1. Quantum One: Lecture 15 1

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3. Completeness Relations, Matrix Elements, and Hermitian Conjugation 3

4. In the last lecture, we introduced a class of operators called ket-bra operators, 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 S. 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. 4

5. 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} 5

6. 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} 6

7. 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} 7

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} 8

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} 9

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} 10

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} 11

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} 12

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} 13

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} 14

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} 15

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} 16

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} 17

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} 18

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} 19

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} 20

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} 21

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} 22

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} 23

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} 24

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} 25

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} 26

27. But we know two representations for a single particle: The states {|i 〉 } form an ONB for S so 〈 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

28. But we know two representations for a single particle: The states {|i 〉 } form an ONB for S so 〈 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

29. But we know two representations for a single particle: The states {|i 〉 } form an ONB for S so 〈 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

30. Similarly, for a single particle: The states {|i 〉 } form an ONB for S so 〈 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

31. Similarly, for a single particle: The states {|i 〉 } form an ONB for S so 〈 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

32. Similarly, for a single particle: The states {|i 〉 } form an ONB for S so 〈 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

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34. Matrix Elements, and the Action of Operators to the Left 34

35. Definition: Matrix Element The matrix element of an operator A between (or connecting) the states |ψ 〉 and |χ 〉 is the scalar where Comments: 1.At the moment the “matrix element” between arbitrary states has been defined, even though we have not actually seen any matrix that this matrix elements is an element of! 2.We will refer to this as the |ψ 〉 - |χ 〉 matrix element of 35

36. Definition: Matrix Element The matrix element of an operator A between (or connecting) the states |ψ 〉 and |χ 〉 is the scalar where Comments: 1.At the moment the “matrix element” between arbitrary states has been defined, even though we have not actually seen any matrix that this matrix element is an element of! 2.We will refer to this as the |ψ 〉 - |χ 〉 matrix element of 36

37. Definition: Matrix Element The matrix element of an operator A between (or connecting) the states |ψ 〉 and |χ 〉 is the scalar where Comments: 1.At the moment the “matrix element” between arbitrary states has been defined, even though we have not actually seen any matrix that this matrix element is an element of! 2.We will refer to this as the |ψ 〉 - |χ 〉 matrix element of 37

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44. Definition: expectation value If the state |φ 〉 is a unit vector, then the matrix element 〈 φ|A|φ 〉 of |φ 〉 with itself is referred to by many authors as the expectation value of the operator A taken with respect to the state |φ 〉. Comments: 1.At this point this is just a mathematical definition of the term “expectation value” and has no physical notion attached to it. 2.The commonly used term is actually very unfortunate, since the expectation value of an observable is almost never the value that you most expect to get, and is often equal to a value that you can actually never expect to get. 44

45. Definition: expectation value If the state |φ 〉 is a unit vector, then the matrix element 〈 φ|A|φ 〉 of |φ 〉 with itself is referred to by many authors as the expectation value of the operator A taken with respect to the state |φ 〉. Comments: 1.At this point this is just a mathematical definition of the term “expectation value” and has no physical notion attached to it. 2.The commonly used term is actually very unfortunate, since the expectation value of an observable is almost never the value that you most expect to get, and is often equal to a value that you can actually never expect to get. 45

46. Definition: expectation value If the state |φ 〉 is a unit vector, then the matrix element 〈 φ|A|φ 〉 of |φ 〉 with itself is referred to by many authors as the expectation value of the operator A taken with respect to the state |φ 〉. Comments: 1.At this point this is just a mathematical definition of the term “expectation value” and has no physical notion attached to it. 2.The commonly used term is actually very unfortunate, since the expectation value of an observable is almost never the value that you most expect to get, and is often equal to a value that you can actually never expect to get. 46

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60. Hermitian Conjugation: The Hermitian adjoint of any scalar turns out to be just its complex conjugate. This is easy to show. Consider the expansion of an arbitrary ket The adjoint of this last equation is which implies that 60

61. Hermitian Conjugation: The Hermitian adjoint of any scalar turns out to be just its complex conjugate. This is easy to show. Consider the expansion of an arbitrary ket The adjoint of this last equation is which implies that 61

62. Hermitian Conjugation: The Hermitian adjoint of any scalar turns out to be just its complex conjugate. This is easy to show. Consider the expansion of an arbitrary ket The adjoint of this last equation is which implies that 62

63. Hermitian Conjugation: The Hermitian adjoint of any scalar turns out to be just its complex conjugate. This is easy to show. Consider the expansion of an arbitrary ket The adjoint of this last equation is which shows that 63

