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The variational principle

The variational principle Quantum 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, . . .

where E0 ≤ 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 boundary conditions), then the expectation value of the Hamiltonian represents an upper bound 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 boundary conditions), then the expectation value of the Hamiltonian represents an upper bound 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 boundary conditions), then the expectation value of the Hamiltonian represents an upper bound 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〉 = 1 On 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〉 = 1 On 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〉 = 1 On 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〉 = 1 On 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

2 mω2x2 .

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

2 mω2x2 .

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−αx 2

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

2 mω2x2 .

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−αx 2

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−αx 2

α parameter

A = 4

√ 2α

π from normalization condition

(ii) calculate 〈H〉 = 〈T 〉+ 〈V 〉

〈T 〉 = ~ 2α

2m

〈V 〉 = mω 2

8α

On how to solve these kind of integrals, 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−αx 2

α parameter

A = 4

√ 2α

π from normalization condition

(ii) calculate 〈H〉 = 〈T 〉+ 〈V 〉

〈T 〉 = ~ 2α

2m

〈V 〉 = mω 2

8α

On how to solve these kind of integrals, 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−

mω 2~ 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−

mω 2~ x

2

exact ground state energy and w.f.

A question

Why did we get the exact energy and w.f.?