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The variational principle states that for any normalizable trial state
ψ
\psi
ψ
, the Rayleigh quotient
E
[
ψ
]
=
⟨
ψ
∣
H
∣
ψ
⟩
/
⟨
ψ
∣
ψ
⟩
E[\psi] = \langle\psi|H|\psi\rangle / \langle\psi|\psi\rangle
E
[
ψ
]
=
⟨
ψ
∣
H
∣
ψ
⟩
/
⟨
ψ
∣
ψ
⟩
satisfies which relation with the true ground-state energy
E
0
E_0
E
0
?
E
[
ψ
]
=
E
0
E[\psi] = E_0
E
[
ψ
]
=
E
0
exactly for every normalizable trial state
E
[
ψ
]
≤
E
0
E[\psi] \leq E_0
E
[
ψ
]
≤
E
0
always, giving a lower bound on the ground-state energy
E
[
ψ
]
≥
E
0
E[\psi] \geq E_0
E
[
ψ
]
≥
E
0
always, with equality if and only if
ψ
\psi
ψ
is the exact ground state
E
[
ψ
]
E[\psi]
E
[
ψ
]
is unrelated to
E
0
E_0
E
0
unless
ψ
\psi
ψ
happens to solve the Schrodinger equation
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