Identical particles - Quantum mechanics 2 - Lecture...

Identical particles

Identical particlesQuantum mechanics 2 - Lecture 1

Igor Lukacevic

UJJS, Dept. of Physics, Osijek

14. prosinca 2011.

Igor Lukacevic Identical particles

Identical particles

Contents

1 Two-particle wave equation

2 Symmetric and antisymmetric solutions

3 Example: He spectrum

4 Antisymmetry principle

5 Example: H molecule

6 Literature

Igor Lukacevic Identical particles

Identical particles

Two-particle wave equation

Contents

1 Two-particle wave equation

2 Symmetric and antisymmetric solutions

3 Example: He spectrum

4 Antisymmetry principle

5 Example: H molecule

6 Literature

Igor Lukacevic Identical particles

Identical particles

Two-particle wave equation

Which interactions exist here and what istheir nature?

Igor Lukacevic Identical particles

Identical particles

Two-particle wave equation

H =1

2m1p21 +

1

2m2p22 + V (r1, r2)

Igor Lukacevic Identical particles

Identical particles

Two-particle wave equation

H =1

2m1p21 +

1

2m2p22 + V (r1, r2)

pk →~i∇k

Igor Lukacevic Identical particles

Identical particles

Two-particle wave equation

H =1

2m1p21 +

1

2m2p22 + V (r1, r2)

pk →~i∇k

− ~2m1

∆1Ψ− ~2m2

∆2Ψ + V (r1, r2)Ψ = EΨ

Igor Lukacevic Identical particles

Identical particles

Two-particle wave equation

H =1

2m1p21 +

1

2m2p22 + V (r1, r2)

pk →~i∇k

− ~2m1

∆1Ψ− ~2m2

∆2Ψ + V (r1, r2)Ψ = EΨ

Ψ = Ψ(x1, y1, z1, x2, y2, z2) visualization is lost

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

Contents

1 Two-particle wave equation

2 Symmetric and antisymmetric solutions

3 Example: He spectrum

4 Antisymmetry principle

5 Example: H molecule

6 Literature

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

Assumption: no interaction!

electron 1 electron 2

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

− ~2m1

∆1Ψ +V1(r1)Ψ− ~2m2

∆2Ψ +V2(r2)Ψ = EΨ Ψ(r1, r2) = u(r1)v(r2)

@@@R

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Separationofvariables

+r1 → 1r2 → 2

?

− ~2m1

∆1u + V1(1)u = E1u

− ~2m2

∆2v + V2(2)v = E2v

}99K E1 + E2 = E

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

No interaction:

probability density Ψ∗Ψ = u(1)∗v(2)∗u(1)v(2) = u(1)∗u(1)v(2)∗v(2)

particles do not correlate

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

Assumption: allow interaction!

nucleus electrons

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

Assumption: allow interaction!

1 e-e: V12 =e2

r12, r12 =

√(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2

2 n-e: V1 = −2e2

r1, V2 = −

2e2

r2

3 total: V = −2e2

r1−

2e2

r2+

e2

r12

S.E. ⇒ − ~2

2m(∆1 + ∆2)Ψ + VΨ = EΨ

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

Assumption: allow interaction!

1 e-e: V12 =e2

r12, r12 =

√(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2

2 n-e: V1 = −2e2

r1, V2 = −

2e2

r2

3 total: V = −2e2

r1−

2e2

r2+

e2

r12

S.E. ⇒ − ~2

2m(∆1 + ∆2)Ψ + VΨ = EΨ

A question

What happens with S.E. if we interchange the coordinates of particles?

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

Assumption: allow interaction!

1 e-e: V12 =e2

r12, r12 =

√(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2

2 n-e: V1 = −2e2

r1, V2 = −

2e2

r2

3 total: V = −2e2

r1−

2e2

r2+

e2

r12

S.E. ⇒ − ~2

2m(∆1 + ∆2)Ψ + VΨ = EΨ

A question

What happens with S.E. if we interchange the coordinates of particles?S.E. is symmetrical wrt that interchange!

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

Assumption: allow interaction!

1 e-e: V12 =e2

r12, r12 =

√(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2

2 n-e: V1 = −2e2

r1, V2 = −

2e2

r2

3 total: V = −2e2

r1−

2e2

r2+

e2

r12

S.E. ⇒ − ~2

2m(∆1 + ∆2)Ψ + VΨ = EΨ

Second question

What about its solutions then?

