The Hellmann-Feynman theorem is a theorem in quantum mechanics , which relates the energy eigenvalues of a time-independent Hamilton operator to the parameters it contains. It is named after its discoverers Hans Hellmann (1936) and Richard Feynman (1939).
In general, the theorem says:
∂
E.
n
∂
λ
=
∫
ψ
n
∗
∂
H
^
∂
λ
ψ
n
d
τ
{\ displaystyle {\ frac {\ partial {E_ {n}}} {\ partial {\ lambda}}} = \ int {\ psi _ {n} ^ {*} {\ frac {\ partial {\ hat {H }}} {\ partial {\ lambda}}} \ psi _ {n} d \ tau}}
H
^
{\ displaystyle {\ hat {H}}}
is the parameterized Hamilton operator ,
E.
n
{\ displaystyle E_ {n}}
is the nth eigenvalue of the Hamilton operator,
ψ
n
{\ displaystyle \ psi _ {n}}
is the nth eigenvector of the Hamilton operator,
λ
{\ displaystyle \ lambda}
is the parameter that interests
and means a complete integration over the entire domain of definition of the eigenvectors.
d
τ
{\ displaystyle d \ tau}
The proof
The proof, if you proceed in a purely formal manner, is quite simple. In Dirac's Bra-Ket notation, the following can be written:
∂
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λ
∂
λ
=
∂
∂
λ
⟨
ψ
(
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|
H
^
λ
|
ψ
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⟩
=
⟨
∂
ψ
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∂
λ
|
H
^
λ
|
ψ
(
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⟩
+
⟨
ψ
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|
H
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λ
|
∂
ψ
(
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∂
λ
⟩
+
⟨
ψ
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|
∂
H
^
λ
∂
λ
|
ψ
(
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⟩
=
E.
λ
⟨
∂
ψ
(
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∂
λ
|
ψ
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⟩
+
E.
λ
⟨
ψ
(
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|
∂
ψ
(
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∂
λ
⟩
+
⟨
ψ
(
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|
∂
H
^
λ
∂
λ
|
ψ
(
λ
)
⟩
=
E.
λ
∂
∂
λ
⟨
ψ
(
λ
)
|
ψ
(
λ
)
⟩
+
⟨
ψ
(
λ
)
|
∂
H
^
λ
∂
λ
|
ψ
(
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⟩
=
⟨
ψ
(
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)
|
∂
H
^
λ
∂
λ
|
ψ
(
λ
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⟩
.
{\ displaystyle {\ begin {aligned} {\ frac {\ partial E _ {\ lambda}} {\ partial \ lambda}} & = {\ frac {\ partial} {\ partial \ lambda}} \ langle \ psi (\ lambda) | {\ hat {H}} _ {\ lambda} | \ psi (\ lambda) \ rangle \\ & = \ langle {\ frac {\ partial \ psi (\ lambda)} {\ partial \ lambda}} | {\ hat {H}} _ {\ lambda} | \ psi (\ lambda) \ rangle + \ langle \ psi (\ lambda) | {\ hat {H}} _ {\ lambda} | {\ frac {\ partial \ psi (\ lambda)} {\ partial \ lambda}} \ rangle + \ langle \ psi (\ lambda) | {\ frac {\ partial {\ hat {H}} _ {\ lambda}} {\ partial \ lambda}} | \ psi (\ lambda) \ rangle \\ & = E _ {\ lambda} \ langle {\ frac {\ partial \ psi (\ lambda)} {\ partial \ lambda}} | \ psi (\ lambda) \ rangle + E _ {\ lambda} \ langle \ psi (\ lambda) | {\ frac {\ partial \ psi (\ lambda)} {\ partial \ lambda}} \ rangle + \ langle \ psi (\ lambda) | { \ frac {\ partial {\ hat {H}} _ {\ lambda}} {\ partial \ lambda}} | \ psi (\ lambda) \ rangle \\ & = E _ {\ lambda} {\ frac {\ partial} {\ partial \ lambda}} \ langle \ psi (\ lambda) | \ psi (\ lambda) \ rangle + \ langle \ psi (\ lambda) | {\ frac {\ partial {\ hat {H}} _ {\ lambda}} {\ partial \ lambda}} | \ psi (\ lambda) \ rangle \\ & = \ langle \ psi (\ lambda) | {\ frac {\ partial {\ hat {H}} _ {\ lambda}} {\ partial \ lambda} } | \ psi (\ lambda) \ rangle. \ end {aligned}}}
there applies:
H
^
λ
|
ψ
(
λ
)
⟩
=
E.
λ
|
ψ
(
λ
)
⟩
,
{\ displaystyle {\ hat {H}} _ {\ lambda} | \ psi (\ lambda) \ rangle = E _ {\ lambda} | \ psi (\ lambda) \ rangle,}
⟨
ψ
(
λ
)
|
ψ
(
λ
)
⟩
=
1
⇒
∂
∂
λ
⟨
ψ
(
λ
)
|
ψ
(
λ
)
⟩
=
0.
{\ displaystyle \ langle \ psi (\ lambda) | \ psi (\ lambda) \ rangle = 1 \ Rightarrow {\ frac {\ partial} {\ partial \ lambda}} \ langle \ psi (\ lambda) | \ psi ( \ lambda) \ rangle = 0.}
For a critical, mathematical consideration of this proof, see.
Individual evidence
↑ Hellmann Introduction to Quantum Chemistry , Deuticke, Leipzig and Vienna 1937 (translation from Russian)
↑ Richard Feynman Forces in Molecules , Physical Review, Volume 56, 1939, pp. 340-343
↑ David Carfì: The pointwise Hellmann-Feynman theorem . In: AAPP Physical, Mathematical, and Natural Sciences . 88, No. 1, 2010, ISSN 1825-1242 . doi : 10.1478 / C1A1001004 .
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