The variational principle - unios.hr...The variational principle The variational principle Quantum...
Transcript of The variational principle - unios.hr...The variational principle The variational principle Quantum...
-
The variational principle
The variational principleQuantum mechanics 2 - Lecture 5
Igor Lukačević
UJJS, Dept. of Physics, Osijek
November 8, 2012
Igor Lukačević The variational principle
-
The variational principle
Contents
1 Theory
2 The ground state of helium
3 The linear variational problem
4 Literature
Igor Lukačević The variational principle
-
The variational principle
Theory
Contents
1 Theory
2 The ground state of helium
3 The linear variational problem
4 Literature
Igor Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 Lukačević The variational principle
-
The variational principle
Theory
Theorem - the variational principle
Given any normalized function ψ̃ (that satisfies the appropriate boundaryconditions), then the expectation value of the Hamiltonian represents an upperbound to the exact ground state energy
〈ψ̃|H|ψ̃〉 ≥ E0 .
Igor Lukačević The variational principle
-
The variational principle
Theory
Theorem - the variational principle
Given any normalized function ψ̃ (that satisfies the appropriate boundaryconditions), then the expectation value of the Hamiltonian represents an upperbound to the exact ground state energy
〈ψ̃|H|ψ̃〉 ≥ E0 .
A question
What if ψ̃ is a ground state w.f.?
Igor Lukačević The variational principle
-
The variational principle
Theory
Theorem - the variational principle
Given any normalized function ψ̃ (that satisfies the appropriate boundaryconditions), then the expectation value of the Hamiltonian represents an upperbound to the exact ground state energy
〈ψ̃|H|ψ̃〉 ≥ E0 .
A question
What if ψ̃ is a ground state w.f.?
〈ψ̃|H|ψ̃〉 = E0
Igor Lukačević The variational principle
-
The variational principle
Theory
Proof
ψ̃ are normalized ⇒ 〈ψ̃|ψ̃〉 = 1
Igor Lukačević The variational principle
-
The variational principle
Theory
Proof
ψ̃ are normalized ⇒ 〈ψ̃|ψ̃〉 = 1On the other hand, (unknown) ψ form a complete set ⇒ |ψ̃〉 =
∑α cα|ψα〉
Igor Lukačević The variational principle
-
The variational principle
Theory
Proof
ψ̃ are normalized ⇒ 〈ψ̃|ψ̃〉 = 1On the other hand, (unknown) ψα form a complete set ⇒ |ψ̃〉 =
∑α cα|ψα〉
So,
〈ψ̃|ψ̃〉 =〈∑
β
cβψβ
∣∣∣∑α
cαψα〉
=∑αβ
c∗βcα 〈ψβ |ψα〉︸ ︷︷ ︸δαβ
=∑α
|cα|2 = 1
Igor Lukačević The variational principle
-
The variational principle
Theory
Proof
ψ̃ are normalized ⇒ 〈ψ̃|ψ̃〉 = 1On the other hand, (unknown) ψα form a complete set ⇒ |ψ̃〉 =
∑α cα|ψα〉
So,
〈ψ̃|ψ̃〉 =〈∑
β
cβψβ
∣∣∣∑α
cαψα〉
=∑αβ
c∗βcα 〈ψβ |ψα〉︸ ︷︷ ︸δαβ
=∑α
|cα|2 = 1
Now
〈ψ̃|H|ψ̃〉 =〈∑
β
cβψβ
∣∣∣H∣∣∣∑α
cαψα〉
︸ ︷︷ ︸∑α cαH|ψα〉︸ ︷︷ ︸
Eα|ψα〉
=∑αβ
c∗βcαEα 〈ψβ |ψα〉︸ ︷︷ ︸δαβ
=∑α
Eα|cα|2
Igor Lukačević The variational principle
-
The variational principle
Theory
Proof
ψ̃ are normalized ⇒ 〈ψ̃|ψ̃〉 = 1On the other hand, (unknown) ψα form a complete set ⇒ |ψ̃〉 =
∑α cα|ψα〉
So,
〈ψ̃|ψ̃〉 =〈∑
β
cβψβ
∣∣∣∑α
cαψα〉
=∑αβ
c∗βcα 〈ψβ |ψα〉︸ ︷︷ ︸δαβ
=∑α
|cα|2 = 1
Now
〈ψ̃|H|ψ̃〉 =〈∑
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
〈ψ̃|H|ψ̃〉 ≥∑α
E0|cα|2 = E0∑α
|cα|2 = E0
Igor Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 〉 = ~2α
2m
〈V 〉 = mω2
8α
On how to solve these kind ofintegrals, see Ref. [5].