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

Assumption: allow interaction!

1 e-e: V12 =e2

r12, r12 =

√(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2

2 n-e: V1 = −2e2

r1, V2 = −

2e2

r2

3 total: V = −2e2

r1−

2e2

r2+

e2

r12

S.E. ⇒ − ~2

2m(∆1 + ∆2)Ψ + VΨ = EΨ

Second question

What about its solutions then?

Ψ(1, 2)

Ψ(2, 1)

Ψ = c1Ψ(1, 2) + c2Ψ(2, 1)

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

Assumption: allow interaction!

1 e-e: V12 =e2

r12, r12 =

√(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2

2 n-e: V1 = −2e2

r1, V2 = −

2e2

r2

3 total: V = −2e2

r1−

2e2

r2+

e2

r12

S.E. ⇒ − ~2

2m(∆1 + ∆2)Ψ + VΨ = EΨ

Second question

What about its solutions then?

Ψ(1, 2)

Ψ(2, 1)

Ψ = c1Ψ(1, 2) + c2Ψ(2, 1)

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

Assumption: allow interaction!

1 e-e: V12 =e2

r12, r12 =

√(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2

2 n-e: V1 = −2e2

r1, V2 = −

2e2

r2

3 total: V = −2e2

r1−

2e2

r2+

e2

r12

S.E. ⇒ − ~2

2m(∆1 + ∆2)Ψ + VΨ = EΨ

Second question

What about its solutions then?

Ψ(1, 2)

Ψ(2, 1)

Ψ = c1Ψ(1, 2) + c2Ψ(2, 1)

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

Assumption: allow interaction!

1 e-e: V12 =e2

r12, r12 =

√(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2

2 n-e: V1 = −2e2

r1, V2 = −

2e2

r2

3 total: V = −2e2

r1−

2e2

r2+

e2

r12

S.E. ⇒ − ~2

2m(∆1 + ∆2)Ψ + VΨ = EΨ

Third question

How many linear combinations arethere and which of them can wechoose?

Ψ(1, 2)

Ψ(2, 1)

Ψ = c1Ψ(1, 2) + c2Ψ(2, 1)

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

A principle

Electrons are identical particles...they cannot be distinguished betweeneachother.

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

A principle

Electrons are identical particles...they cannot be distinguished betweeneachother.

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

A principle

Electrons are identical particles...they cannot be distinguished betweeneachother.

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

A principle

Electrons are identical particles...they cannot be distinguished betweeneachother.

electron 1 electron 2

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

A principle

Electrons are identical particles...they cannot be distinguished betweeneachother.

electron 2 electron 1

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

A principle

Electrons are identical particles...they cannot be distinguished betweeneachother.=⇒ probability density must be unchanged if we interchage their coordinates

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

A principle

Electrons are identical particles...they cannot be distinguished betweeneachother.=⇒ probability density must be unchanged if we interchage their coordinates

Fourth question

Which linear combinations satisfy this condition?

Igor Lukacevic Identical particles

Identical particles

Symmetric and antisymmetric solutions

A principle

Electrons are identical particles...they cannot be distinguished betweeneachother.=⇒ probability density must be unchanged if we interchage their coordinates

Fourth question

Which linearcombinations satisfythis condition?

There are two possibilities (Hund & Wigner)

Ψ+ = Ψ(1, 2) + Ψ(2, 1) - symmetric solutionΨ− = Ψ(1, 2)−Ψ(2, 1) - antisymmetric solution

Igor Lukacevic Identical particles

Identical particles

Example: He spectrum

Contents

1 Two-particle wave equation

2 Symmetric and antisymmetric solutions

3 Example: He spectrum

4 Antisymmetry principle

5 Example: H molecule

6 Literature

Igor Lukacevic Identical particles

Identical particles

Example: He spectrum

Energy values are relative to the ground state of

He+, i.e. one has to subtract 54.4 eV.

Igor Lukacevic Identical particles

Identical particles

Example: He spectrum

Energy values are relative to the ground state of

He+, i.e. one has to subtract 54.4 eV.

Bohr’s theory cannot explain this, although it was known that toparahelium and ortohelium belonged the singlet (antiparallel spin) andtriplet (parallel spin) states

Igor Lukacevic Identical particles

Identical particles

Example: He spectrum

Energy values are relative to the ground state of

He+, i.e. one has to subtract 54.4 eV.

