not a very good approximation to a Gaussian exponential.
What improvements can you suggest?
17.8.6 The Schrửdinger equation for a central potential may be written as Lu(r)+ ¯h2l(l+1)
2Mr2 u(r)=Eu(r).
Thel(l+1)term, the angular momentum barrier, comes from splitting off the angu- lar dependence (Section 9.3). Treating this term as a perturbation, use your variational technique to show thatE > E0, whereE0is the energy eigenvalue ofLu0=E0u0cor- responding tol=0. This means that the minimum energy state will havel=0, zero angular momentum.
Hint. You can expandu(r)asu0(r)+∞
i=1ciui, whereLui=Eiui,Ei> E0. 17.8.7 In the matrix eigenvector, eigenvalue equation
Ari=λiri,
whereλis ann×nHermitian matrix. For simplicity, assume that itsnreal eigenvalues (Section 3.5) are distinct,λ1being the largest. If r is an approximation to r1,
r=r1+ n i=2
δiri, show that
r†Ar r†r ≤λ1
and that the error inλ1is of the order|δi|2. Take|δi| ≪1.
Hint. Thenri form a complete orthogonal set spanning then-dimensional (complex) space.
17.8.8 The variational solution of Example 17.8.1 may be refined by takingy=x(1−x)+ a2x2(1−x)2. Using the numerical quadrature, calculateλapprox=F[y(x)], Eq. (17.128), for a fixed value ofa2. Varya2to minimizeλ. Calculate the value ofa2that minimizes λ and calculate λ itself, both to five significant figures. Compare your eigenvalue λ withπ2.
Additional Readings
Bliss, G. A., Calculus of Variations. The Mathematical Association of America. LaSalle, IL: Open Court Pub- lishing Co. (1925). As one of the older texts, this is still a valuable reference for details of problems such as minimum-area problems.
Courant, R., and H. Robbins, What Is Mathematics? 2nd ed. New York: Oxford University Press (1996). Chapter VII contains a fine discussion of the calculus of variations, including soap film solutions to minimum-area problems.
Lanczos, C., The Variational Principles of Mechanics, 4th ed. Toronto: University of Toronto Press (1970), reprinted, Dover (1986). This book is a very complete treatment of variational principles and their applica- tions to the development of classical mechanics.
Sagan, H., Boundary and Eigenvalue Problems in Mathematical Physics. New York: Wiley (1961), reprinted, Dover (1989). This delightful text could also be listed as a reference for Sturm–Liouville theory, Legendre and Bessel functions, and Fourier Series. Chapter 1 is an introduction to the calculus of variations, with applications to mechanics. Chapter 7 picks up the calculus of variations again and applies it to eigenvalue problems.
Sagan, H., Introduction to the Calculus of Variations. New York: McGraw-Hill (1969), reprinted, Dover (1983).
This is an excellent introduction to the modern theory of the calculus of variations, which is more sophisticated and complete than his 1961 text. Sagan covers sufficiency conditions and relates the calculus of variations to problems of space technology.
Weinstock, R., Calculus of Variations. New York: McGraw-Hill (1952); New York: Dover (1974). A detailed, systematic development of the calculus of variations and applications to Sturm–Liouville theory and physical problems in elasticity, electrostatics, and quantum mechanics.
Yourgrau, W., and S. Mandelstam, Variational Principles in Dynamics and Quantum Theory, 3rd ed. Philadelphia:
Saunders (1968); New York: Dover (1979). This is a comprehensive, authoritative treatment of variational principles. The discussions of the historical development and the many metaphysical pitfalls are of particular interest.
N ONLINEAR M ETHODS
AND C HAOS
Our mind would lose itself in the complexity of the world if that complexity were not harmonious; like the short–sighted, it would only see the details, and would be obliged to forget each of these details before examining the next, because it would be incapable of taking in the whole. The only facts worthy of our attention are those which introduce order into this complexity and so make it accessible to us.
HENRIPOINCARÉ