1. Quantum One

3. Hermitian, Anti-Hermitian, and Unitary Operators

4. In the last segment, we saw how the completeness relation for ONBs allows one to easily obtain representation dependent equations from representation independent ones, and vice versa.
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 relation in the space of kets, with the corresponding relation in the space of bras, and vice versa, and we developed some rules for “taking the Hermitian adjoint” of any expression formulated in the Dirac notation.
In the this lecture, we use this idea to introduce some important new concepts.
We start with the definition of a Hermitian operator.

5. In the last segment, we saw how the completeness relation for ONBs allows one to easily obtain representation dependent equations from representation independent ones, and vice versa.
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 relation in the space of kets, with the corresponding relation in the space of bras, and vice versa, and we developed some rules for “taking the Hermitian adjoint” of any expression formulated in the Dirac notation.
In the this lecture, we use this idea to introduce some important new concepts.
We start with the definition of a Hermitian operator.

6. In the last segment, we saw how the completeness relation for ONBs allows one to easily obtain representation dependent equations from representation independent ones, and vice versa.
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 relation in the space of kets, with the corresponding relation in the space of bras, and vice versa, and we developed some rules for “taking the Hermitian adjoint” of any expression formulated in the Dirac notation.
In the this lecture, we use this idea to introduce some important new concepts.
We start with the definition of a Hermitian operator.

7. In the last segment, we saw how the completeness relation for ONBs allows one to easily obtain representation dependent equations from representation independent ones, and vice versa.
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 relation in the space of kets, with the corresponding relation in the space of bras, and vice versa, and we developed some rules for “taking the Hermitian adjoint” of any expression formulated in the Dirac notation.
In the this lecture, we use this idea to introduce some important new concepts.
We start with the definition of a Hermitian operator.

8. Definition: An operator A is said to be Hermitian or self adjoint if A is equal to its Hermitian adjoint, i.e., if
A = A⁺.
In terms of matrix elements, the property
which is true for any operator, reduces for Hermitian operators to the relation
As a special case this implies that
Thus, the expectation values of Hermitian operators are strictly real.

9. Definition: An operator A is said to be Hermitian or self adjoint if A is equal to its Hermitian adjoint, i.e., if
A = A⁺.
In terms of matrix elements, the property
which is true for any operator, reduces for Hermitian operators to the relation
As a special case this implies that
Thus, the expectation values of Hermitian operators are strictly real.

10. Definition: An operator A is said to be Hermitian or self adjoint if A is equal to its Hermitian adjoint, i.e., if
A = A⁺.
In terms of matrix elements, the property
which is true for any operator, reduces for Hermitian operators to the relation
As a special case this implies that
Thus, the expectation values of Hermitian operators are strictly real.

11. Definition: An operator A is said to be Hermitian or self adjoint if A is equal to its Hermitian adjoint, i.e., if
A = A⁺.
In terms of matrix elements, the property
which is true for any operator, reduces for Hermitian operators to the relation
As a special case this implies that
Thus, the expectation values of Hermitian operators are strictly real.

12. Definition: An operator A is said to be Hermitian or self adjoint if A is equal to its Hermitian adjoint, i.e., if
A = A⁺.
In terms of matrix elements, the property
which is true for any operator, reduces for Hermitian operators to the relation
As a special case this implies that
Thus, the expectation values of Hermitian operators are strictly real.

13. Definition: An operator A is said to be Hermitian or self adjoint if A is equal to its Hermitian adjoint, i.e., if
A = A⁺.
In terms of matrix elements, the property
which is true for any operator, reduces for Hermitian operators to the relation
As a special case this implies that
Thus, the expectation values of Hermitian operators are strictly real.

14. Definition: An operator A is said to be Hermitian or self adjoint if A is equal to its Hermitian adjoint, i.e., if
A = A⁺.
In terms of matrix elements, the property
which is true for any operator, reduces for Hermitian operators to the relation
As a special case this implies that
Thus, the expectation values of Hermitian operators are strictly real.
In a related definition, we now introduce the idea of anti-Hermitian operators.

15. Definition: An operator A is said to be anti-Hermitian if it is equal to the negative of its Hermitian adjoint, i.e., if
In terms of matrix elements, the property
which is true for any operator, reduces for anti-Hermitian operators to the relation
As a special case this implies that
Thus, expectation values of anti-Hermitian operators are strictly imaginary.

