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Variational theorem (Read 5084 times)
Gerrit-Jan Linker
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Variational theorem
18.02.08 at 21:05:52
 
Variational theorem
 
The variational theorem is: <Φ|H|Φ> ≥ E0
 
where:  
H is the time independent Hamiltonian
E0 is the exact Energy of the ground state (the lowest eigenvalue of H)
Φ is a normalised trial wave function satisfying the same boundary conditions as the axact wave function.
 
In words the theorem says that the mean value of H from that wavefunction Φ is alway above or equal to the exact energy of the ground state of the system.  
 
When the exact wave function ψ0 is used <ψ0|H|ψ0> = <ψ0|E00> = E0
 
 
 
Derivation:
 
Let us use the trial wave function Φ again.
 
Let F = <Φ|H - E0|Φ>
 
We can expand Φ in the basis set of eigen functions {ψi} of H:
Φ = ∑iciψi
 
When we use that in the above formula we get:
F = <Φ|H - E0|Φ> = <∑iciψi|H - E0|∑jcjψj>
= ∑ijc*icji|H - E0j>
= ∑ijc*icj(<ψi|H|ψj> - <ψi|E0j>)
= ∑ijc*icj(<ψi|Ejj> - <ψi|E0j>)
= ∑ijc*icj(Ejij> - E0ij>)
= ∑ijc*icj(Ejδij - E0δij)
= ∑ijc*icjδij(Ej - E0)
= ∑ic*ici(Ei - E0)
= ∑i|ci|2(Ei - E0)
 
We arrive at F = ∑i|ci|2(Ei - E0)
Since Ei ≥ E0 it follows that F > 0
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« Last Edit: 18.02.08 at 21:32:13 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
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