Bohr’s theory cannot explain this, although it was known that toparahelium and ortohelium belonged the singlet (antiparallel spin) andtriplet (parallel spin) states

Solution Heisenberg in 1926.

Igor Lukacevic Identical particles

Identical particles

Example: He spectrum

Assumptions:

un = electron wave functions in state n! (n, l ,m)

again, no interaction ⇒{

Ψ = u0(1)un(2)E = E0 + En

e-e interaction is a small perturbation

Igor Lukacevic Identical particles

Identical particles

Example: He spectrum

Assumptions:

un = electron wave functions in state n! (n, l ,m)

again, no interaction ⇒{

Ψ = u0(1)un(2)E = E0 + En

e-e interaction is a small perturbation

What do we need, if we want to estimate the e-e interaction energy?

Igor Lukacevic Identical particles

Identical particles

Example: He spectrum

Assumptions:

un = electron wave functions in state n! (n, l ,m)

again, no interaction ⇒{

Ψ = u0(1)un(2)E = E0 + En

e-e interaction is a small perturbation

What do we need, if we want to estimate the e-e interaction energy?

Ψ+ = 1√

2[u0(1)un(2) + u0(2)un(1)]

Ψ− = 1√2

[u0(1)un(2)− u0(2)un(1)]

Igor Lukacevic Identical particles

Identical particles

Example: He spectrum

Assumptions:

un = electron wave functions in state n! (n, l ,m)

again, no interaction ⇒{

Ψ = u0(1)un(2)E = E0 + En

e-e interaction is a small perturbation

What do we need, if we want to estimate the e-e interaction energy?

Ψ+ = 1√

2[u0(1)un(2) + u0(2)un(1)]

Ψ− = 1√2

[u0(1)un(2)− u0(2)un(1)]

1√2

are normalization

factors

Igor Lukacevic Identical particles

Identical particles

Example: He spectrum

Expectation of e-e interaction energy

E ′ =

∫ ∫Ψ∗

e2

r12Ψdτ1dτ2

Igor Lukacevic Identical particles

Identical particles

Example: He spectrum

Expectation of e-e interaction energy

E ′ =

∫ ∫Ψ∗

e2

r12Ψdτ1dτ2

E ′ = Eex ± Ecorr , Eex > 0, Ecorr > 0

For calculation details seeRefs. [1] and [3].

Igor Lukacevic Identical particles

Identical particles

Example: He spectrum

Expectation of e-e interaction energy

E ′ =

∫ ∫Ψ∗

e2

r12Ψdτ1dτ2

E ′ = Eex ± Ecorr , Eex > 0, Ecorr > 0

For calculation details seeRefs. [1] and [3].

Terms

+ 99K paraterms− 99K ortoterms

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Contents

1 Two-particle wave equation

2 Symmetric and antisymmetric solutions

3 Example: He spectrum

4 Antisymmetry principle

5 Example: H molecule

6 Literature

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Spin s =

{+ 1

2, spin up, parallel to the outer mag. field

− 12, spin down, antiparallel to the outer mag. field

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Spin s =

{+ 1

2, spin up, parallel to the outer mag. field

− 12, spin down, antiparallel to the outer mag. field

wave function:

Ψ(x , y , z , s) =

{Ψ(x , y , z ,+ 1

2) = Ψ+(x , y , z)

Ψ(x , y , z ,− 12) = Ψ−(x , y , z)

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Spin s =

{+ 1

2, spin up, parallel to the outer mag. field

− 12, spin down, antiparallel to the outer mag. field

wave function:

Ψ(x , y , z , s) =

{Ψ(x , y , z ,+ 1

2) = Ψ+(x , y , z) = Ψ(x , y , z)α

Ψ(x , y , z ,− 12) = Ψ−(x , y , z) = Ψ(x , y , z)β

α� spin wave function for parallel spinβ � spin wave function for antiparallel spin

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Assumptions

z we have 2 electrons: α(1) and β(2)

z their spins are independent

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Assumptions

z we have 2 electrons: α(1) and β(2)

z their spins are independent

A question

What will total spin wave functions look like?

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Assumptions

z we have 2 electrons: α(1) and β(2)

z their spins are independent

Total spin wave functions

Spin orientationelectron 1 electron 2

α(1)α(2) ↑ ↑β(1)β(2) ↓ ↓α(1)β(2) ↑ ↓β(1)α(2) ↓ ↑

Second question

Can you guess any other?

