The variational principle
The variational principleQuantum mechanics 2 - Lecture 5
Igor Lukacevic
UJJS, Dept. of Physics, Osijek
November 5, 2012
Igor Lukacevic The variational principle
The variational principle
Contents
1 Theory
2 The ground state of helium
3 Literature
Igor Lukacevic The variational principle
The variational principle
Theory
Contents
1 Theory
2 The ground state of helium
3 Literature
Igor Lukacevic 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 Lukacevic 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 Lukacevic 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 Lukacevic The variational principle
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
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
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
The variational principle
Theory
Proof
ψ are normalized ⇒ 〈ψ|ψ〉 = 1
Igor Lukacevic The variational principle
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
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
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
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
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
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
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
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
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
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
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
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− mω
2~ 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
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
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
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
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
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
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
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
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
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
The variational principle
The ground state of helium
Contents
1 Theory
2 The ground state of helium
3 Literature
Igor Lukacevic The variational principle
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Igor Lukacevic The variational principle
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
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
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
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
Igor Lukacevic The variational principle
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
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
The variational principle
Literature
Contents
1 Theory
2 The ground state of helium
3 Literature
Igor Lukacevic 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, 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