1. Let’s consider the consequences of this commutator further
[A,B] = 0

2. Here is the S equation for the H atom
Let’s consider the consequences of this commutator further
[A,B] = 0
Here is the S equation for the H atom

3. Here is the S equation for the H atom
Let’s consider the consequences of this commutator further
[A,B] = 0
Here is the S equation for the H atom

4. Here is the S equation for the H atom
Let’s consider the consequences of this commutator further
[A,B] = 0
Here is the S equation for the H atom 1 2 B A

5. Here is the S equation for the H atom
Let’s consider the consequences of this commutator further
[A,B] = 0
Here is the S equation for the H atom 1 2 B A R

6. Here is the S equation for the H atom
Let’s consider the consequences of this commutator further
[A,B] = 0
Here is the S equation for the H atom 1
r12 2 B A R

7. Here is the S equation for the H atom
Let’s consider the consequences of this commutator further
[A,B] = 0
Here is the S equation for the H atom 1
r12 2
r1A B A R

8. Here is the S equation for the H atom
Let’s consider the consequences of this commutator further
[A,B] = 0
Here is the S equation for the H atom 1
r12 2
r1A
r2B B A R

9. Here is the S equation for the H atom
Let’s consider the consequences of this commutator further
[A,B] = 0
Here is the S equation for the H atom 1
r12 2
r1A
r2B
r1B B A R

10. Here is the S equation for the H atom
Let’s consider the consequences of this commutator further
[A,B] = 0
Here is the S equation for the H atom 1
r12 2
r1A
r2A
r2B
r1B B A R

11. Let’s consider the consequences of this commutator further [A,B] = 0
Here is the S equation for the H atom
Now let us consider something that may seem a bit odd
the permutation operators P12 or PAB
and their effect on H

12. Let’s consider the consequences of this commutator further [A,B] = 0
Here is the S equation for the H atom
Now let us consider something that may seem a bit odd
the permutation operators P12 or PAB
and their effect on H
P12 permutes the coordinates of particles 1 and 2

13. Let’s consider the consequences of this commutator further [A,B] = 0
Here is the S equation for the H atom
Now let us consider something that may seem a bit odd
the permutation operators P12 or PAB
and their effect on H
P12 permutes the coordinates of particles 1 and 2
the electrons

14. Let’s consider the consequences of this commutator further [A,B] = 0
Here is the S equation for the H atom
Now let us consider something that may seem a bit odd
the permutation operators P12 or PAB
and their effect on H
P12 permutes the coordinates of particles 1 and 2
the electrons
PAB permutes the coordinates of particle A and B

15. Let’s consider the consequences of this commutator further [A,B] = 0
Here is the S equation for the H atom
Now let us consider something that may seem a bit odd
the permutation operators P12 or PAB
and their effect on H
P12 permutes the coordinates of particles 1 and 2
the electrons
PAB permutes the coordinates of particle A and B
the protons

17. I think it is obvious that the Hamiltonian is unaffected and so
Pij and the Hamiltonian commute ie [H, Pij] = 0

18. I think it is obvious that the Hamiltonian is unaffected and so
Pij and the Hamiltonian commute ie [H, Pij] = 0
Thus
P12 Ψ(1,2) = pΨ(1,2)

19. I think it is obvious that the Hamiltonian is unaffected and so
Pij and the Hamiltonian commute ie [H, Pij] = 0
Thus
P12 Ψ(1,2) = pΨ(1,2)
but also
P12 Ψ(1,2) = Ψ(2,1)

20. I think it is obvious that the Hamiltonian is unaffected and so
Pij and the Hamiltonian commute ie [H, Pij] = 0
Thus
P12 Ψ(1,2) = pΨ(1,2)
but also
P12 Ψ(1,2) = Ψ(2,1)
and
P12P12 Ψ(1,2) = P12 Ψ(2,1) = Ψ(1,2)

21. I think it is obvious that the Hamiltonian is unaffected and so
Pij and the Hamiltonian commute ie [H, Pij] = 0
Thus
P12 Ψ(1,2) = pΨ(1,2)
but also
P12 Ψ(1,2) = Ψ(2,1)
and
P12P12 Ψ(1,2) = P12 Ψ(2,1) = Ψ(1,2)
So as
P12P12 Ψ(1,2) = p2Ψ(1,2)

22. I think it is obvious that the Hamiltonian is unaffected and so
Pij and the Hamiltonian commute ie [H, Pij] = 0
Thus
P12 Ψ(1,2) = pΨ(1,2)
but also
P12 Ψ(1,2) = Ψ(2,1)
and
P12P12 Ψ(1,2) = P12 Ψ(2,1) = Ψ(1,2)
So as
P12P12 Ψ(1,2) = p2Ψ(1,2)
p2 = 1 and p = ± 1