64. Hermitian Conjugation: We now extend this idea to operators. If in S the operator A maps the ket |χ 〉 onto the ket |ψ 〉 then the adjoint of A must have the corresponding effect in the adjoint space. Thus, we define an operator A⁺ (which is pronounced "A adjoint" or "A dagger") Such that if A|χ 〉 = |ψ 〉 then 〈 χ|A⁺ = 〈 ψ| This is the relationship we were looking for in the question posed above. It shows that when we "flip things around" we have to replace operators by their adjoints to get valid statements. Thus we can write [ A|χ 〉 ]⁺ = 〈 χ|A⁺ 64

65. Hermitian Conjugation: We now extend this idea to operators. If in S the operator A maps the ket |χ 〉 onto the ket |ψ 〉 then the adjoint of A must have the corresponding effect in the adjoint space. Thus, we define an operator A⁺ (which is pronounced "A adjoint" or "A dagger") Such that if A|χ 〉 = |ψ 〉 then 〈 χ|A⁺ = 〈 ψ| This is the relationship we were looking for in the question posed above. It shows that when we "flip things around" we have to replace operators by their adjoints to get valid statements. Thus we can write [ A|χ 〉 ]⁺ = 〈 χ|A⁺ 65

66. Hermitian Conjugation: We now extend this idea to operators. If in S the operator A maps the ket |χ 〉 onto the ket |ψ 〉 then the adjoint of A must have the corresponding effect in the adjoint space. Thus, we define an operator A⁺ (which is pronounced "A adjoint" or "A dagger") Such that if A|χ 〉 = |ψ 〉 then 〈 χ|A⁺ = 〈 ψ| This is the relationship we were looking for in the question posed above. It shows that when we "flip things around" we have to replace operators by their adjoints to get valid statements. Thus we can write [ A|χ 〉 ]⁺ = 〈 χ|A⁺ 66

67. Hermitian Conjugation: We now extend this idea to operators. If in S the operator A maps the ket |χ 〉 onto the ket |ψ 〉 then the adjoint of A must have the corresponding effect in the adjoint space. Thus, we define an operator A⁺ (which is pronounced "A adjoint" or "A dagger") Such that if A|χ 〉 = |ψ 〉 then 〈 χ|A⁺ = 〈 ψ| This is the relationship we were looking for in the question posed above. It shows that when we "flip things around" we have to replace operators by their adjoints to get valid statements. Thus we can write [ A|χ 〉 ]⁺ = 〈 χ|A⁺ 67

68. Hermitian Conjugation: We now extend this idea to operators. If in S the operator A maps the ket |χ 〉 onto the ket |ψ 〉 then the adjoint of A must have the corresponding effect in the adjoint space. Thus, we define an operator A⁺ (which is pronounced "A adjoint" or "A dagger") Such that if A|χ 〉 = |ψ 〉 then 〈 χ|A⁺ = 〈 ψ| This is the relationship we were looking for in the question posed above. It shows that when we "flip things around" we have to replace operators by their adjoints to get valid statements. Thus we can write [ A|χ 〉 ]⁺ = 〈 χ|A⁺ 68

69. Hermitian Conjugation: We now extend this idea to operators. If in S the operator A maps the ket |χ 〉 onto the ket |ψ 〉 then the adjoint of A must have the corresponding effect in the adjoint space. Thus, we define an operator A⁺ (which is pronounced "A adjoint" or "A dagger") Such that if A|χ 〉 = |ψ 〉 then 〈 χ|A⁺ = 〈 ψ| This is the relationship we were looking for in the question posed earlier. It shows that when we "flip things around" we have to replace operators by their adjoints to get valid statements. Thus we can write [ A|χ 〉 ]⁺ = 〈 χ|A⁺ 69

70. Hermitian Conjugation: We now extend this idea to operators. If in S the operator A maps the ket |χ 〉 onto the ket |ψ 〉 then the adjoint of A must have the corresponding effect in the adjoint space. Thus, we define an operator A⁺ (which is pronounced "A adjoint" or "A dagger") Such that if A|χ 〉 = |ψ 〉 then 〈 χ|A⁺ = 〈 ψ| This is the relationship we were looking for in the question posed earlier. It shows that when we "flip things around" we have to replace operators by their adjoints to get valid statements. Thus we can write [ A|χ 〉 ]⁺ = 〈 χ|A⁺ 70

71. Hermitian Conjugation: A few moments of study of the adjoint process allows the following rules to be developed: To take the adjoint of any product of operators, numbers, bra's, ket's etc., 1)replace all elements by their adjoints (bra's are replaced by ket's, operators by their adjoints, numbers by their conjugates), and 2)reverse the order of all elements in the original product. Once this operation is performed, any numbers can be commuted past any operators or vectors to simplify the expression. As an example, note that 71