Igor Lukačević The variational principle
-
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 〉 = ~2α
2m
〈V 〉 = mω2
8α
On how to solve these kind ofintegrals, see Ref. [5].
〈H〉 = ~2α
2m+
mω2
8α
Igor Lukačević The variational principle
-
The variational principle
Theory
Example: One-dimensional harmonic oscilator
(iii) minimize 〈H〉 wrt parameter α
d
dα〈H〉 = 0 =⇒ α = mω
2~
Igor Lukačević The variational principle
-
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~ x
2
Igor Lukačević The variational principle
-
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~ x
2
exact ground state energy and w.f.
A question
Why did we get the exact energy and w.f.?
Igor Lukačević The variational principle
-
The variational principle
Theory
Example: One-dimensional harmonic oscilator
b] Do the same, but with trial function ψ(x) = Bxe−βx2
Igor Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 = mω2~
Igor Lukačević The variational principle
-
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 = mω2~
(iv) get minimal values
〈H〉min =3
2~ω
ψmin(x) =
√2√π
(mω~
)3/4xe−
mω2~ x
2
Igor Lukačević The variational principle
-
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 = mω2~
(iv) get minimal values
〈H〉min =3
2~ω
ψmin(x) =
√2√π
(mω~
)3/4xe−
mω2~ x
2
exact 1st excited stateenergy and w.f.
Igor Lukačević The variational principle
-
The variational principle
Theory
Example: One-dimensional harmonic oscilator
In conclusion...
ψb]trial (x) = Bxe
−βx2
ψa]trial (x) = ψ
gsexact(x) = Ae
−αx2
}=⇒
〈ψ
b]trial (x)|ψ
gsexact(x)
〉= 0
Igor Lukačević The variational principle
-
The variational principle
Theory
Example: One-dimensional harmonic oscilator
In conclusion...
ψb]trial (x) = Bxe
−βx2
ψa]trial (x) = ψ
gsexact(x) = Ae
−αx2
}=⇒
〈ψ
b]trial (x)|ψ
gsexact(x)
〉= 0
Also, 〈H〉b]min accounts for 1st excited state
Igor Lukačević The variational principle
-
The variational principle
Theory
Example: One-dimensional harmonic oscilator
In conclusion...
ψb]trial (x) = Bxe
−βx2
ψa]trial (x) = ψ
gsexact(x) = Ae
−αx2
}=⇒
〈ψ
b]trial (x)|ψ
gsexact(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 Lukačević The variational principle
-
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 Lukačević The variational principle
-
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4π�0
(2
r1+
2
r2− 1|~r1 −~r2|
)
A question
What does each of these terms mean?
Igor Lukačević The variational principle
-
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4π�0
(2
r1+
2
r2− 1|~r1 −~r2|
)
A question
What does each of these terms mean?
Igor Lukačević The variational principle
-
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4π�0
(2
r1+
2
r2− 1|~r1 −~r2|
)
kinetic energy of electrons 1 and 2
Igor Lukačević The variational principle
-
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4π�0
(2
r1+
2
r2− 1|~r1 −~r2|
)
electrostatic attraction between the nucleusand electrons 1 and 2
Igor Lukačević The variational principle
-
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4π�0
(2
r1+
2
r2− 1|̃r1 − r̃2|
)
electrostatic repulsion between the electrons1 and 2
Igor Lukačević The variational principle
-
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4π�0
(2
r1+
2
r2− 1|~r1 −~r2|
)
Our mission to calculate the ground state energy Egs
E expgs = −78.975 eV
Igor Lukačević The variational principle
-
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4π�0
(2
r1+
2
r2− 1|~r1 −~r2|
)
A question
Can you identify the troublesome term in H?
Igor Lukačević The variational principle
-
The variational principle
The ground state of helium
H = − ~2
2m(∆1 + ∆2)
− e2
4π�0
(2
r1+
2
r2− 1|~r1 −~r2|
)
A question
Can you identify the troublesome term in H?
Vee =e2
4π�0
1
|~r1 −~r2|
Igor Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 Lukačević The variational principle
-
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
4π�0
)(23
a3π
)2 ∫e−4
r1+r2a
|~r1 −~r2|d~r1d~r2
Igor Lukačević The variational principle
-
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
4π�0
)(23
a3π
)2 ∫e−4
r1+r2a
|~r1 −~r2|d~r1d~r2
A question
What do you expect for 〈Vee〉 and why?
Igor Lukačević The variational principle
-
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
4π�0
)(23
a3π
)2 ∫e−4
r1+r2a
|~r1 −~r2|d~r1d~r2 = 34 eV
HW
Calculate〈Vee〉 usingRefs. [2] and[5].
Igor Lukačević The variational principle
-
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
4π�0
)(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 Lukačević The variational principle
-
The variational principle
The ground state of helium
Zeff effective nuclearcharge
Trial w.f.
ψ1(~r1,~r2) =Z 3effa3π
e−Zeffr1+r2
a
Igor Lukačević The variational principle
-
The variational principle
The ground state of helium
Zeff effective nuclearcharge
Trial w.f.
ψ1(~r1,~r2) =Z3effa3π
e−Zeffr1+r2
a
Zeff - variationalparameter
Igor Lukačević The variational principle
-
The variational principle
The ground state of helium
Let us rewrite the Hamiltonian:
H = − ~2
2m(∆1 + ∆2)−
e2
4π�0
(Zeffr1
+Zeffr2
)+
e2
4π�0
[Zeff − 2
r1+
Zeff − 2r2
− 1|~r1 −~r2|
]
Igor Lukačević The variational principle
-
The variational principle
The ground state of helium
Let us rewrite the Hamiltonian:
H = − ~2
2m(∆1 + ∆2)−
e2
4π�0
(Zeffr1
+Zeffr2
)+
e2
4π�0
[Zeff − 2
r1+
Zeff − 2r2
− 1|~r1 −~r2|
]
=⇒ 〈H〉 =[−2Z 2eff +
27
4Zeff
]E1
For calculation details, see Ref. [2].
Igor Lukačević The variational principle
-
The variational principle
The ground state of helium
Let us rewrite the Hamiltonian:
H = − ~2
2m(∆1 + ∆2)−
e2
4π�0
(Zeffr1
+Zeffr2
)+
e2
4π�0
[Zeff − 2
r1+
Zeff − 2r2
− 1|~r1 −~r2|
]
=⇒ 〈H〉 =[−2Z 2eff +
27
4Zeff
]E1
Now minimizing 〈H〉 we getZmineff = 1.69
Igor Lukačević The variational principle
-
The variational principle
The ground state of helium
Let us rewrite the Hamiltonian:
H = − ~2
2m(∆1 + ∆2)−
e2
4π�0
(Zeffr1
+Zeffr2
)+
e2
4π�0
[Zeff − 2
r1+
Zeff − 2r2
− 1|~r1 −~r2|
]
=⇒ 〈H〉 =[−2Z 2eff +
27
4Zeff
]E1
Now minimizing 〈H〉 we getZmineff = 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 Lukačević The variational principle
-
The variational principle
The linear variational problem
Contents
1 Theory
2 The ground state of helium
3 The linear variational problem
4 Literature
Igor Lukačević The variational principle
-
The variational principle
The linear variational problem
ψ̃ normalized trial function depends on α1, α2 . . .
〈ψ̃|H|ψ̃〉 very complex function of α1, α2 . . .
Igor Lukačević The variational principle
-
The variational principle
The linear variational problem
ψ̃ normalized trial function depends on α1, α2 . . .
〈ψ̃|H|ψ̃〉 very complex function of α1, α2 . . .
Suppose
|ψ̃〉 =N∑
i=1
ci |ψi 〉 , 〈ψi |ψj〉 = δij
Igor Lukačević The variational principle
-
The variational principle
The linear variational problem
ψ̃ normalized trial function depends on α1, α2 . . .
〈ψ̃|H|ψ̃〉 very complex function of α1, α2 . . .
Suppose
|ψ̃〉 =N∑
i=1
ci |ψi 〉 , 〈ψi |ψj〉 = δij
=⇒ (H)ij = Hij = 〈ψi |H|ψj〉 matrix representation in basis {|ψi 〉}
Igor Lukačević The variational principle
-
The variational principle
The linear variational problem
ψ̃ normalized trial function depends on α1, α2 . . .
〈ψ̃|H|ψ̃〉 very complex function of α1, α2 . . .
Suppose
|ψ̃〉 =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 Lukačević The variational principle
-
The variational principle
The linear variational problem
ψ̃ normalized trial function depends on α1, α2 . . .
〈ψ̃|H|ψ̃〉 very complex function of α1, α2 . . .
