Mathematical Methods for Physics and Engineering - Matematica.NET

22.8 ADJUSTMENT OF PARAMETERSIt is easily verified that functions (b), (c) and (d) all satisfy (22.30) but, so far as mimickingthe correct solution is concerned, we would expect from the figure that (b) would besuperior to the other two. The three evaluations are straightforward, using (22.22) and(22.23):∫ 10λ b =(2 − 2x)2 dx∫ 1(2x − 0 x2 ) 2 dx = 4/38/15 =2.50∫ 10λ c =(3x2 − 6x +3) 2 dx∫ 10 (x3 − 3x 2 +3x) 2 dx = 9/59/14 =2.80∫ 10λ d =(π2 /4) sin 2 (πx) dx∫ 1= π2 /80 sin4 (πx/2) dx 3/8 =3.29.We expected all evaluations to yield estimates greater than the lowest eigenvalue, 2.47,and this is indeed so. From these trials alone we are able to say (only) that λ 0 ≤ 2.50.As expected, the best approximation (b) to the true eigenfunction yields the lowest, andtherefore the best, upper bound on λ 0 . ◭We may generalise the work of this section to other differential equations ofthe form Ly = λρy, whereL = L † . In particular, one findswhere I and J are now given byI =∫ baλ min ≤ I J ≤ λ max,y ∗ (Ly) dx and J =∫ baρy ∗ ydx. (22.31)It is straightforward to show that, for the special case of the Sturm–Liouvilleequation, for whichLy = −(py ′ ) ′ − qy,the expression for I in (22.31) leads to (22.22).22.8 Adjustment of parametersInstead of trying to estimate λ 0 by selecting a large number of different trialfunctions, we may also use trial functions that include one or more parameterswhich themselves may be adjusted to give the lowest value to λ = I/J andhence the best estimate of λ 0 . The justification for this method comes from theknowledge that no matter what form of function is chosen, nor what values areassigned to the parameters, provided the boundary conditions are satisfied λ cannever be less than the required λ 0 .To illustrate this method an example from quantum mechanics will be used.The time-independent Schrödinger equation is formally written as the eigenvalueequation Hψ = Eψ, whereH is a linear operator, ψ the wavefunction describinga quantum mechanical system and E the energy of the system. The energy795