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Proof of the variational principle

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Show that the average of the Hamiltonian is an upper bound to the ground state energy.

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We give a proof of the variational principle for the ground state energy.

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Let |psi_n> be the normalized eigenstates with eigenvalue E_n of the Hamiltonian H:

H|psi_n> = E_n |psi_n> (1)

If |psi> is any arbitrary state we can always expand it in terms of the eigenstates:

|psi> = sum from n = 0 to infinity of c_n |psi_n> (2)

The expansion coefficients c_n are given by:

c_n = <psi_n|psi>

If |psi> is normalized then:

<psi|psi> =1

Inserting (2) in here and using that |psi_n> are orthonormal ...

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