Quantum One

1. Quantum One

3. Completeness Relations, Matrix Elements, and Hermitian Conjugation

4. In the last lecture, we introduced a class of operators called ket-braoperators, whose action on arbitrary states is “self-evident”. We used operators of this type to define another important class of operators called projection operators, which obey an idempotency condition, and that generally project an arbitrary state onto some subspace of . By considering complete sums of orthogonal projectors, we deduced the completeness relations for discrete and continuous ONBs, which provide a decomposition of the identity operator in terms of a “complete set of states”. As it turns out, these completeness relations provide useful tools for generating representation dependent equations, from their representation independent counterparts.

5. In the last lecture, we introduced a class of operators called ket-braoperators, whose action on arbitrary states is “self-evident”. We used operators of this type to define another important class of operators called projection operators, which obey an idempotency condition, and that generally project an arbitrary state onto some subspace of . By considering complete sums of orthogonal projectors, we deduced the completeness relations for discrete and continuous ONBs, which provide a decomposition of the identity operator in terms of a “complete set of states”. As it turns out, these completeness relations provide useful tools for generating representation dependent equations, from their representation independent counterparts.

6. In the last lecture, we introduced a class of operators called ket-braoperators, whose action on arbitrary states is “self-evident”. We used operators of this type to define another important class of operators called projection operators, which obey an idempotency condition, and that generally project an arbitrary state onto some subspace of . By considering complete sums of orthogonal projectors, we deduced the completeness relations for discrete and continuous ONBs, which provide a decomposition of the identity operator in terms of a “complete set of states”. As it turns out, these completeness relations provide useful tools for generating representation dependent equations, from their representation independent counterparts.

7. In the last lecture, we introduced a class of operators called ket-braoperators, whose action on arbitrary states is “self-evident”. We used operators of this type to define another important class of operators called projection operators, which obey an idempotency condition, and that generally project an arbitrary state onto some subspace of . By considering complete sums of orthogonal projectors, we deduced the completeness relations for discrete and continuous ONBs, which provide a decomposition of the identity operator in terms of a “complete set of states”. As it turns out, these completeness relations provide useful tools for generating representation dependent equations, from their representation independent counterparts.

8. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

9. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

10. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

11. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

12. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

13. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

14. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

15. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

16. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

17. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

18. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

19. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

20. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

21. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

22. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

23. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

24. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

25. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

26. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

27. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

28. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

29. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

30. Consider, e.g., that if {|i〉} form an ONB for S then 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

31. But we know two representations for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

32. But we know two representations for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

33. But we know two representations for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

34. But we know two representations for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

35. But we know two representations for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

36. But we know two representations for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

37. But we know two representations for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

38. But we know two representations for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

39. But we know two representations for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

40. Similarly, for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

41. Similarly, for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

42. Similarly, for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

43. Similarly, for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

44. Similarly, for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

45. Similarly, for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

46. Similarly, for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

47. Similarly, for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

48. Similarly, for a single particle: The states {|i〉} form an ONB for Sso 〈i|j〉=δ_{ij} ∑_{i}|i〉〈i|=1 and we can write: |χ〉=1|χ〉=∑_{i}|i〉〈i|χ〉=∑_{i}χ_{i}|i〉 〈ψ|χ〉=〈ψ|(1|χ〉)=∑_{i}〈ψ|i〉〈i|χ〉=∑_{i}ψ_{i}^{∗}χ_{i}

50. Matrix Elements, and the Action of Operators to the Left