THE VARIATION THEOREM

Given the system, those Hamiltonian operator Ĥ is time independent and whose lower - energy eigenvalue is E, if ø is any normalized, and well- behaving functions of the coordinates of the system's particles that satisfying the boundary conditions of the problem, then

∫ ø* Ĥ ø dτ ≥ E₁,      ø normalized..........(1)

                      The variation theorem allowing use to calculate, a Upper bond for the system's and ground state energy.

Equation (1) to prove, and we can expand ø in terms of the complete, with orthonormal set of eigenfunction of Ĥ, the stationary - state eigenfunction ψk :

∑ = ɑk ψk ...........(2)                      where

k

Ĥψk = Ekψk......... (3)

Note that all the expansion equation (2) requires that ø obeyed the same boundary conditions as the ψk's, and substitution of equation (2) into the left side of equation (1) gives

∫ ø* Ĥø dτ = ∫ ∑ ɑₖ*ψαₖ* Ĥ ∑ ɑⱼψⱼ dτ = ∫ ∑ 

                        ₖ                     ⱼ                     ₖ

ɑₖ*ψₖ* ∑ ɑⱼ Ĥψⱼ dτ

              ⱼ

Using the eigenvalue equation (3) and assuming the validity of interchanging the integration and the infinite summations, we can get

∫ ø* Ĥø dτ = ∫ ∑ ɑₖ* ψₖ* ∑ ɑⱼ Eⱼ ψⱼ dτ  = ∑ ∑ 

                        ₖ               ⱼ                         ₖ  ⱼ

ɑₖ* ɑⱼEⱼ ∫ ψₖ* ψⱼ dτ

= ∑ ∑ ɑₖ* ɑⱼEⱼ δₖⱼ 

    ₖ ⱼ

Where the orthonormality of the eigenfunction ψₖ was used. We performed the sum over j, and, as usual, the kronecker delta makes all terms of zero except the one with j = k, gives 

∫ ø * Ĥ ø dτ = ∑ aₖ* aₖ Eₖ = ∑ ∣ ɑₖl² Eₖ.....(4)

                        ₖ                    ₖ

Since E₁ is the lower - energy eigenvalue of Ĥ , we have Eₖ ≥ E₁. Since ∣ ɑₖ∣² is never negative (-), we can multiply the inequality of Eₖ ≥ E₁ by ∣ ɑₖ∣² without changing the direction of the inequality sign to get ∣ ɑₖ∣² Eₖ ≥ ∣ ɑₖ∣² E₁. Therefore, ∑ₖ ∣ɑₖ∣² Eₖ ≥ ∑ₖ ∣ɑₖ∣² E₁, and uses of equation (4) gives 

∫ø* Ĥø dτ = ∑ ∣ɑₖ ∣² Eₖ ≥ ∑ ∣ɑₖ∣² E₁ = E₁ ∑ ∣ɑₖ∣²

                     ₖ                   ₖ                      ₖ

........(5)

Because ø is normalized, and we have ∫ ø*ø dτ = 1. Substitution of the expansion equation (2) into the normalization conditions gives

1 = ∫ ø*ø dτ = ∫ ∑ ɑ*ₖψ*ₖ ∑ ɑⱼψⱼ ∫ ψ*ₖ ψⱼ dτ = 

                          ₖ              ⱼ 

∑ ∑ ɑ*ₖ ɑⱼ δₖⱼ     1 = ∑ ∣ ɑₖ∣².............(6)

ₖ  ⱼ                             ₖ

Note that in derived equation (4) and (6) we must essentially repeated the derivation of equation respectively use of equation (6) and (5) gives the variation theorem equation (1):

∫ ø* Ĥø dτ ≥ E₁,       ø normalized ...... (7)

Suppose we have a function ø that is not normalized. To apply the variation theorem, we multiply ø by a normalization constant N so that Nø is normalized. Replacing ø by Nø  in equation (7) we have, 

      ∣ N∣² ∫ ø* Ĥø dτ ≥ E₁...........(8)

N is determined by ∫ (Nø)* Nø dτ = ∣N∣² ∫ ø* ø dτ = 1 : so ∣N∣² =1 / ∫ø* ø dτ and equation (8) becomes, 

.....(10)

Where ø is any well - behaving functions for not necessarily normalized and that satisfying the boundary conditions of the problem. 

The function ø is called " trial variation function" and the integral in equation (8) or the ratio of integrals in equation (9) is called the " variational integral". To arrive at a good approximation to the ground - state energy E₁, we can try many trial variation functions and looking for the one that giving the lowest value of the variational integral. From equation (1) the lower the value of the variational integral, the best the approximation we have to E₁.

One way to disprove the quantum mechanics could be to find a trial variation function that made the vibrational integral lesser than E₁ for the some system where E₁ is known. 

Let ψ₁ be the true ground - state wave function :

Ĥ ψ₁ = E₁ ψ₁...........(10)

If we happened to be lucky enough to hit upon the variation function that was equal to ψ₁, then using equation (10) in (1), we can see that the variational integral will be equal to E₁. Thus the ground - state wave function gives the minimum value of the variational integral, the closer the trial variational function will approach the true ground - state wave function. However, it turns out that the variational integral approaches E₁ a lot faster than the trial variation functions approaches ψ₁, and it's possible to get a rather a good approximation to E₁ using a rather poor ø. 

                  In practice, one usually puts many more parameters into the trial function ø and then varies this parameters so as to the minimized the variational integral. Successfully use of the variation method depends on the ability to make a good choice for the trial function. 

Let us look at some examples of variation method, the real utility of the method is for problems to which we can don't know about the true solutions, we will considering problems that are almost exactly solvable, so that we can do judge the very accuracy of our results.