23. I think it is obvious that the Hamiltonian is unaffected and so
Pij and the Hamiltonian commute ie [H, Pij] = 0
Thus
P12 Ψ(1,2) = pΨ(1,2)
but also
P12 Ψ(1,2) = Ψ(2,1)
and
P12P12 Ψ(1,2) = P12 Ψ(2,1) = Ψ(1,2)
So as
P12P12 Ψ(1,2) = p2Ψ(1,2)
p2 = 1 and p = ± 1
So two different types of quantum particles exist … those for which the total wave function on interchange stays the same i.e. p = +1 or changes sign p = – 1

24. It is found empirically that p = +1 for integral spin particles

25. It is found empirically that p = +1 for integral spin particles D 1
Photons 1

26. It is found empirically that p = +1 for integral spin particles D 1
Photons 1
and follow Bose-Einstein statistics and are called Bosons

27. It is found empirically that p = +1 for integral spin particles D 1
Photons 1
and follow Bose-Einstein statistics and are called Bosons
p = –1 for half-integral spin particles

28. It is found empirically that p = +1 for integral spin particles D 1
Photons 1
and follow Bose-Einstein statistics and are called Bosons
and
p = –1 for half-integral spin particles
electrons ½
protons ½
chlorine nuclei 3/2

29. It is found empirically that p = +1 for integral spin particles D 1
Photons 1
and follow Bose-Einstein statistics and are called Bosons
and
p = –1 for half-integral spin particles
electrons ½
protons ½
chlorine nuclei 3/2
and follow Fermi-Dirac statistics and are called Fermions

30.  ↑ ↑  and  ↓ ↓  are already symmetric
↑ ↓ H2
Protons ↑
↑ ↑
↑ ↓
↓ ↑
↓ ↓ ↓
 ↑ ↑  and  ↓ ↓  are already symmetric
Opposing off-diagonals can form a symmetric and an antisymmetric combination
 ↑ ↓  ±  ↓ ↑ 

31. Three symmetric and one antisymmetric wavefunctions
α β H2
Protons
α α
α β α β
β α
β β
α α
α β
+ β α
β β
α β
– β α
Three symmetric and one antisymmetric wavefunctions

32.  ↑ ↑  → →  ↓ ↓  are already symmetric
↑ → ↓ N2
I = 1
particles
↑ ↑
↑ →
↑ ↓ ↑ →
→ ↑ →→
→ ↓
↓ ↑
↓ →
↓ ↓ ↓
 ↑ ↑  → →  ↓ ↓  are already symmetric
Opposing off-diagonals can form symmetric and antisymmetric combinations eg
 ↑ →  ±  → ↑ 

33. +1+1 0 0 −1+1 are already symmetricN2
I = 1
particles +1 -1
+1+1 0 0 −1+1 are already symmetric
Opposing off diagonals can form symmetric and antisymmetric combinations eg
+10 ± 0+1

34. +1+1 0 0 −1+1 are already symmetricN2
I = 1
particles +1 -1
+1+1 0 0 −1+1 are already symmetric
Opposing off-diagonals can form symmetric and antisymmetric combinations
+10 ± 0+1
in pairs

35. Spin I I
I -1
1 - I
- I
2I+1 I
I -1
1 - I
- I
2I+1

36. There are (2I+1)2 - (2I+1) off-diagonal functions which can form
There are (2I+1) x (2I+1) functions all-together of which 2I+1 are diagonal and thus already symmetric.
There are (2I+1)2 - (2I+1) off-diagonal functions
which can form
½[(2I+1)2 - (2I+1)] symmetric combinations
and
½[(2I+1)2 - (2I+1)] antisymmetic combinations

37. 2I+1 = n
Symmetric ½(n2 – n) + n
Antisymmetric ½(n2 –n)
S ½(n2 – n) + n =
A ½(n2 –n)
S ½(n – 1) + 1 =
A ½(n – 1)
S I + 1 =
A I

39. I I -1
1 - I
- I
2I+1 I
I -1
1 - I
- I
2I+1

40. It is found empirically that p = +1 for integral spin particles D 1
Photons 1
and follow Bose-Einstein statistics and are called Bosons
p = –1 for half-integral spin particles
electrons ½
protons ½
chlorine nuclei 3/2

41. Here is the S equation for the H atom
Let’s consider the consequences of this commutator further
[A,B] = 0
Here is the S equation for the H atom

42. Men’n”
Men’n”

45.  ↑ ↑  and  ↓ ↓  are already symmetric
↑ ↓ ↑
↑ ↑
↑ ↓
↓ ↑
↓ ↓ ↓
 ↑ ↑  and  ↓ ↓  are already symmetric
Opposing off-diagonals can form a symmetric and an antisymmetric combination
 ↑ ↓  ±  ↓ ↑ 