Variational method ↩

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created: 2022-01-08 15:31:58
modified: 2022-01-11 17:14:19

The variational method is a way of approximating the ground-state energy (lowest possible energy eigenstate) in quantum mechanics. We can get the minimal eigenvalue of a Hermitian Hamiltonian operator from the minimalization of the energy functional:

Where is the Hamiltonian operator and usually is an ansatz for the ground state wave function, which might depend on some real parameter (). The wave function is under the normalization condition:

In order to find the minimum of , we take under consideration the normalization condition with a Lagrange multiplier and we introduce the following functional:

Where is the Lagrange multiplier. Let's calculate the functional at up to first order in :

From this, we neglect the second order terms . Then, the variation of :

Now consider the variation for :

Now let be such a vector that for a small change in it, the change in is zero, so is stationary for the change in :

And from this, follows:

So the approximated energy of the system is, using the vector that satisfies :

Using a parameter in

If we use a real parameter in , then the variation condition becomes:

And the energy approximation becomes:

Of course wile enforcing the normalization condition: