1. Quantum One

3. Canonical Commutation Relations

4. In the last segment, we used the completeness relation for continuous ONBs to develop ket-bra expansions, and integral representations of linear operators.
We then saw how the integral kernel associated with these representations can be used to directly compute quantities related to the operators they represent.
We also introduced the notion of diagonality of an operator in a given representation, and developed expansions for the basic operators of a single quantum particle in the representations in which they are diagonal.
Finally, we saw how differential operators can also be expressed as ket-bra expansions with integral kernels that involve derivatives of the delta function.
We begin the current segment by deriving what are referred to as canonical commutation relations.

5. In the last segment, we used the completeness relation for continuous ONBs to develop ket-bra expansions, and integral representations of linear operators.
We then saw how the integral kernel associated with these representations can be used to directly compute quantities related to the operators they represent.
We also introduced the notion of diagonality of an operator in a given representation, and developed expansions for the basic operators of a single quantum particle in the representations in which they are diagonal.
Finally, we saw how differential operators can also be expressed as ket-bra expansions with integral kernels that involve derivatives of the delta function.
We begin the current segment by deriving what are referred to as canonical commutation relations.

6. In the last segment, we used the completeness relation for continuous ONBs to develop ket-bra expansions, and integral representations of linear operators.
We then saw how the integral kernel associated with these representations can be used to directly compute quantities related to the operators they represent.
We also introduced the notion of diagonality of an operator in a given representation, and developed expansions for the basic operators of a single quantum particle in the representations in which they are diagonal.
Finally, we saw how differential operators can also be expressed as ket-bra expansions with integral kernels that involve derivatives of the delta function.
We begin the current segment by deriving what are referred to as canonical commutation relations.

7. In the last segment, we used the completeness relation for continuous ONBs to develop ket-bra expansions, and integral representations of linear operators.
We then saw how the integral kernel associated with these representations can be used to directly compute quantities related to the operators they represent.
We also introduced the notion of diagonality of an operator in a given representation, and developed expansions for the basic operators of a single quantum particle in the representations in which they are diagonal.
Finally, we saw how differential operators can also be expressed as ket-bra expansions with integral kernels that involve derivatives of the delta function.
We begin the current segment by deriving what are referred to as canonical commutation relations.

8. In the last segment, we used the completeness relation for continuous ONBs to develop ket-bra expansions, and integral representations of linear operators.
We then saw how the integral kernel associated with these representations can be used to directly compute quantities related to the operators they represent.
We also introduced the notion of diagonality of an operator in a given representation, and developed expansions for the basic operators of a single quantum particle in the representations in which they are diagonal.
Finally, we saw how differential operators can also be expressed as ket-bra expansions with integral kernels that involve derivatives of the delta function.
We begin the current segment by deriving what are referred to as canonical commutation relations.

9. Canonical Commutation Relations It is clear that there is a close relationship between the position operator and the wavevector operator
This relationship is often expressed in terms of commutation relations between the different Cartesian components of these operators.
Note first that the Cartesian components of the position operator commute with one another, i.e.,
Since this is true for each element of an ONB we deduce the operator identity
This extends to the operator Z as well, so we can generally write

10. Canonical Commutation Relations It is clear that there is a close relationship between the position operator and the wavevector operator
This relationship is often expressed in terms of commutation relations between the different Cartesian components of these operators.
Note first that the Cartesian components of the position operator commute with one another, i.e.,
Since this is true for each element of an ONB we deduce the operator identity
This extends to the operator Z as well, so we can generally write

11. Canonical Commutation Relations It is clear that there is a close relationship between the position operator and the wavevector operator
This relationship is often expressed in terms of commutation relations between the different Cartesian components of these operators.
Note first that the Cartesian components of the position operator commute with one another, i.e.,
Since this is true for each element of an ONB we deduce the operator identity
This extends to the operator Z as well, so we can generally write

12. Canonical Commutation Relations It is clear that there is a close relationship between the position operator and the wavevector operator
This relationship is often expressed in terms of commutation relations between the different Cartesian components of these operators.
Note first that the Cartesian components of the position operator commute with one another, i.e.,
Since this is true for each element of an ONB we deduce the operator identity
This extends to the operator Z as well, so we can generally write

13. Canonical Commutation Relations It is clear that there is a close relationship between the position operator and the wavevector operator
This relationship is often expressed in terms of commutation relations between the different Cartesian components of these operators.
Note first that the Cartesian components of the position operator commute with one another, i.e.,
Since this is true for each element of an ONB we deduce the operator identity
This extends to the operator Z as well, so we can generally write

14. Canonical Commutation Relations
A similar argument applied to basis states of the momentum representation shows that the Cartesian components of the wavevector or momentum operator also commute with one another, i.e.,
On the other hand, the Cartesian components of the position operator do not generally commute with the Cartesian components of the wavevector operator.
To see this it is useful to work in a specific representation.
We will work in the position representation.

15. Canonical Commutation Relations
A similar argument applied to basis states of the momentum representation shows that the Cartesian components of the wavevector or momentum operator also commute with one another, i.e.,
On the other hand, the Cartesian components of the position operator do not generally commute with the Cartesian components of the wavevector operator.
To see this it is useful to work in a specific representation.
We will work in the position representation.

16. Canonical Commutation Relations
A similar argument applied to basis states of the momentum representation shows that the Cartesian components of the wavevector or momentum operator also commute with one another, i.e.,
On the other hand, the Cartesian components of the position operator do not generally commute with the Cartesian components of the wavevector operator.
To see this it is useful to work in a specific representation.
We will work in the position representation.

17. Canonical Commutation Relations
A similar argument applied to basis states of the momentum representation shows that the Cartesian components of the wavevector or momentum operator also commute with one another, i.e.,
On the other hand, the Cartesian components of the position operator do not generally commute with the Cartesian components of the wavevector operator.
To see this it is useful to work in a specific representation.
We will work in the position representation.

18. Canonical Commutation Relations
A similar argument applied to basis states of the momentum representation shows that the Cartesian components of the wavevector or momentum operator also commute with one another, i.e.,
On the other hand, the Cartesian components of the position operator do not generally commute with the Cartesian components of the wavevector operator.
To see this it is useful to work in a specific representation.
We will work in the position representation.

19. Canonical Commutation Relations
In the position representation, we note that for an arbitrary state represented by the wave function
On the other hand,
where we have used the standard relation
Combining these two terms, we find that in the position representation

20. Canonical Commutation Relations
In the position representation, we note that for an arbitrary state represented by the wave function
On the other hand,
where we have used the standard relation
Combining these two terms, we find that in the position representation

21. Canonical Commutation Relations
In the position representation, we note that for an arbitrary state represented by the wave function
On the other hand,
where we have used the standard relation
Combining these two terms, we find that in the position representation

22. Canonical Commutation Relations
In the position representation, we note that for an arbitrary state represented by the wave function
On the other hand,
where we have used the standard relation
Combining these two terms, we find that in the position representation

23. Canonical Commutation Relations
In the position representation, we note that for an arbitrary state represented by the wave function
On the other hand,
where we have used the standard relation
Combining these two terms, we find that in the position representation

24. Canonical Commutation Relations
In the position representation, we note that for an arbitrary state represented by the wave function
On the other hand,
where we have used the standard relation
Combining these two terms, we find that in the position representation

25. Canonical Commutation Relations
In the position representation, we note that for an arbitrary state represented by the wave function
On the other hand,
where we have used the standard relation
Combining these two terms, we find that in the position representation

26. Canonical Commutation Relations
In the position representation, we note that for an arbitrary state represented by the wave function
On the other hand,
where we have used the standard relation
Combining these two terms, we find that in the position representation

27. Canonical Commutation Relations
In the position representation, we note that for an arbitrary state represented by the wave function
On the other hand,
where we have used the standard relation
Subtracting these two terms, we find that in the position representation

28. Canonical Commutation Relations
Being true for all states and all basis states of the position representation, we deduce the operator identity
which often appears in terms of the momentum operator, in the form first derived by Max Born, i.e.,

29. Canonical Commutation Relations
Being true for all states and all basis states of the position representation, we deduce the operator identity
which often appears in terms of the momentum operator, in the form first derived by Max Born, i.e.,

30. Canonical Commutation Relations
Being true for all states and all basis states of the position representation, we deduce the operator identity
which often appears in terms of the momentum operator, in the form first derived by Max Born, i.e.,

31. Canonical Commutation Relations
Being true for all states and all basis states of the position representation, we deduce the operator identity
which often appears in terms of the momentum operator, in the form first derived by Max Born, i.e.,

32. Canonical Commutation Relations
Being true for all states and all basis states of the position representation, we deduce the operator identity
which often appears in terms of the momentum operator, in the form first obtained by Max Born, i.e.,

35. Changing Representations: Unitary Operators

36. In an earlier segment we defined unitary operators, which satisfy
and saw that unitary operators always map an ONB for S onto some other ONB for the same space.
In fact, given any two ONBs for S, there exists a unitary operator U that maps one set onto another, and whose adjoint maps the second set back onto the first.
To see this let and be two different ONBs for S so that

37. In an earlier segment we defined unitary operators, which satisfy
and saw that unitary operators always map an ONB for S onto some other ONB for the same space.
In fact, given any two ONBs for S, there exists a unitary operator U that maps one set onto another, and whose adjoint maps the second set back onto the first.
To see this let and be two different ONBs for S so that

38. In an earlier segment we defined unitary operators, which satisfy
and saw that unitary operators always map an ONB for S onto some other ONB for the same space.
In fact, given any two ONBs for S, there exists a unitary operator U that maps one set onto another, and whose adjoint maps the second set back onto the first.
To see this let and be two different ONBs for S so that

39. In an earlier segment we defined unitary operators, which satisfy
and saw that unitary operators always map an ONB for S onto some other ONB for the same space.
In fact, given any two ONBs for S, there exists a unitary operator U that maps one set onto another, and whose adjoint maps the second set back onto the first.
To see this let and be two different ONBs for S so that

40. In an earlier segment we defined unitary operators, which satisfy
and saw that unitary operators always map an ONB for S onto some other ONB for the same space.
In fact, given any two ONBs for S, there exists a unitary operator U that maps one set onto another, and whose adjoint maps the second set back onto the first.
To see this let and be two different ONBs for S so that

41. Let U be a linear operator, defined so that
Then the matrix elements of U in the |ψ_{i}〉 representation are
just the expansion coefficients for the basis vectors in terms of the other basis vectors
From these expansion coefficients, we can construct a ket-bra expansion for U.
Identifying the identify operator on the left, this reduces to a single sum

42. Let U be a linear operator, defined so that
Then the matrix elements of U in the representation are
just the expansion coefficients for the basis vectors in terms of the other basis vectors
From these expansion coefficients, we can construct a ket-bra expansion for U.
Identifying the identify operator on the left, this reduces to a single sum

43. Let U be a linear operator, defined so that
Then the matrix elements of U in the representation are
just the expansion coefficients for the basis vectors in terms of the basis vectors
From these expansion coefficients, we can construct a ket-bra expansion for U.
Identifying the identify operator on the left, this reduces to a single sum

44. Let U be a linear operator, defined so that
Then the matrix elements of U in the representation are
just the expansion coefficients for the basis vectors in terms of the basis vectors
From these expansion coefficients, we can construct a ket-bra expansion for U.
Identifying the identify operator on the left, this reduces to a single sum

45. Let U be a linear operator, defined so that
Then the matrix elements of U in the representation are
just the expansion coefficients for the basis vectors in terms of the basis vectors
From these expansion coefficients, we can construct a ket-bra expansion for U.
Identifying the identify operator on the left, this reduces to a single sum

46. It follows that if
then
so that that
So the operators U and U⁺ that connect these two sets of basis states are unitary Note, moreover that
so that

47. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary Note, moreover that

48. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary Note, moreover that

49. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary Note, moreover that

50. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary Note, moreover that

51. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary Note, moreover that

52. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary Note, moreover that

53. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary. