lecture 5 variational principle
TRANSCRIPT
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The variational principle
The variational principleQuantum mechanics 2 - Lecture 5
Igor Lukacevic
UJJS, Dept. of Physics, Osijek
November 8, 2012
Igor Lukacevic The variational principle
![Page 2: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/2.jpg)
The variational principle
Contents
1 Theory
2 The ground state of helium
3 The linear variational problem
4 Literature
Igor Lukacevic The variational principle
![Page 3: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/3.jpg)
The variational principle
Theory
Contents
1 Theory
2 The ground state of helium
3 The linear variational problem
4 Literature
Igor Lukacevic The variational principle
![Page 4: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/4.jpg)
The variational principle
Theory
What is a problem we would like to solve?
To find approximate solutions of eigenvalue problem
Oφ(x) = ωφ(x)
Igor Lukacevic The variational principle
![Page 5: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/5.jpg)
The variational principle
Theory
What is a problem we would like to solve?
To find approximate solutions of eigenvalue problem
Oφ(x) = ωφ(x)
A question
Can you remember any eigenvalue problems?
Igor Lukacevic The variational principle
![Page 6: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/6.jpg)
The variational principle
Theory
What is a problem we would like to solve?
To find approximate solutions of Oφ(x) = ωφ(x).
A question
Can you remember any eigenvalue problems?
Hψα = Eαψα , α = 0, 1, . . .
whereE0 ≤ E1 ≤ E2 ≤ · · · ≤ Eα ≤ · · · , 〈ψα|ψβ〉 = δαβ
Igor Lukacevic The variational principle
![Page 7: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/7.jpg)
The variational principle
Theory
Theorem - the variational principle
Given any normalized function ψ (that satisfies the appropriate boundaryconditions), then the expectation value of the Hamiltonian represents an upperbound to the exact ground state energy
〈ψ|H|ψ〉 ≥ E0 .
Igor Lukacevic The variational principle
![Page 8: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/8.jpg)
The variational principle
Theory
Theorem - the variational principle
Given any normalized function ψ (that satisfies the appropriate boundaryconditions), then the expectation value of the Hamiltonian represents an upperbound to the exact ground state energy
〈ψ|H|ψ〉 ≥ E0 .
A question
What if ψ is a ground state w.f.?
Igor Lukacevic The variational principle
![Page 9: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/9.jpg)
The variational principle
Theory
Theorem - the variational principle
Given any normalized function ψ (that satisfies the appropriate boundaryconditions), then the expectation value of the Hamiltonian represents an upperbound to the exact ground state energy
〈ψ|H|ψ〉 ≥ E0 .
A question
What if ψ is a ground state w.f.?
〈ψ|H|ψ〉 = E0
Igor Lukacevic The variational principle
![Page 10: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/10.jpg)
The variational principle
Theory
Proof
ψ are normalized ⇒ 〈ψ|ψ〉 = 1
Igor Lukacevic The variational principle
![Page 11: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/11.jpg)
The variational principle
Theory
Proof
ψ are normalized ⇒ 〈ψ|ψ〉 = 1
On the other hand, (unknown) ψ form a complete set ⇒ |ψ〉 =∑α cα|ψα〉
Igor Lukacevic The variational principle
![Page 12: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/12.jpg)
The variational principle
Theory
Proof
ψ are normalized ⇒ 〈ψ|ψ〉 = 1
On the other hand, (unknown) ψα form a complete set ⇒ |ψ〉 =∑α cα|ψα〉
So,
〈ψ|ψ〉 =⟨∑
β
cβψβ
∣∣∣∑α
cαψα⟩
=∑αβ
c∗βcα 〈ψβ |ψα〉︸ ︷︷ ︸δαβ
=∑α
|cα|2 = 1
Igor Lukacevic The variational principle
![Page 13: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/13.jpg)
The variational principle
Theory
Proof
ψ are normalized ⇒ 〈ψ|ψ〉 = 1
On the other hand, (unknown) ψα form a complete set ⇒ |ψ〉 =∑α cα|ψα〉
So,
〈ψ|ψ〉 =⟨∑
β
cβψβ
∣∣∣∑α
cαψα⟩
=∑αβ
c∗βcα 〈ψβ |ψα〉︸ ︷︷ ︸δαβ
=∑α
|cα|2 = 1
Now
〈ψ|H|ψ〉 =⟨∑
β
cβψβ
∣∣∣H∣∣∣∑α
cαψα⟩
︸ ︷︷ ︸∑α cαH|ψα〉︸ ︷︷ ︸
Eα|ψα〉
=∑αβ
c∗βcαEα 〈ψβ |ψα〉︸ ︷︷ ︸δαβ
=∑α
Eα|cα|2
Igor Lukacevic The variational principle
![Page 14: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/14.jpg)
The variational principle
Theory
Proof
ψ are normalized ⇒ 〈ψ|ψ〉 = 1
On the other hand, (unknown) ψα form a complete set ⇒ |ψ〉 =∑α cα|ψα〉
So,
〈ψ|ψ〉 =⟨∑
β
cβψβ
∣∣∣∑α
cαψα⟩
=∑αβ
c∗βcα 〈ψβ |ψα〉︸ ︷︷ ︸δαβ
=∑α
|cα|2 = 1
Now
〈ψ|H|ψ〉 =⟨∑
β
cβψβ
∣∣∣H∣∣∣∑α
cαψα⟩
︸ ︷︷ ︸∑α cαH|ψα〉︸ ︷︷ ︸
Eα|ψα〉
=∑αβ
c∗βcαEα 〈ψβ |ψα〉︸ ︷︷ ︸δαβ
=∑α
Eα|cα|2
But Eα ≥ E0 , ∀α, hence
〈ψ|H|ψ〉 ≥∑α
E0|cα|2 = E0∑α
|cα|2 = E0
Igor Lukacevic The variational principle
![Page 15: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/15.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
a] Find the ground state energy and w.f. of one-dimensional harmonic oscilator:
H = − ~2
2m∆ +
1
2mω2x2 .
Igor Lukacevic The variational principle
![Page 16: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/16.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
a] Find the ground state energy and w.f. of one-dimensional harmonic oscilator:
H = − ~2
2m∆ +
1
2mω2x2 .
How to do this using the variational principle...
(i) pick a trial function which somehow resembles the exact ground state w.f.:
ψ(x) = Ae−αx2
α parameter
A =4
√2α
π from normalization condition (do it for HW)
Igor Lukacevic The variational principle
![Page 17: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/17.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
a] Find the ground state energy and w.f. of one-dimensional harmonic oscilator:
H = − ~2
2m∆ +
1
2mω2x2 .
How to do this using the variational principle...
(i) pick a trial function which somehow resembles the exact ground state w.f.:
ψ(x) = Ae−αx2
α parameter
A =4
√2α
π from normalization condition
(ii) calculate 〈H〉 = 〈T 〉+ 〈V 〉
Igor Lukacevic The variational principle
![Page 18: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/18.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
How to do this using the variational principle...
(i) pick a trial function which somehow resembles the exact ground state w.f.:
ψ(x) = Ae−αx2
α parameter
A =4
√2α
π from normalization condition
(ii) calculate 〈H〉 = 〈T 〉+ 〈V 〉
〈T 〉 =~2α2m
〈V 〉 =mω2
8α
On how to solve these kind ofintegrals, see Ref. [5].
Igor Lukacevic The variational principle
![Page 19: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/19.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
How to do this using the variational principle...
(i) pick a trial function which somehow resembles the exact ground state w.f.:
ψ(x) = Ae−αx2
α parameter
A =4
√2α
π from normalization condition
(ii) calculate 〈H〉 = 〈T 〉+ 〈V 〉
〈T 〉 =~2α2m
〈V 〉 =mω2
8α
On how to solve these kind ofintegrals, see Ref. [5].
〈H〉 =~2α2m
+mω2
8α
Igor Lukacevic The variational principle
![Page 20: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/20.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
(iii) minimize 〈H〉 wrt parameter α
d
dα〈H〉 = 0 =⇒ α =
mω
2~
Igor Lukacevic The variational principle
![Page 21: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/21.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
(iii) minimize 〈H〉 wrt parameter α
d
dα〈H〉 = 0 =⇒ α =
mω
2~
(iv) insert back into 〈H〉 and ψ(x):
〈H〉min =1
2~ω
ψmin(x) = 4
√mω
π~e−
mω2~ x2
Igor Lukacevic The variational principle
![Page 22: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/22.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
(iii) minimize 〈H〉 wrt parameter α
d
dα〈H〉 = 0 =⇒ α =
mω
2~
(iv) insert back into 〈H〉 and ψ(x):
〈H〉min =1
2~ω
ψmin(x) = 4
√mω
π~e−
mω2~ x2
exact ground state energy and w.f.
A question
Why did we get the exact energy and w.f.?
Igor Lukacevic The variational principle
![Page 23: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/23.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
b] Do the same, but with trial function ψ(x) = Bxe−βx2
Igor Lukacevic The variational principle
![Page 24: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/24.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
b] Do the same, but with trial function ψ(x) = Bxe−βx2
(i) normalization gives B =
√2√π
(mω~
)3/4
Igor Lukacevic The variational principle
![Page 25: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/25.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
b] Do the same, but with trial function ψ(x) = Bxe−βx2
(i) normalization gives B =
√2√π
(mω~
)3/4(ii) calculate 〈H〉 =
3~3
2mβ +
3mω2
8
1
β
Igor Lukacevic The variational principle
![Page 26: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/26.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
b] Do the same, but with trial function ψ(x) = Bxe−βx2
(i) normalization gives B =
√2√π
(mω~
)3/4(ii) calculate 〈H〉 =
3~3
2mβ +
3mω2
8
1
β
(iii) minimize 〈H〉 =⇒ β =mω
2~
Igor Lukacevic The variational principle
![Page 27: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/27.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
b] Do the same, but with trial function ψ(x) = Bxe−βx2
(i) normalization gives B =
√2√π
(mω~
)3/4(ii) calculate 〈H〉 =
3~3
2mβ +
3mω2
8
1
β
(iii) minimize 〈H〉 =⇒ β =mω
2~(iv) get minimal values
〈H〉min =3
2~ω
ψmin(x) =
√2√π
(mω~
)3/4xe−
mω2~ x2
Igor Lukacevic The variational principle
![Page 28: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/28.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
b] Do the same, but with trial function ψ(x) = Bxe−βx2
(i) normalization gives B =
√2√π
(mω~
)3/4(ii) calculate 〈H〉 =
3~3
2mβ +
3mω2
8
1
β
(iii) minimize 〈H〉 =⇒ β =mω
2~(iv) get minimal values
〈H〉min =3
2~ω
ψmin(x) =
√2√π
(mω~
)3/4xe−
mω2~ x2
exact 1st excited stateenergy and w.f.
Igor Lukacevic The variational principle
![Page 29: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/29.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
In conclusion...
ψb]trial (x) = Bxe−βx2
ψa]trial (x) = ψgs
exact(x) = Ae−αx2
}=⇒
⟨ψ
b]trial (x)|ψgs
exact(x)⟩
= 0
Igor Lukacevic The variational principle
![Page 30: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/30.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
In conclusion...
ψb]trial (x) = Bxe−βx2
ψa]trial (x) = ψgs
exact(x) = Ae−αx2
}=⇒
⟨ψ
b]trial (x)|ψgs
exact(x)⟩
= 0
Also, 〈H〉b]min accounts for 1st excited state
Igor Lukacevic The variational principle
![Page 31: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/31.jpg)
The variational principle
Theory
Example: One-dimensional harmonic oscilator
In conclusion...
ψb]trial (x) = Bxe−βx2
ψa]trial (x) = ψgs
exact(x) = Ae−αx2
}=⇒
⟨ψ
b]trial (x)|ψgs
exact(x)⟩
= 0
Also, 〈H〉b]min accounts for 1st excited state
Corollary
If 〈ψ|ψgs〉 = 0, then 〈H〉 ≥ Efes , where Efes is the energy of the 1st excitedstate.
Igor Lukacevic The variational principle
![Page 32: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/32.jpg)
The variational principle
The ground state of helium
Contents
1 Theory
2 The ground state of helium
3 The linear variational problem
4 Literature
Igor Lukacevic The variational principle
![Page 33: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/33.jpg)
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4πε0
(2
r1+
2
r2− 1
|~r1 −~r2|
)
A question
What does each of these terms mean?
Igor Lukacevic The variational principle
![Page 34: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/34.jpg)
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4πε0
(2
r1+
2
r2− 1
|~r1 −~r2|
)
A question
What does each of these terms mean?
Igor Lukacevic The variational principle
![Page 35: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/35.jpg)
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4πε0
(2
r1+
2
r2− 1
|~r1 −~r2|
)
kinetic energy of electrons 1 and 2
Igor Lukacevic The variational principle
![Page 36: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/36.jpg)
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4πε0
(2
r1+
2
r2− 1
|~r1 −~r2|
)
electrostatic attraction between the nucleusand electrons 1 and 2
Igor Lukacevic The variational principle
![Page 37: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/37.jpg)
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4πε0
(2
r1+
2
r2− 1
|r1 − r2|
)
electrostatic repulsion between the electrons1 and 2
Igor Lukacevic The variational principle
![Page 38: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/38.jpg)
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4πε0
(2
r1+
2
r2− 1
|~r1 −~r2|
)
Our mission to calculate the ground state energy Egs
E expgs = −78.975 eV
Igor Lukacevic The variational principle
![Page 39: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/39.jpg)
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4πε0
(2
r1+
2
r2− 1
|~r1 −~r2|
)
A question
Can you identify the troublesome term in H?
Igor Lukacevic The variational principle
![Page 40: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/40.jpg)
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4πε0
(2
r1+
2
r2− 1
|~r1 −~r2|
)
A question
Can you identify the troublesome term in H?
Vee =e2
4πε0
1
|~r1 −~r2|
Igor Lukacevic The variational principle
![Page 41: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/41.jpg)
The variational principle
The ground state of helium
For start, let us ignore Vee
A question
Can you “guess” what happens then with H, how ψ looks like and what’s theenergy?
Igor Lukacevic The variational principle
![Page 42: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/42.jpg)
The variational principle
The ground state of helium
For start, let us ignore Vee
A question
Can you “guess” what happens then with H, how ψ looks like and what’s theenergy?
H = H1 + H2
ψ0(~r1,~r2) = ψ100(~r1)ψ100(~r2) =23
a3πe−2
r1+r2a
E0 = 8E1 = −109 eV
Igor Lukacevic The variational principle
![Page 43: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/43.jpg)
The variational principle
The ground state of helium
Now, let us account for Vee :
ψ0 99K trial w.f. (is this justifiable?)
〈H〉 = 8E1 + 〈Vee〉
〈Vee〉 =
(e2
4πε0
)(23
a3π
)2 ∫e−4
r1+r2a
|~r1 −~r2|d~r1d~r2
Igor Lukacevic The variational principle
![Page 44: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/44.jpg)
The variational principle
The ground state of helium
Now, let us account for Vee :
ψ0 99K trial w.f. (is this justifiable?)
〈H〉 = 8E1 + 〈Vee〉
〈Vee〉 =
(e2
4πε0
)(23
a3π
)2 ∫e−4
r1+r2a
|~r1 −~r2|d~r1d~r2
A question
What do you expect for 〈Vee〉 and why?
Igor Lukacevic The variational principle
![Page 45: Lecture 5 Variational Principle](https://reader034.vdocument.in/reader034/viewer/2022051215/55cf8ede550346703b96802d/html5/thumbnails/45.jpg)
The variational principle
The ground state of helium
Now, let us account for Vee :
ψ0 99K trial w.f. (is this justifiable?)
〈H〉 = 8E1 + 〈Vee〉
〈Vee〉 =
(e2
4πε0
)(23
a3π
)2 ∫e−4
r1+r2a
|~r1 −~r2|d~r1d~r2 = 34 eV
HW
Calculate〈Vee〉 usingRefs. [2] and[5].
Igor Lukacevic The variational principle
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The variational principle
The ground state of helium
Now, let us account for Vee :
ψ0 99K trial w.f. (is this justifiable?)
〈H〉 = 8E1 + 〈Vee〉
〈Vee〉 =
(e2
4πε0
)(23
a3π
)2 ∫e−4
r1+r2a
|~r1 −~r2|d~r1d~r2 = 34 eV
〈H〉 = −109 eV + 34 eV = −75 eV
E expgs = −79 eV
Rel. error 5.1%
Igor Lukacevic The variational principle
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The variational principle
The ground state of helium
Zeff effective nuclearcharge
Trial w.f.
ψ1(~r1,~r2) =Z 3
eff
a3πe−Zeff
r1+r2a
Igor Lukacevic The variational principle
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The variational principle
The ground state of helium
Zeff effective nuclearcharge
Trial w.f.
ψ1(~r1,~r2) =Z3
eff
a3πe−Zeff
r1+r2a
Zeff - variationalparameter
Igor Lukacevic The variational principle
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The variational principle
The ground state of helium
Let us rewrite the Hamiltonian:
H = − ~2
2m(∆1 + ∆2)− e2
4πε0
(Zeff
r1+
Zeff
r2
)+
e2
4πε0
[Zeff − 2
r1+
Zeff − 2
r2− 1
|~r1 −~r2|
]
Igor Lukacevic The variational principle
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The variational principle
The ground state of helium
Let us rewrite the Hamiltonian:
H = − ~2
2m(∆1 + ∆2)− e2
4πε0
(Zeff
r1+
Zeff
r2
)+
e2
4πε0
[Zeff − 2
r1+
Zeff − 2
r2− 1
|~r1 −~r2|
]
=⇒ 〈H〉 =
[−2Z 2
eff +27
4Zeff
]E1
For calculation details, see Ref. [2].
Igor Lukacevic The variational principle
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The variational principle
The ground state of helium
Let us rewrite the Hamiltonian:
H = − ~2
2m(∆1 + ∆2)− e2
4πε0
(Zeff
r1+
Zeff
r2
)+
e2
4πε0
[Zeff − 2
r1+
Zeff − 2
r2− 1
|~r1 −~r2|
]
=⇒ 〈H〉 =
[−2Z 2
eff +27
4Zeff
]E1
Now minimizing 〈H〉 we getZmin
eff = 1.69
Igor Lukacevic The variational principle
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The variational principle
The ground state of helium
Let us rewrite the Hamiltonian:
H = − ~2
2m(∆1 + ∆2)− e2
4πε0
(Zeff
r1+
Zeff
r2
)+
e2
4πε0
[Zeff − 2
r1+
Zeff − 2
r2− 1
|~r1 −~r2|
]
=⇒ 〈H〉 =
[−2Z 2
eff +27
4Zeff
]E1
Now minimizing 〈H〉 we getZmin
eff = 1.69
Which gives
〈H〉min = Emin = −77.5 eV ,Egs − Emin
Egs= 1.87%
Note:For more precise results see E. A. Hylleraas, Z. Phys. 65, 209 (1930) or C. L. Pekeris,
Phys. Rev. 115, 1216 (1959).
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
Contents
1 Theory
2 The ground state of helium
3 The linear variational problem
4 Literature
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
ψ normalized trial function depends on α1, α2 . . .
〈ψ|H|ψ〉 very complex function of α1, α2 . . .
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
ψ normalized trial function depends on α1, α2 . . .
〈ψ|H|ψ〉 very complex function of α1, α2 . . .
Suppose
|ψ〉 =N∑
i=1
ci |ψi 〉 , 〈ψi |ψj〉 = δij
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
ψ normalized trial function depends on α1, α2 . . .
〈ψ|H|ψ〉 very complex function of α1, α2 . . .
Suppose
|ψ〉 =N∑
i=1
ci |ψi 〉 , 〈ψi |ψj〉 = δij
=⇒ (H)ij = Hij = 〈ψi |H|ψj〉 matrix representation in basis {|ψi 〉}
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
ψ normalized trial function depends on α1, α2 . . .
〈ψ|H|ψ〉 very complex function of α1, α2 . . .
Suppose
|ψ〉 =N∑
i=1
ci |ψi 〉 , 〈ψi |ψj〉 = δij
=⇒ (H)ij = Hij = 〈ψi |H|ψj〉 matrix representation in basis {|ψi 〉}
H hermitian{|ψi 〉} real
}⇒ H symmetric
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
ψ normalized trial function depends on α1, α2 . . .
〈ψ|H|ψ〉 very complex function of α1, α2 . . .
Suppose
|ψ〉 =N∑
i=1
ci |ψi 〉 , 〈ψi |ψj〉 = δij
=⇒ (H)ij = Hij = 〈ψi |H|ψj〉 matrix representation in basis {|ψi 〉}
H hermitian{|ψi 〉} real
}⇒ H symmetric
ψ normalized ⇒∑
i
c2i = 1
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
ψ normalized trial function depends on α1, α2 . . .
〈ψ|H|ψ〉 very complex function of α1, α2 . . .
Suppose
|ψ〉 =N∑
i=1
ci |ψi 〉 , 〈ψi |ψj〉 = δij
=⇒ (H)ij = Hij = 〈ψi |H|ψj〉 matrix representation in basis {|ψi 〉}
H hermitian{|ψi 〉} real
}⇒ H symmetric
ψ normalized ⇒∑
i
c2i = 1
the expectation value depends on cij :
=⇒ 〈ψ|H|ψ〉 =∑
ij
cijHij
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
ψ normalized ⇒∑
i
c2i = 1
the expectation value depends on cij :
=⇒ 〈ψ|H|ψ〉 =∑
ij
cijHij
Unfortunately,∂
∂ck〈ψ|H|ψ〉 = 0 , k = 1, 2, . . . ,N
is unsolvable for ck are mutually dependent.
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
ψ normalized ⇒∑
i
c2i = 1
the expectation value depends on cij :
=⇒ 〈ψ|H|ψ〉 =∑
ij
cijHij
Unfortunately,∂
∂ck〈ψ|H|ψ〉 = 0 , k = 1, 2, . . . ,N
is unsolvable for ck are mutually dependent.
Lagrange’s method of undetermined multipliers
L(c1, . . . , cN ,E) = 〈ψ|H|ψ〉 − E(〈ψ|ψ〉 − 1
)=∑
ij
cicjHij − E
(∑i
c2i − 1
)
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
Unfortunately,∂
∂ck〈ψ|H|ψ〉 = 0 , k = 1, 2, . . . ,N
is unsolvable for ck are mutually dependent.
Lagrange’s method of undetermined multipliers
L(c1, . . . , cN ,E) = 〈ψ|H|ψ〉 − E(〈ψ|ψ〉 − 1
)=∑
ij
cicjHij − E
(∑i
c2i − 1
)
〈ψ|H|ψ〉 and L are minimal for same ck
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
Let us now choose c1, c2, . . . , cN−1 as independent
⇒ cN is given by∑
i
c2i = 1
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
Let us now choose c1, c2, . . . , cN−1 as independent
⇒ cN is given by∑
i
c2i = 1
Then we have∂L∂ck
= 0 , k = 1, 2, . . . ,N − 1
but not necessarily∂L∂cN
= 0
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
Let us now choose c1, c2, . . . , cN−1 as independent
⇒ cN is given by∑
i
c2i = 1
Then we have∂L∂ck
= 0 , k = 1, 2, . . . ,N − 1
but not necessarily∂L∂cN
= 0
But we still have undetermined multiplier E , so now we choose it so that
∂L∂ck
= 0 , k = 1, 2, . . . ,N − 1,N
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
Let us now choose c1, c2, . . . , cN−1 as independent
⇒ cN is given by∑
i
c2i = 1
Then we have∂L∂ck
= 0 , k = 1, 2, . . . ,N − 1
but not necessarily∂L∂cN
= 0
But we still have undetermined multiplier E , so now we choose it so that
∂L∂ck
= 0 , k = 1, 2, . . . ,N − 1,N
On the other hand
∂L∂ck
=∑
j
cjHkj +∑
i
ciHik − 2Eck
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
Then we have∂L∂ck
= 0 , k = 1, 2, . . . ,N − 1
but not necessarily∂L∂cN
= 0
But we still have undetermined multiplier E , so now we choose it so that
∂L∂ck
= 0 , k = 1, 2, . . . ,N − 1,N
On the other hand
∂L∂ck
=∑
j
cjHkj +∑
i
ciHik︸ ︷︷ ︸equal, since Hij=Hji
−2Eck
So, ∑j
Hijcj − Eci = 0
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
Or in matrix formHc = Ec
A question
What represents this equation?
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
Or in matrix formHc = Ec
⇒ Hcα = Eαcα , α = 0, 1, . . . ,N − 1 , (cα)†cβ =∑
i
cαi cβi = δαβ
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
Or in matrix formHc = Ec
⇒ Hcα = Eαcα , α = 0, 1, . . . ,N − 1 , (cα)†cβ =∑
i
cαi cβi = δαβ
Eαβ = Eαδαβ , Ciα = cαi =⇒ HC = EC
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
Or in matrix formHc = Ec
⇒ Hcα = Eαcα , α = 0, 1, . . . ,N − 1 , (cα)†cβ =∑
i
cαi cβi = δαβ
Eαβ = Eαδαβ , Ciα = cαi =⇒ HC = EC
Solving gives N orthonormal solutions
|ψα〉 =N∑
i=1
cαi |ψi 〉 , α = 0, 1, . . . ,N − 1
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
Or in matrix formHc = Ec
⇒ Hcα = Eαcα , α = 0, 1, . . . ,N − 1 , (cα)†cβ =∑
i
cαi cβi = δαβ
Eαβ = Eαδαβ , Ciα = cαi =⇒ HC = EC
Solving gives N orthonormal solutions
|ψα〉 =N∑
i=1
cαi |ψi 〉 , α = 0, 1, . . . ,N − 1
What about E ’s:〈ψβ |H|ψα〉 = Eαδαβ
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
HC = EC
Solving gives N orthonormal solutions
|ψα〉 =N∑
i=1
cαi |ψi 〉 , α = 0, 1, . . . ,N − 1
What about E ’s:〈ψβ |H|ψα〉 = Eαδαβ
For example,E0 = 〈ψ0|H|ψ0〉 ≥ E0
A question
What’s the meaning of other E ’s?
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
HC = EC
Solving gives N orthonormal solutions
|ψα〉 =N∑
i=1
cαi |ψi 〉 , α = 0, 1, . . . ,N − 1
What about E ’s:〈ψβ |H|ψα〉 = Eαδαβ
For example,E0 = 〈ψ0|H|ψ0〉 ≥ E0
A question
What’s the meaning of other E ’s? Eα ≥ Eα , α = 1, 2, . . .
Igor Lukacevic The variational principle
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The variational principle
The linear variational problem
In conclusion
Solving the matrix eigenvalue problem
HC = EC ,
by diagonalization, is equivalent to the variational principle in a subspacespanned by {|ψi 〉 , i = 1, 2, . . . ,N}.
Igor Lukacevic The variational principle
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The variational principle
Literature
Contents
1 Theory
2 The ground state of helium
3 The linear variational problem
4 Literature
Igor Lukacevic The variational principle
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The variational principle
Literature
Literature
1 A. Szabo, N. Ostlund, Modern Quantum Chemistry, Introduction toAdvanced Electronic Structure theory, Dover Publications, New York,1996.
2 D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., PearsonEducation, Inc., Upper Saddle River, NJ, 2005.
3 I. Supek, Teorijska fizika i struktura materije, II. dio, Skolska knjiga,Zagreb, 1989.
4 Y. Peleg, R. Pnini, E. Zaarur, Shaum’s Outline of Theory and Problems ofQuantum Mechanics, McGraw-Hill, 1998.
5 I. N. Bronstejn, K. A. Semendjajev, Matematicki prirucnik, Tehnickaknjiga, Zagreb, 1991.
Igor Lukacevic The variational principle