16. Definition: An operator A is said to be anti-Hermitian if it is equal to the negative of its Hermitian adjoint, i.e., if
In terms of matrix elements, the property
which is true for any operator, reduces for anti-Hermitian operators to the relation
As a special case this implies that
Thus, expectation values of anti-Hermitian operators are strictly imaginary.

17. Definition: An operator A is said to be anti-Hermitian if it is equal to the negative of its Hermitian adjoint, i.e., if
In terms of matrix elements, the property
which is true for any operator, reduces for anti-Hermitian operators to the relation
As a special case this implies that
Thus, expectation values of anti-Hermitian operators are strictly imaginary.

18. Definition: An operator A is said to be anti-Hermitian if it is equal to the negative of its Hermitian adjoint, i.e., if
In terms of matrix elements, the property
which is true for any operator, reduces for anti-Hermitian operators to the relation
As a special case this implies that
Thus, expectation values of anti-Hermitian operators are strictly imaginary.

19. Definition: An operator A is said to be anti-Hermitian if it is equal to the negative of its Hermitian adjoint, i.e., if
In terms of matrix elements, the property
which is true for any operator, reduces for anti-Hermitian operators to the relation
As a special case this implies that
Thus, expectation values of anti-Hermitian operators are strictly imaginary.

20. Definition: An operator A is said to be anti-Hermitian if it is equal to the negative of its Hermitian adjoint, i.e., if
In terms of matrix elements, the property
which is true for any operator, reduces for anti-Hermitian operators to the relation
As a special case this implies that
Thus, expectation values of anti-Hermitian operators are strictly imaginary.

21. Definition: An operator A is said to be anti-Hermitian if it is equal to the negative of its Hermitian adjoint, i.e., if
In terms of matrix elements, the property
which is true for any operator, reduces for anti-Hermitian operators to the relation
As a special case this implies that
Thus, expectation values of anti-Hermitian operators are strictly imaginary.

22. Note that if A is any operator, it may be written in the form
where A_{H}=(1/2)(A+A⁺) (the Hermitian part of A) is Hermitian
and A_{A}=(1/2)(A-A⁺) (the anti-Hermitian part of A) is anti- Hermitian.

23. Note that if A is any operator, it may be written in the form
where A_{H}=(1/2)(A+A⁺) (the Hermitian part of A) is Hermitian
and A_{A}=(1/2)(A-A⁺) (the anti-Hermitian part of A) is anti- Hermitian.

24. Note that if A is any operator, it may be written in the form
where A_{H}=(1/2)(A+A⁺) (the Hermitian part of A) is Hermitian
and A_{A}=(1/2)(A-A⁺) (the anti-Hermitian part of A) is anti- Hermitian.

25. Note that if A is any operator, it may be written in the form
where A_{H}=(1/2)(A+A⁺) (the Hermitian part of A) is Hermitian, since
and A_{A}=(1/2)(A-A⁺) (the anti-Hermitian part of A) is anti- Hermitian.

26. Note that if A is any operator, it may be written in the form
where A_{H}=(1/2)(A+A⁺) (the Hermitian part of A) is Hermitian, since
and A_{A}=(1/2)(A-A⁺) (the anti-Hermitian part of A) is anti- Hermitian.

27. Note that if A is any operator, it may be written in the form
where A_{H}=(1/2)(A+A⁺) (the Hermitian part of A) is Hermitian, since
and A_{A}=(1/2)(A-A⁺) (the anti-Hermitian part of A) is anti- Hermitian.

28. Note that if A is any operator, it may be written in the form
where A_{H}=(1/2)(A+A⁺) (the Hermitian part of A) is Hermitian, since
and where A_{A}=(1/2)(A-A⁺) (the anti-Hermitian part of A) is anti-Hermitian.

29. Note that if A is any operator, it may be written in the form
where A_{H}=(1/2)(A+A⁺) (the Hermitian part of A) is Hermitian, since
and where A_{A}=(1/2)(A-A⁺) (the anti-Hermitian part of A) is anti-Hermitian

30. Note that if A is any operator, it may be written in the form
where A_{H}=(1/2)(A+A⁺) (the Hermitian part of A) is Hermitian, since
and A_{A}=(1/2)(A-A⁺) (the anti-Hermitian part of A) is anti-Hermitian, since

31. Note that if A is any operator, it may be written in the form
where A_{H}=(1/2)(A+A⁺) (the Hermitian part of A) is Hermitian
and A_{A}=(1/2)(A-A⁺) (the anti-Hermitian part of A) is anti-Hermitian, since

32. Note that if A is any operator, it may be written in the form
where A_{H}=(1/2)(A+A⁺) (the Hermitian part of A) is Hermitian
and A_{A}=(1/2)(A-A⁺) (the anti-Hermitian part of A) is anti-Hermitian, since

33. Thus an arbitrary operator can be decomposed into Hermitian and anti-Hermitian parts,
very similar to the way that an arbitrary complex number
z =x + iy
can be decomposed in real and imaginary parts
The concept of the Hermitian adjoint also allows us to define a very important class of operators referred to as unitary operators.

