Computational Physics

30 The variational method for the Schrödinger equation

change inEvanishes to first order:

δE≡0. (3.2)

Defining

P=〈ψ|H|ψ〉 and

Q=〈ψ|ψ〉,

(3.3)

we can write the changeδEin the energy to first order inδψas

δE=

〈ψ+δψ|H|ψ+δψ〉

〈ψ+δψ|ψ+δψ〉

−

〈ψ|H|ψ〉

〈ψ|ψ〉

≈

〈δψ|H|ψ〉−(P/Q)〈δψ|ψ〉

Q

+

〈ψ|H|δψ〉−(P/Q)〈ψ|δψ〉

Q

. (3.4)

As this should vanish for anarbitrarybut small change inψ, we find, using
E=P/Q:
Hψ=Eψ, (3.5)

together with the Hermitian conjugate of this equation, which is equivalent.
In variational calculus, stationary states of the energy-functional are foundwithin
a subspace of the Hilbert space. An important example is linear variational calculus,
in which the subspace is spanned by a set of basis vectors|χp〉,p=1,...,N.We
take these to be orthonormal at first, that is,

〈χp|χq〉=δpq, (3.6)

whereδpqis the Kronecker delta-function which is 0 unlessp=q, and in that case,
it is 1.
For a state
|ψ〉=

∑

p

Cp|χp〉, (3.7)

the energy-functional is given by

E=

∑N

p,q= 1 C

∗

pCqHpq

∑N

p,q= 1 C∗pCqδpq

(3.8)

with
Hpq=〈χp|H|χq〉. (3.9)

The stationary states follow from the condition that the derivative of this functional
with respect to theCpvanishes, which leads to

∑N

q= 1

(Hpq−Eδpq)Cq= 0 forp=1,...,N. (3.10)