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Total spin wave functions

Spin orientationelectron 1 electron 2

α(1)α(2) ↑ ↑β(1)β(2) ↓ ↓α(1)β(2) ↑ ↓β(1)α(2) ↓ ↑

Total spin wave functions

S MS

α(1)α(2) 1 1α(1)β(2) + α(2)β(1) 1 0 symmetric ⇒ triplet

β(1)β(2) 1 -1

α(1)β(2)− α(2)β(1) 0 0 antisymmetric ⇒ singlet

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Ok, let us now construct the total wave function:

♠ spatial wave function

♠ spin wave function

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Ok, let us now construct the total wave function:

♠ spatial wave function

♠ spin wave function

Remember

Ψr =

{u(1)v(2) + u(2)v(1), symmetricu(1)v(2)− u(2)v(1), antisymmetric

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Ok, let us now construct the total wave function:

♠ spatial wave function

♠ spin wave function

Total wave function

Ψr,s =

u(1)v(2) + u(2)v(1)×

α(1)α(2) symmα(1)β(2) + α(2)β(1) symmβ(1)β(2) symmα(1)β(2)− α(2)β(1) antisymm

u(1)v(2)− u(2)v(1)×

α(1)α(2) antisymmα(1)β(2) + α(2)β(1) antisymmβ(1)β(2) antisymmα(1)β(2)− α(2)β(1) symm

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Total wave function

Ψr,s =

u(1)v(2) + u(2)v(1)×

α(1)α(2) symmα(1)β(2) + α(2)β(1) symmβ(1)β(2) symmα(1)β(2)− α(2)β(1) antisymm

u(1)v(2)− u(2)v(1)×

α(1)α(2) antisymmα(1)β(2) + α(2)β(1) antisymmβ(1)β(2) antisymmα(1)β(2)− α(2)β(1) symm

A question

Which of these w.f. come in nature?

A hint...

Use Pauli’s exclusion principle.

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Total wave function

Ψr,s =

u(1)v(2) + u(2)v(1)×

α(1)α(2) symmα(1)β(2) + α(2)β(1) symmβ(1)β(2) symmα(1)β(2)− α(2)β(1) antisymm

u(1)v(2)− u(2)v(1)×

α(1)α(2) antisymmα(1)β(2) + α(2)β(1) antisymmβ(1)β(2) antisymmα(1)β(2)− α(2)β(1) symm

A question

Which of these w.f. come in nature?

A hint...

Use Pauli’s exclusion principle.

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Possible total wave functions

Ψr,s =

u(1)v(2) + u(2)v(1)× α(1)β(2)− α(2)β(1) antisymm

u(1)v(2)− u(2)v(1)×

α(1)α(2) antisymmα(1)β(2) + α(2)β(1) antisymmβ(1)β(2) antisymm

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Possible total wave functions

Ψr,s =

u(1)v(2) + u(2)v(1)× α(1)β(2)− α(2)β(1) antisymm

u(1)v(2)− u(2)v(1)×

α(1)α(2) antisymmα(1)β(2) + α(2)β(1) antisymmβ(1)β(2) antisymm

Antisymmetry principle (generalized Pauli’s principle)

Total wave function of electrons has to be antisymmetric, wrt the interchangeof their (spatial and/or spin) coordinates.

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

If we turn back to helium...

He electron wave functions

ΨHer,s =

u(1)v(2) + u(2)v(1)× α(1)β(2)− α(2)β(1) parahelium

u(1)v(2)− u(2)v(1)×

α(1)α(2)α(1)β(2) + α(2)β(1) ortoheliumβ(1)β(2)

Antisymmetry principle (generalized Pauli’s principle)

Total wave function of electrons has to be antisymmetric, wrt the interchangeof their (spatial and/or spin) coordinates.

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Generalizations

♣ to N-particle systems Ψ(1, 2, 3, . . .) = −Ψ(2, 1, 3, . . .)

♣ to all fermions

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Generalizations

♣ to N-particle systems Ψ(1, 2, 3, . . .) = −Ψ(2, 1, 3, . . .)

♣ to all fermions

A question

What about symmetric wave functions Ψ(1, 2, 3, . . .) = Ψ(2, 1, 3, . . .)?They describe the particles with spin 0, 1, 2, . . .

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Generalizations

♣ to N-particle systems Ψ(1, 2, 3, . . .) = −Ψ(2, 1, 3, . . .)