34. Definition: An operator U is said to be unitary if its adjoint is equal to its inverse:
Thus, for a unitary operator U⁺=U⁻¹,
or equivalently,
Unitary operators (or the transformations they induce) play the same role in quantum mechanical Hilbert spaces that orthogonal transformations play in R³.
Indeed, note that if the states form an ONB for S, and if
then

35. Definition: An operator U is said to be unitary if its adjoint is equal to its inverse:
Thus, for a unitary operator U⁺=U⁻¹,
or equivalently,
Unitary operators (or the transformations they induce) play the same role in quantum mechanical Hilbert spaces that orthogonal transformations play in R³.
Indeed, note that if the states form an ONB for S, and if
then

36. Definition: An operator U is said to be unitary if its adjoint is equal to its inverse:
Thus, for a unitary operator U⁺=U⁻¹,
or equivalently,
Unitary operators (or the transformations they induce) play the same role in quantum mechanical Hilbert spaces that orthogonal transformations play in R³.
Indeed, note that if the states form an ONB for S, and if
then

37. Definition: An operator U is said to be unitary if its adjoint is equal to its inverse:
Thus, for a unitary operator U⁺=U⁻¹,
or equivalently,
Unitary operators (or the transformations they induce) play the same role in quantum mechanical Hilbert spaces that orthogonal transformations play in R³.
Indeed, note that if the states form an ONB for S, and if
then

38. Definition: An operator U is said to be unitary if its adjoint is equal to its inverse:
Thus, for a unitary operator U⁺=U⁻¹,
or equivalently,
Unitary operators (or the transformations they induce) play the same role in quantum mechanical Hilbert spaces that orthogonal transformations play in R³.
Indeed, note that if the states form an ONB for S, and if
then

39. Definition: An operator U is said to be unitary if its adjoint is equal to its inverse:
Thus, for a unitary operator U⁺=U⁻¹,
or equivalently,
Unitary operators (or the transformations they induce) play the same role in quantum mechanical Hilbert spaces that orthogonal transformations play in R³.
Indeed, note that if the states form an ONB for S, and if
then

40. Definition: An operator U is said to be unitary if its adjoint is equal to its inverse:
Thus, for a unitary operator U⁺=U⁻¹,
or equivalently,
Unitary operators (or the transformations they induce) play the same role in quantum mechanical Hilbert spaces that orthogonal transformations play in R³.
Indeed, note that if the states form an ONB for S, and if
then

41. Definition: An operator U is said to be unitary if its adjoint is equal to its inverse:
Thus, for a unitary operator U⁺=U⁻¹,
or equivalently,
Unitary operators (or the transformations they induce) play the same role in quantum mechanical Hilbert spaces that orthogonal transformations play in R³.
Indeed, note that if the states form an ONB for S, and if
then

42. Definition: An operator U is said to be unitary if its adjoint is equal to its inverse:
Thus, for a unitary operator U⁺=U⁻¹,
or equivalently,
Unitary operators (or the transformations they induce) play the same role in quantum mechanical Hilbert spaces that orthogonal transformations play in R³.
Indeed, note that if the states form an ONB for S, and if
then

43. Definition: An operator U is said to be unitary if its adjoint is equal to its inverse:
Thus, for a unitary operator U⁺=U⁻¹,
or equivalently,
Unitary operators (or the transformations they induce) play the same role in quantum mechanical Hilbert spaces that orthogonal transformations play in R³.
Indeed, note that if the states form an ONB for S, and if
then

44. Thus, unitary operators map complete any complete orthonormal basis of states onto another complete orthonormal basis for the same space, and generally preserve vector relationships in state space, the way that orthogonal transformations do in R³.

46. Ket-Bra and Matrix Representation of Operators

47. Matrix Representation of Operators : Let {|n〉} be an ONB for the space S and let A be an operator acting in the space.
From the trivial identity
we write

48. Matrix Representation of Operators : Let {|n〉} be an ONB for the space S and let A be an operator acting in the space.
From the trivial identity
we write

49. Matrix Representation of Operators : Let {|n〉} be an ONB for the space S and let A be an operator acting in the space.
From the trivial identity
we write

50. Matrix Representation of Operators : Let {|n〉} be an ONB for the space S and let A be an operator acting in the space.
From the trivial identity or
This gives what we call a ket-bra expansion for this operator in this representation, in which appear the matrix elements
of A connecting the basis states |n〉 and |n′〉.

51. Matrix Representation of Operators : Let {|n〉} be an ONB for the space S and let A be an operator acting in the space.
From the trivial identity or
This gives what is called a ket-bra expansion for this operator in this representation, and completely specifies the linear operator A in terms of its matrix elements
connecting the basis states of this representation.

52. Matrix Representation of Operators
The operator , therefore, is completely determined by its matrix elements in any ONB.
Thus, suppose that
for some states and
The expansion coefficients for the states and clearly related.
Note that if
then
which can be written

53. Matrix Representation of Operators
The operator , therefore, is completely determined by its matrix elements in any ONB.
Thus, suppose that
for some states and
The expansion coefficients for the states and clearly related.
Note that if
then
which can be written

54. Matrix Representation of Operators
The operator , therefore, is completely determined by its matrix elements in any ONB.
Thus, suppose that
for some states and
The expansion coefficients for the states and clearly related.
Note that if
then
which can be written

55. Matrix Representation of Operators
The operator , therefore, is completely determined by its matrix elements in any ONB.
Thus, suppose that
for some states and
The expansion coefficients for the states and clearly related.
Note that if
then
which can be written

56. Matrix Representation of Operators
The operator , therefore, is completely determined by its matrix elements in any ONB.
Thus, suppose that
for some states and
The expansion coefficients for the states and clearly related.
Note that if
then
which can be written

57. Matrix Representation of Operators
The operator , therefore, is completely determined by its matrix elements in any ONB.
Thus, suppose that
for some states and
The expansion coefficients for the states and clearly related.
Note that if
then
which can be written

58. Matrix Representation of Operators
The operator , therefore, is completely determined by its matrix elements in any ONB.
Thus, suppose that
for some states and
The expansion coefficients for the states and clearly related.
Note that if
then
which can be written

59. Matrix Representation of Operators
The operator , therefore, is completely determined by its matrix elements in any ONB.
Thus, suppose that
for some states and
The expansion coefficients for the states and clearly related.
Note that if
then
which can be written

60. Matrix Representation of Operators
The operator , therefore, is completely determined by its matrix elements in any ONB.
Thus, suppose that
for some states and
The expansion coefficients for the states and clearly related.
Note that if
then
which can be written

61. Matrix Representation of Operators
But this is of the form of a matrix multiplying a column vector
i.e.,
is just the ith component of the equation that one obtains through the operation

62. Matrix Representation of Operators
But this is of the form of a matrix multiplying a column vector
i.e.,
is just the i-th component of the equation that one obtains through the operation

63. Matrix Representation of Operators
But this is of the form of a matrix multiplying a column vector
i.e.,
is just the i-th component of the equation that one obtains through the operation

64. Matrix Representation of Operators
But this is of the form of a matrix multiplying a column vector
i.e.,
is just the i-th component of the equation that one obtains through the operation

65. Matrix Representation of Operators
Thus we see that in the row-vector-column vector representation induced by any discrete ONB, an operator A is naturally represented by a square matrix , i.e.,
with entries that are just the matrix elements of A connecting the different basis states of that representation.
This matrix representation facilitates computing quantities related to A itself.

66. Matrix Representation of Operators
Thus we see that in the row-vector-column vector representation induced by any discrete ONB, an operator A is naturally represented by a square matrix , i.e.,
with entries that are just the matrix elements of A connecting the different basis states of that representation.
This matrix representation facilitates computing quantities related to A itself.

67. Matrix Representation of Operators
Thus we see that in the row-vector-column vector representation induced by any discrete ONB, an operator A is naturally represented by a square matrix , i.e.,
with entries that are just the matrix elements of A connecting the different basis states of that representation.
This matrix representation facilitates computing quantities related to A itself.

68. Matrix Representation of Operators
Thus we see that in the row-vector-column vector representation induced by any discrete ONB, an operator A is naturally represented by a square matrix , i.e.,
with entries that are just the matrix elements of A connecting the different basis states of that representation.
This matrix representation facilitates computing quantities related to A itself.

69. Matrix Representation of Operators
Thus we see that in the row-vector-column vector representation induced by any discrete ONB, an operator A is naturally represented by a square matrix , i.e.,
with entries that are just the matrix elements of A connecting the different basis states of that representation.
This matrix representation facilitates computing quantities related to A itself.

70. Matrix Representation of Operators
Consider the matrix element of A between arbitrary states and
Inserting our expansion for A this becomes
where and so that we can write
which is readily expressed in terms of the product of a row-vector, a square matrix, and a column-vector, i.e.,

71. Matrix Representation of Operators
Consider the matrix element of A between arbitrary states and
Inserting our expansion for A this becomes
where and so that we can write
which is readily expressed in terms of the product of a row-vector, a square matrix, and a column-vector, i.e.,

72. Matrix Representation of Operators
Consider the matrix element of A between arbitrary states and
Inserting our expansion for A this becomes
where and so that we can write
which is readily expressed in terms of the product of a row-vector, a square matrix, and a column-vector, i.e.,

73. Matrix Representation of Operators
Consider the matrix element of A between arbitrary states and
Inserting our expansion for A this becomes
But and so that we can write
which is readily expressed in terms of the product of a row-vector, a square matrix, and a column-vector, i.e.,

74. Matrix Representation of Operators
Consider the matrix element of A between arbitrary states and
Inserting our expansion for A this becomes
But and so we can write this as
which is readily expressed in terms of the product of a row-vector, a square matrix, and a column-vector, i.e.,

75. Matrix Representation of Operators
Consider the matrix element of A between arbitrary states and
Inserting our expansion for A this becomes
But and so we can write this as
which is readily expressed in terms of the product of a row-vector, a square matrix, and a column-vector, i.e.,

76. Matrix Representation of Operators
Consider the matrix element of A between arbitrary states and
Inserting our expansion for A this becomes
But and so we can write this as
which is readily expressed in terms of the product of a row-vector, a square matrix, and a column-vector.

77. Matrix Representation of Operators
That is, the expression
is equivalent to

78. Matrix Representation of Operators
That is, the expression
is equivalent to

79. Matrix Representation of Operators As another example, consider the operator product of
and
The product operator C=AB has a similar expansion, i.e.,
where so

80. Matrix Representation of Operators As another example, consider the operator product of
and
The product operator C=AB has a similar expansion, i.e.,
where so

81. Matrix Representation of Operators As another example, consider the operator product of
and
The product operator C=AB has a similar expansion, i.e.,
where so

82. Matrix Representation of Operators As another example, consider the operator product of
and
The product operator C=AB has a similar expansion, i.e.,
where so

83. Matrix Representation of Operators As another example, consider the operator product of
and
The product operator C=AB has a similar expansion, i.e.,
where so

84. Matrix Representation of Operators As another example, consider the operator product of
and
The product operator C=AB has a similar expansion, i.e.,
where so

85. Matrix Representation of Operators As another example, consider the operator product of
and
The product operator C=AB has a similar expansion, i.e.,
where so

86. Matrix Representation of Operators As another example, consider the operator product of
and
The product operator C=AB has a similar expansion, i.e.,
where so

87. Matrix Representation of Operators
But this is just of the form of a matrix multiplication, i.e.,
is equivalent to
which can write as where as before, [A] stands for “the matrix representing ”, etc.

88. Matrix Representation of Operators
But this is just of the form of a matrix multiplication, i.e.,
is equivalent to
which can write as where as before, [A] stands for “the matrix representing ”, etc.

89. Matrix Representation of Operators
But this is just of the form of a matrix multiplication, i.e.,
is equivalent to
which can write as where as before, [A] stands for “the matrix representing ”, etc.

90. Matrix Representation of Operators
But this is just of the form of a matrix multiplication, i.e.,
is equivalent to
which can write as where as before, [A] stands for “the matrix representing ”, etc.

91. Matrix Representation of Operators
But this is just of the form of a matrix multiplication, i.e.,
is equivalent to
which can write as where as before, [A] stands for “the matrix representing ”, etc.

92. Matrix Representation of Operators As a final example, consider the matrix representing the adjoint of an operator. If
then by the two-part rule we developed for taking the adjoint, it follows that
We can now switch the summation indices to find that
from which we deduce that

93. Matrix Representation of Operators As a final example, consider the matrix representing the adjoint of an operator. If
then by the two-part rule we developed for taking the adjoint, it follows that
We can now switch the summation indices to find that
from which we deduce that

94. Matrix Representation of Operators As a final example, consider the matrix representing the adjoint of an operator. If
then by the two-part rule we developed for taking the adjoint, it follows that
We can now switch the summation indices to find that
from which we deduce that

95. Matrix Representation of Operators As a final example, consider the matrix representing the adjoint of an operator. If
then by the two-part rule we developed for taking the adjoint, it follows that
We can now switch the summation indices to find that
from which we deduce that

96. Matrix Representation of Operators As a final example, consider the matrix representing the adjoint of an operator. If
then by the two-part rule we developed for taking the adjoint, it follows that
We can now switch the summation indices to find that
from which we deduce that

97. Matrix Representation of Operators As a final example, consider the matrix representing the adjoint of an operator. If
then by the two-part rule we developed for taking the adjoint, it follows that
We can now switch the summation indices to find that
from which we deduce that

98. Thus, the matrix
representing is the complex-conjugate transpose of the matrix representing .
We thus define the adjoint of a matrix to be this combined operation (which applies to single column/row vectors as well, obviously).
A Hermitian operator is equal to its adjoint, so that the matrix elements representing such an operator obey the relation
Thus, Hermitian operators are represented by Hermitian matrices, which satisfy

99. Thus, the matrix
representing is the complex-conjugate transpose of the matrix representing .
We thus define the adjoint of a matrix to be this combined operation (which applies to single column/row vectors as well, obviously).
A Hermitian operator is equal to its adjoint, so that the matrix elements representing such an operator obey the relation
Thus, Hermitian operators are represented by Hermitian matrices, which satisfy

100. Thus, the matrix
representing is the complex-conjugate transpose of the matrix representing .
We thus define the adjoint of a matrix to be this combined operation (which applies to single column/row vectors as well, obviously).
A Hermitian operator is equal to its adjoint, so that the matrix elements representing such an operator obey the relation
Thus, Hermitian operators are represented by Hermitian matrices, which satisfy

101. Thus, the matrix
representing is the complex-conjugate transpose of the matrix representing .
We thus define the adjoint of a matrix to be this combined operation (which applies to single column/row vectors as well, obviously).
A Hermitian operator is equal to its adjoint, so that the matrix elements representing such an operator obey the relation
Thus, Hermitian operators are represented by Hermitian matrices, which satisfy

102. It is an interesting fact that neither the transpose nor the complex conjugate of an operator are well defined concepts; i.e.,
Given a linear operator A there is no operator that can be uniquely identified with the transpose of A.
Although one can form the transpose of the matrix [A] representing A in any basis, the operator associated with the transposed matrix will not correspond to the operator associated with the transpose of the matrix representing A in other bases.
Similarly, complex conjugation can be performed on matrices, or on matrix elements, but it is not an operation that is defined for the operators themselves.
The Hermitian adjoint, which in a sense combines these two representation dependent operations, yields an operator that is independent of representation.

103. It is an interesting fact that neither the transpose nor the complex conjugate of an operator are well defined concepts; i.e.,
Given a linear operator A there is no operator that can be uniquely identified with the transpose of .
Although one can form the transpose of the matrix [A] representing A in any basis, the operator associated with the transposed matrix will not correspond to the operator associated with the transpose of the matrix representing the linear operator in any other representation.
Similarly, complex conjugation can be performed on matrices, or on matrix elements, but it is not an operation that is defined for the operators themselves.
The Hermitian adjoint, which in a sense combines these two representation dependent operations, yields an operator that is independent of representation.

104. It is an interesting fact that neither the transpose nor the complex conjugate of an operator are well defined concepts; i.e.,
Given a linear operator A there is no operator that can be uniquely identified with the transpose of .
Although one can form the transpose of the matrix [A] representing A in any basis, the operator associated with the transposed matrix will not correspond to the operator associated with the transpose of the matrix representing the linear operator in any other representation.
Similarly, complex conjugation can be performed on matrices, or on matrix elements, but it is not an operation that is defined for the operators themselves.
The Hermitian adjoint, which in a sense combines these two representation dependent operations, yields an operator that is independent of representation.

105. It is an interesting fact that neither the transpose nor the complex conjugate of an operator are well defined concepts; i.e.,
Given a linear operator A there is no operator that can be uniquely identified with the transpose of .
Although one can form the transpose of the matrix [A] representing A in any basis, the operator associated with the transposed matrix will not correspond to the same linear operator associated with the transpose of the matrix representing the linear operator in any other representation.
Similarly, complex conjugation can be performed on matrices, or on matrix elements, but it is not an operation that is defined for the operators themselves.
The Hermitian adjoint, which in a sense combines these two representation dependent operations, yields an operator that is independent of representation.

106. It is an interesting fact that neither the transpose nor the complex conjugate of an operator are well defined concepts; i.e.,
Given a linear operator A there is no operator that can be uniquely identified with the transpose of .
Although one can form the transpose of the matrix [A] representing A in any basis, the operator associated with the transposed matrix will not correspond to the same linear operator associated with the transpose of the matrix representing the linear operator in any other representation.
Similarly, complex conjugation can be performed on matrices, or on matrix elements, but it is not an operation that is defined for the operators themselves.
The Hermitian adjoint, which in a sense combines these two representation dependent operations, yields an operator that is independent of representation.

107. It is an interesting fact that neither the transpose nor the complex conjugate of an operator are well defined concepts; i.e.,
Given a linear operator A there is no operator that can be uniquely identified with the transpose of .
Although one can form the transpose of the matrix [A] representing A in any basis, the operator associated with the transposed matrix will not correspond to the same linear operator associated with the transpose of the matrix representing the linear operator in any other representation.
Similarly, complex conjugation can be performed on matrices, or on matrix elements, but it is not an operation that is defined for the operators themselves.
The Hermitian adjoint, which in a sense combines these two representation dependent operations, yields an operator that is independent of representation.

108. It is an interesting fact that neither the transpose nor the complex conjugate of an operator are well defined concepts; i.e.,
Given a linear operator A there is no operator that can be uniquely identified with the transpose of .
Although one can form the transpose of the matrix [A] representing A in any basis, the operator associated with the transposed matrix will not correspond to the same linear operator associated with the transpose of the matrix representing the linear operator in any other representation.
Similarly, complex conjugation can be performed on matrices, or on matrix elements, but it is not an operation that is defined for linear operators themselves.
The Hermitian adjoint, which in a sense combines these two representation dependent operations, yields an operator that is independent of representation.

109. It is an interesting fact that neither the transpose nor the complex conjugate of an operator are well defined concepts; i.e.,
Given a linear operator A there is no operator that can be uniquely identified with the transpose of .
Although one can form the transpose of the matrix [A] representing A in any basis, the operator associated with the transposed matrix will not correspond to the same linear operator associated with the transpose of the matrix representing the linear operator in any other representation.
Similarly, complex conjugation can be performed on matrices, or on matrix elements, but it is not an operation that is defined for linear operators themselves.
The Hermitian adjoint, which in a sense combines these two representation dependent operations, yields an operator that is independent of representation.

110. This again emphasizes one of the basic themes, which is that bras, kets, and operators are not row vectors, column vectors, and matrices.
The former may be represented by the latter, but they are conceptually different things, and it always important to keep that in mind.
Matrix representations of this form were developed extensively by Heisenberg and gave rise to the term ``matrix mechanics'', in analogy to the ``wave mechanics'' developed by Schro”dinger, which focuses on a wave function representation for the underlying space.
Clearly, however, whether one has a wave mechanical or matrix mechanical representation depends simply upon the choice of basis (i.e., discrete or continuous) in which one is working.

111. This again emphasizes one of the basic themes, which is that bras, kets, and operators are not row vectors, column vectors, and matrices.
The former may be represented by the latter, but they are conceptually different things, and it always important to keep that in mind.
Matrix representations of this form were developed extensively by Heisenberg and gave rise to the term ``matrix mechanics'', in analogy to the ``wave mechanics'' developed by Schro”dinger, which focuses on a wave function representation for the underlying space.
Clearly, however, whether one has a wave mechanical or matrix mechanical representation depends simply upon the choice of basis (i.e., discrete or continuous) in which one is working.

112. This again emphasizes one of the basic themes, which is that bras, kets, and operators are not row vectors, column vectors, and matrices.
The former may be represented by the latter, but they are conceptually different things, and it always important to keep that in mind.
Matrix representations of this form were developed extensively by Heisenberg and gave rise to the term ``matrix mechanics'', in analogy to the ``wave mechanics'' developed by Schro”dinger, which focuses on a wave function representation for the underlying space.
Clearly, however, whether one has a wave mechanical or matrix mechanical representation depends simply upon the choice of basis (i.e., discrete or continuous) in which one is working.

113. This again emphasizes one of the basic themes, which is that bras, kets, and operators are not row vectors, column vectors, and matrices.
The former may be represented by the latter, but they are conceptually different things, and it always important to keep that in mind.
Matrix representations of this form were developed extensively by Heisenberg and gave rise to the term ``matrix mechanics'', in analogy to the ``wave mechanics'' developed by Schrödinger, which focuses on a wave function representation for the underlying space.
Clearly, however, whether one has a wave mechanical or matrix mechanical representation depends simply upon the choice of basis (i.e., discrete or continuous) in which one is working.

114. This again emphasizes one of the basic themes, which is that bras, kets, and operators are not row vectors, column vectors, and matrices.
The former may be represented by the latter, but they are conceptually different things, and it always important to keep that in mind.
Matrix representations of this form were developed extensively by Heisenberg and gave rise to the term ``matrix mechanics'', in analogy to the ``wave mechanics'' developed by Schrödinger, which focuses on a wave function representation for the underlying space.
Clearly, however, whether one has a (i) wave mechanical or (ii) matrix mechanical representation of the space depends simply upon the choice of basis in which one is working i.e., whether it is (i) continuous or (ii) discrete.

115. In this segment, we defined what we mean by Hermitian operators, anti- Hermitian operators, and unitary operators, and saw how any operator can be expressed in terms of its Hermitian and anti-Hermitian parts.
We then used the completeness relation for a discrete ONB to develop ket-bra expansions, and matrix representations of general linear operators. .
We then saw how these matrix representations can be used to directly compute quantities related to the operators they represent.
Finally, we saw how to represent the matrix corresponding to the adjoint of an operator, and how Hermitian operators are represented by Hermitian matrices.
In the next lecture, we show these discrete representations can be extended to continuously indexed bases sets, where linear operators can be represented by ket-bra expansions in integral form.

116. In this segment, we defined what we mean by Hermitian operators, anti- Hermitian operators, and unitary operators, and saw how any operator can be expressed in terms of its Hermitian and anti-Hermitian parts.
We then used the completeness relation for a discrete ONB to develop ket-bra expansions, and matrix representations of general linear operators. .
We then saw how these matrix representations can be used to directly compute quantities related to the operators they represent.
Finally, we saw how to represent the matrix corresponding to the adjoint of an operator, and how Hermitian operators are represented by Hermitian matrices.
In the next lecture, we show these discrete representations can be extended to continuously indexed bases sets, where linear operators can be represented by ket-bra expansions in integral form.

117. In this segment, we defined what we mean by Hermitian operators, anti- Hermitian operators, and unitary operators, and saw how any operator can be expressed in terms of its Hermitian and anti-Hermitian parts.
We then used the completeness relation for a discrete ONB to develop ket-bra expansions, and matrix representations of general linear operators.
We then saw how these matrix representations can be used to directly compute quantities related to the operators they represent.
Finally, we saw how to represent the matrix corresponding to the adjoint of an operator, and how Hermitian operators are represented by Hermitian matrices.
In the next lecture, we show these discrete representations can be extended to continuously indexed bases sets, where linear operators can be represented by ket-bra expansions in integral form.

118. In this segment, we defined what we mean by Hermitian operators, anti- Hermitian operators, and unitary operators, and saw how any operator can be expressed in terms of its Hermitian and anti-Hermitian parts.
We then used the completeness relation for a discrete ONB to develop ket-bra expansions, and matrix representations of general linear operators.
We then saw how these matrix representations can be used to directly compute quantities related to the operators they represent.
Finally, we saw how to represent the matrix corresponding to the adjoint of an operator, and how Hermitian operators are represented by Hermitian matrices.
In the next lecture, we show these discrete representations can be extended to continuously indexed bases sets, where linear operators can be represented by ket-bra expansions in integral form.

119. In this segment, we defined what we mean by Hermitian operators, anti- Hermitian operators, and unitary operators, and saw how any operator can be expressed in terms of its Hermitian and anti-Hermitian parts.
We then used the completeness relation for a discrete ONB to develop ket-bra expansions, and matrix representations of general linear operators.
We then saw how these matrix representations can be used to directly compute quantities related to the operators they represent.
Finally, we saw how to represent the matrix corresponding to the adjoint of an operator, and how Hermitian operators are represented by Hermitian matrices.
In the next segment, we show how expressions and ideas developed for discrete representations can be extended to continuously-indexed bases sets, and how linear operators can be represented by ket-bra expansions in integral form.