Computational Physics

34 The variational method for the Schrödinger equation

Table 3.1.Energy levels of the infinitely deep potential well.

N= 5 N= 8 N= 12 N= 16 Exact

2.4674 2.4674 2.4674 2.4674 2.4674

9.8754 9.8696 9.8696 9.8696 9.8696

22.2934 22.2074 22.2066 22.2066 22.2066

50.1246 39.4892 39.4784 39.4784 39.4784

87.7392 63.6045 61.6862 61.6850 61.6850

The first four columns show the variational energy levels for various

numbers of basis statesN. The last column shows the exact values.

The exact levels are approached from above as in Figure 3.1.

CheckCompare the results with the analytic solutions. These are given by

ψn(x)=

{

cos(knx) nodd

sin(knx) neven and positive

(3.19)

withkn=nπ/2,n=1, 2,..., and the corresponding energies are given by

En=k^2 n=

n^2 π^2

4

. (3.20)

For each eigenvectorC, the function

∑N

p= 1 Cpχp(x)should approximate an

eigenfunction(3.19). They can be compared by displaying both graphically.

Carry out the comparison for various numbers of basis states. The variational

levels are shown inTable 3.1, together with the analytical results.

3.2.2 Variational calculation for the hydrogen atom

As we shall see in the next two chapters, one of the main problems of electronic
structure calculations is the treatment of the electron–electron interactions. Here
we develop a program for solving the Schrödinger equation for an electron in a
hydrogen atom for which the many-electron problem does not arise, so that a direct
variational treatment of the problem is possible which can be compared with the
analytical solution [ 3 , 4 ].
The program described here is the first in a series leading to a program for
calculating the electronic structure of the hydrogen molecule. The extension to the
H+ 2 ion can be found in the next chapter in Problem 4.8 and a program for the
hydrogen molecule is considered in Problem 4.12.
The electronic Schrödinger equation for the hydrogen atom reads:
[
−

^2

2 m

∇^2 −

1

4 π 0

1

r

]

ψ(r)=Eψ(r) (3.21)