♣ to all fermions

A question

What about symmetric wave functions Ψ(1, 2, 3, . . .) = Ψ(2, 1, 3, . . .)?They describe the particles with spin 0, 1, 2, . . .

Igor Lukacevic Identical particles

Identical particles

Antisymmetry principle

Generalizations

♣ to N-particle systems Ψ(1, 2, 3, . . .) = −Ψ(2, 1, 3, . . .)

♣ to all fermions

In conclusion

Spin Symmetry Statistics

0,1,2,. . . symmetric Einstein-Bose

1

2,

3

2, . . . antisymmetric Fermi-Dirac

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

Contents

1 Two-particle wave equation

2 Symmetric and antisymmetric solutions

3 Example: He spectrum

4 Antisymmetry principle

5 Example: H molecule

6 Literature

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

Let us first look at rotational spectrum of H molecule.(Mecke - first experimental observation; Heisenberg & Hund - theoreticalexplanation in 1928.)

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

Let us first look at rotational spectrum of H molecule.(Mecke - first experimental observation; Heisenberg & Hund - theoreticalexplanation in 1928.)

The movement of nuclei determines the following properties:

vibration

rotation

spin

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

Let us first look at rotational spectrum of H molecule.(Mecke - first experimental observation; Heisenberg & Hund - theoreticalexplanation in 1928.)

The movement of nuclei determinesthe following properties:

vibration

rotation

spin

These properties are independent

Ψ = ψvibψrotψspin

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

These properties are independent

Ψ = ψvibψrotψspin

A question

What happens with this w.f. if weinterchange the coordinates?

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

These properties are independent

Ψ = ψvibψrotψspin

A question

What happens with this w.f. if weinterchange the coordinates?

vibration

rotation

spin

ψvib = ψvib(r12), r12 = |r2 − r1|

~r → −~r =⇒ ψvib = ψvib

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

These properties are independent

Ψ = ψvibψrotψspin

A question

What happens with this w.f. if weinterchange the coordinates?

vibration

rotation

spinψrot 99K Y

ml =

{Y m

l , l even−Y m

l , l odd

~r → −~r =⇒ ψrot = (−1)lψrot

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

These properties are independent

Ψ = ψvibψrotψspin

A question

What happens with this w.f. if weinterchange the coordinates?

vibration

rotation

spin

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

Quantum # l Spin orientation Total spin Statistical weigth NameEven ↑↓ 0 1 para H2

Odd ↑↑ 1 3 orto H2

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

Quantum # l Spin orientation Total spin Statistical weigth NameEven ↑↓ 0 1 para H2

Odd ↑↑ 1 3 orto H2

Note

There can be no transitions between the states with different symmetrycharacter.

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

Quantum # l Spin orientation Total spin Statistical weigth NameEven ↑↓ 0 1 para H2

Odd ↑↑ 1 3 orto H2

Note

There can be no transitions between the states with different symmetrycharacter. para H2 : orto H2 = 1 : 3

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

Quantum # l Spin orientation Total spin Statistical weigth NameEven ↑↓ 0 1 para H2

Odd ↑↑ 1 3 orto H2

Note

There can be no transitions between the states with different symmetrycharacter. para H2 : orto H2 = 1 : 3

A question

What happens at T = 0 K?

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

Quantum # l Spin orientation Total spin Statistical weigth NameEven ↑↓ 0 1 para H2

Odd ↑↑ 1 3 orto H2

Note

There can be no transitions between the states with different symmetrycharacter. para H2 : orto H2 = 1 : 3

A question

What happens at T = 0 K? All hydrogen molecules go to para-state.

Igor Lukacevic Identical particles

Identical particles

Example: H molecule

Another confirmation (Dennison 1928.)

Igor Lukacevic Identical particles

Identical particles

Literature

Contents

1 Two-particle wave equation

2 Symmetric and antisymmetric solutions

3 Example: He spectrum

4 Antisymmetry principle

5 Example: H molecule

6 Literature

Igor Lukacevic Identical particles

Identical particles

Literature

Literature

1 I. Supek, Teorijska fizika i struktura materije, II. dio, Skolska knjiga,Zagreb, 1989.

2 D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., PearsonEducation, Inc., Upper Saddle River, NJ, 2005.

3 A. Szabo, N. Ostlund, Modern Quantum Chemistry, Introduction toAdvanced Electronic Structure theory, Dover Publications, New York,1996.

Igor Lukacevic Identical particles