72. Hermitian Conjugation: A few moments of study of the adjoint process allows the following rules to be developed: To take the adjoint of any product of operators, numbers, bra's, ket's etc., 1)replace all elements by their adjoints (bra's are replaced by ket's, operators by their adjoints, numbers by their conjugates), and 2)reverse the order of all elements in the original product. Once this operation is performed, any numbers can be commuted past any operators or vectors to simplify the expression. As an example, note that 72

73. Hermitian Conjugation: A few moments of study of the adjoint process allows the following rules to be developed: To take the adjoint of any product of operators, numbers, bra's, ket's etc., 1)replace all elements by their adjoints (bra's are replaced by ket's, operators by their adjoints, numbers by their conjugates), and 2)reverse the order of all elements in the original product. Once this operation is performed, any numbers can be commuted past any operators or vectors to simplify the expression. As an example, note that 73

74. Hermitian Conjugation: A few moments of study of the adjoint process allows the following rules to be developed: To take the adjoint of any product of operators, numbers, bra's, ket's etc., 1)replace all elements by their adjoints (bra's are replaced by ket's, operators by their adjoints, numbers by their conjugates), and 2)reverse the order of all elements in the original product. Once this operation is performed, any numbers can be commuted past any operators or vectors to simplify the expression. As an example, note that 74

75. Hermitian Conjugation: A few moments of study of the adjoint process allows the following rules to be developed: To take the adjoint of any product of operators, numbers, bra's, ket's etc., 1)replace all elements by their adjoints (bra's are replaced by ket's, operators by their adjoints, numbers by their conjugates), and 2)reverse the order of all elements in the original product. Once this operation is performed, any numbers can be commuted past any operators or vectors to simplify the expression. As an example, note that 75

76. Hermitian Conjugation: A short list of properties of the Hermitian adjoint are given below: 76

77. Hermitian Conjugation: A short list of properties of the Hermitian adjoint are given below: 77

78. Hermitian Conjugation: A short list of properties of the Hermitian adjoint are given below: 78

79. Hermitian Conjugation: A short list of properties of the Hermitian adjoint are given below: 79

80. Hermitian Conjugation: A short list of properties of the Hermitian adjoint are given below: So, e.g., ifthen 80

81. Hermitian Conjugation: A short list of properties of the Hermitian adjoint are given below: So, e.g., ifthen 81

82. In this lecture, we saw how the completeness relation for ONBs allows one to obtain representation dependent equations from representation independent ones. We also defined the matrix elements of an operator between different states, and extended the action of operators so that they could act either to the right on kets or to the left on bras. In the process, we were led to the notion of Hermitian conjugation, which allows us to identify any object in the space of kets, with its corresponding object in the space of bras, and vice versa, and develop some rules for “taking the Hermitian adjoint” of any expression formulated in the Dirac notation. In the next lecture, we use this notion to introduce the notion of Hermitian, anti- Hermitian, and Unitary operators. 82

83. In this lecture, we saw how the completeness relation for ONBs allows one to obtain representation dependent equations from representation independent ones. We also defined the matrix elements of an operator between different states, and extended the action of operators so that they could act either to the right on kets or to the left on bras. In the process, we were led to the notion of Hermitian conjugation, which allows us to identify any object in the space of kets, with its corresponding object in the space of bras, and vice versa, and develop some rules for “taking the Hermitian adjoint” of any expression formulated in the Dirac notation. In the next lecture, we use this notion to introduce the notion of Hermitian, anti- Hermitian, and Unitary operators. 83

84. In this lecture, we saw how the completeness relation for ONBs allows one to obtain representation dependent equations from representation independent ones. We also defined the matrix elements of an operator between different states, and extended the action of operators so that they could act either to the right on kets or to the left on bras. In the process, we were led to the notion of Hermitian conjugation, which allows us to identify any object in the space of kets, with its corresponding object in the space of bras, and vice versa, and developed some rules for “taking the Hermitian adjoint” of any expression formulated in the Dirac notation. In the next lecture, we use this notion to introduce the notion of Hermitian, anti- Hermitian, and Unitary operators. 84

85. In this lecture, we saw how the completeness relation for ONBs allows one to obtain representation dependent equations from representation independent ones. We also defined the matrix elements of an operator between different states, and extended the action of operators so that they could act either to the right on kets or to the left on bras. In the process, we were led to the notion of Hermitian conjugation, which allows us to identify any object in the space of kets, with its corresponding object in the space of bras, and vice versa, and developed some rules for “taking the Hermitian adjoint” of any expression formulated in the Dirac notation. In the next lecture, we use this idea to introduce the concept of Hermitian, anti- Hermitian, and Unitary operators, and to develop matrix representations of linear operators. 85

86. 86