The Student Room Group

Minimising energy (quantum mechanics)

The question is:

There are 1 electron and N identical nuclei. The Hamiltonian in the presence of the nuclei is H. We want to find the ground state of the electron. The ground state of the electron if it was bound to the j-th nucleus only, is |j> and its energy E'.
We can write the trial wavefunction as:
|Ψ> = Σ a_j |j>
where |1>, |2>...|N> forms the basis.
Show that if you minimise the energy:
E = <Ψ|H|Ψ> / <Ψ|Ψ>
wrt to a_j, it gives a wavefunction given by:
H . a = E a, where a is the vector {a_1, a_2...a_n} and H is a NxN matrix.
You may assume that <j|k> = δ_jk

Here's what I've done so far:

The hamiltonian: p^2/2m + Σ V_j
where the sum goes from j =1 to N.

I then subbed in the wavefunction into the energy equation giving:
E = Σ (a_j)* <j| H | Σ a_j |j> / Σ (a_j)* <j| Σ a_j |j>
E = Σ (a_j)*(a_j) <j|H|j> / Σ (a_j)*(a_j) <j|j>
Since <j|j> = 1, then:
E = Σ (a_j)*(a_j) <j|H|j> / Σ (a_j)*(a_j)
But now I dont know how to proceed...how would I go about minimising this energy???
I know I have to differentiate δE/δa_j = 0 but Im not sure how to. Its my first time using dirac notation and im pretty confused about what i can or cannot cancel/sub in or differentiate...
Original post by IDontKnowReally
The question is:

There are 1 electron and N identical nuclei. The Hamiltonian in the presence of the nuclei is H. We want to find the ground state of the electron. The ground state of the electron if it was bound to the j-th nucleus only, is |j> and its energy E'.
We can write the trial wavefunction as:
|Ψ> = Σ a_j |j>
where |1>, |2>...|N> forms the basis.
Show that if you minimise the energy:
E = <Ψ|H|Ψ> / <Ψ|Ψ>
wrt to a_j, it gives a wavefunction given by:
H . a = E a, where a is the vector {a_1, a_2...a_n} and H is a NxN matrix.
You may assume that <j|k> = δ_jk

Here's what I've done so far:

The hamiltonian: p^2/2m + Σ V_j
where the sum goes from j =1 to N.

I then subbed in the wavefunction into the energy equation giving:
E = Σ (a_j)* <j| H | Σ a_j |j> / Σ (a_j)* <j| Σ a_j |j>
E = Σ (a_j)*(a_j) <j|H|j> / Σ (a_j)*(a_j) <j|j>
Since <j|j> = 1, then:
E = Σ (a_j)*(a_j) <j|H|j> / Σ (a_j)*(a_j)
But now I dont know how to proceed...how would I go about minimising this energy???
I know I have to differentiate δE/δa_j = 0 but Im not sure how to. Its my first time using dirac notation and im pretty confused about what i can or cannot cancel/sub in or differentiate...

bump
Original post by IDontKnowReally
The question is:

There are 1 electron and N identical nuclei. The Hamiltonian in the presence of the nuclei is H. We want to find the ground state of the electron. The ground state of the electron if it was bound to the j-th nucleus only, is |j> and its energy E'.
We can write the trial wavefunction as:
|Ψ> = Σ a_j |j>
where |1>, |2>...|N> forms the basis.
Show that if you minimise the energy:
E = <Ψ|H|Ψ> / <Ψ|Ψ>
wrt to a_j, it gives a wavefunction given by:
H . a = E a, where a is the vector {a_1, a_2...a_n} and H is a NxN matrix.
You may assume that <j|k> = δ_jk

Here's what I've done so far:

The hamiltonian: p^2/2m + Σ V_j
where the sum goes from j =1 to N.

I then subbed in the wavefunction into the energy equation giving:
E = Σ (a_j)* <j| H | Σ a_j |j> / Σ (a_j)* <j| Σ a_j |j>
E = Σ (a_j)*(a_j) <j|H|j> / Σ (a_j)*(a_j) <j|j>
Since <j|j> = 1, then:
E = Σ (a_j)*(a_j) <j|H|j> / Σ (a_j)*(a_j)
But now I dont know how to proceed...how would I go about minimising this energy???
I know I have to differentiate δE/δa_j = 0 but Im not sure how to. Its my first time using dirac notation and im pretty confused about what i can or cannot cancel/sub in or differentiate...


Did you paraphrase the question? If yes, post the actual question.

I know I have to differentiate δE/δa_j = 0 but Im not sure how to.


Treat aj as variable x and do the partial differentiation using the quotient rule.

If possible, use latex to display the equations.

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