Identical particles - Quantum mechanics 2 - Lecture...
Transcript of Identical particles - Quantum mechanics 2 - Lecture...
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Identical particles
Identical particlesQuantum mechanics 2 - Lecture 1
Igor Lukacevic
UJJS, Dept. of Physics, Osijek
14. prosinca 2011.
Igor Lukacevic Identical particles
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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
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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
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Identical particles
Two-particle wave equation
Which interactions exist here and what istheir nature?
Igor Lukacevic Identical particles
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Identical particles
Two-particle wave equation
H =1
2m1p21 +
1
2m2p22 + V (r1, r2)
Igor Lukacevic Identical particles
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Identical particles
Two-particle wave equation
H =1
2m1p21 +
1
2m2p22 + V (r1, r2)
pk →~i∇k
Igor Lukacevic Identical particles
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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
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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
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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
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Identical particles
Symmetric and antisymmetric solutions
Assumption: no interaction!
electron 1 electron 2
Igor Lukacevic Identical particles
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Identical particles
Symmetric and antisymmetric solutions
− ~2m1
∆1Ψ +V1(r1)Ψ− ~2m2
∆2Ψ +V2(r2)Ψ = EΨ Ψ(r1, r2) = u(r1)v(r2)
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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
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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
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Identical particles
Symmetric and antisymmetric solutions
Assumption: allow interaction!
nucleus electrons
Igor Lukacevic Identical particles
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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
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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
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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
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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
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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
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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
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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
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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
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Identical particles
Symmetric and antisymmetric solutions
A principle
Electrons are identical particles...they cannot be distinguished betweeneachother.
Igor Lukacevic Identical particles
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Identical particles
Symmetric and antisymmetric solutions
A principle
Electrons are identical particles...they cannot be distinguished betweeneachother.
Igor Lukacevic Identical particles
![Page 24: Identical particles - Quantum mechanics 2 - Lecture 1fizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/qm2/Lecture_1_Identical...Identical particles Quantum mechanics 2](https://reader033.fdocuments.us/reader033/viewer/2022041817/5e5bdbd0f046e2760a4f87f3/html5/thumbnails/24.jpg)
Identical particles
Symmetric and antisymmetric solutions
A principle
Electrons are identical particles...they cannot be distinguished betweeneachother.
Igor Lukacevic Identical particles
![Page 25: Identical particles - Quantum mechanics 2 - Lecture 1fizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/qm2/Lecture_1_Identical...Identical particles Quantum mechanics 2](https://reader033.fdocuments.us/reader033/viewer/2022041817/5e5bdbd0f046e2760a4f87f3/html5/thumbnails/25.jpg)
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Identical particles
Example: He spectrum
Expectation of e-e interaction energy
E ′ =
∫ ∫Ψ∗
e2
r12Ψdτ1dτ2
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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].
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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
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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
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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
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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)
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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
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Identical particles
Antisymmetry principle
Assumptions
z we have 2 electrons: α(1) and β(2)
z their spins are independent
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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?
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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?
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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
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Identical particles
Antisymmetry principle
Ok, let us now construct the total wave function:
♠ spatial wave function
♠ spin wave function
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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
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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
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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.
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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.
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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
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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.
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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.
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Identical particles
Antisymmetry principle
Generalizations
♣ to N-particle systems Ψ(1, 2, 3, . . .) = −Ψ(2, 1, 3, . . .)
♣ to all fermions
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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, . . .
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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, . . .
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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
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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
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Identical particles
Example: H molecule
Let us first look at rotational spectrum of H molecule.(Mecke - first experimental observation; Heisenberg & Hund - theoreticalexplanation in 1928.)
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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
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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
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Identical particles
Example: H molecule
These properties are independent
Ψ = ψvibψrotψspin
A question
What happens with this w.f. if weinterchange the coordinates?
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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
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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
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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
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Identical particles
Example: H molecule
Quantum # l Spin orientation Total spin Statistical weigth NameEven ↑↓ 0 1 para H2
Odd ↑↑ 1 3 orto H2
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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.
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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
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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
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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
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Identical particles
Example: H molecule
Another confirmation (Dennison 1928.)
Igor Lukacevic Identical particles
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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
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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