Suppose
|ψ̃〉 =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
ψ̃ normalized ⇒∑
i
c2i = 1
Igor Lukačević The variational principle
-
The variational principle
The linear variational problem
ψ̃ normalized trial function depends on α1, α2 . . .
〈ψ̃|H|ψ̃〉 very complex function of α1, α2 . . .
Suppose
|ψ̃〉 =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
ψ̃ normalized ⇒∑
i
c2i = 1
the expectation value depends on cij :
=⇒ 〈ψ̃|H|ψ̃〉 =∑
ij
cijHij
Igor Lukačević The variational principle
-
The variational principle
The linear variational problem
ψ̃ normalized ⇒∑
i
c2i = 1
the expectation value depends on cij :
=⇒ 〈ψ̃|H|ψ̃〉 =∑
ij
cijHij
Unfortunately,∂
∂ck〈ψ̃|H|ψ̃〉 = 0 , k = 1, 2, . . . ,N
is unsolvable for ck are mutually dependent.
Igor Lukačević The variational principle
-
The variational principle
The linear variational problem
ψ̃ normalized ⇒∑
i
c2i = 1
the expectation value depends on cij :
=⇒ 〈ψ̃|H|ψ̃〉 =∑
ij
cijHij
Unfortunately,∂
∂ck〈ψ̃|H|ψ̃〉 = 0 , k = 1, 2, . . . ,N
is unsolvable for ck are mutually dependent.
Lagrange’s method of undetermined multipliers
L(c1, . . . , cN ,E) = 〈ψ̃|H|ψ̃〉 − E(〈ψ̃|ψ̃〉 − 1
)=∑
ij
cicjHij − E
(∑i
c2i − 1
)
Igor Lukačević The variational principle
-
The variational principle
The linear variational problem
Unfortunately,∂
∂ck〈ψ̃|H|ψ̃〉 = 0 , k = 1, 2, . . . ,N
is unsolvable for ck are mutually dependent.
Lagrange’s method of undetermined multipliers
L(c1, . . . , cN ,E) = 〈ψ̃|H|ψ̃〉 − E(〈ψ̃|ψ̃〉 − 1
)=∑
ij
cicjHij − E
(∑i
c2i − 1
)
〈ψ̃|H|ψ̃〉 and L are minimal for same ck
Igor Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 Lukačević The variational principle
-
The variational principle
The linear variational problem
Or in matrix formHc = Ec
A question
What represents this equation?
Igor Lukačević The variational principle
-
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 Lukačević The variational principle
-
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 Lukačević The variational principle
-
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
|ψ̃α〉 =N∑
i=1
cαi |ψi 〉 , α = 0, 1, . . . ,N − 1
Igor Lukačević The variational principle
-
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
|ψ̃α〉 =N∑
i=1
cαi |ψi 〉 , α = 0, 1, . . . ,N − 1
What about E ’s:〈ψ̃β |H|ψ̃α〉 = Eαδαβ
Igor Lukačević The variational principle
-
The variational principle
The linear variational problem
HC = EC
Solving gives N orthonormal solutions
|ψ̃α〉 =N∑
i=1
cαi |ψi 〉 , α = 0, 1, . . . ,N − 1
What about E ’s:〈ψ̃β |H|ψ̃α〉 = Eαδαβ
For example,E0 = 〈ψ̃0|H|ψ̃0〉 ≥ E0
A question
What’s the meaning of other E ’s?
Igor Lukačević The variational principle
-
The variational principle
The linear variational problem
HC = EC
Solving gives N orthonormal solutions
|ψ̃α〉 =N∑
i=1
cαi |ψi 〉 , α = 0, 1, . . . ,N − 1
What about E ’s:〈ψ̃β |H|ψ̃α〉 = Eαδαβ
For example,E0 = 〈ψ̃0|H|ψ̃0〉 ≥ E0
A question
What’s the meaning of other E ’s? Eα ≥ Eα , α = 1, 2, . . .
Igor Lukačević The variational principle
-
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 Lukačević The variational principle
-
The variational principle
Literature
Contents
1 Theory
2 The ground state of helium
3 The linear variational problem
4 Literature
Igor Lukačević The variational principle
-
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, Školska 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. Bronštejn, K. A. Semendjajev, Matematički priručnik, Tehničkaknjiga, Zagreb, 1991.
Igor Lukačević The variational principle
TheoryThe ground state of heliumThe linear variational problemLiterature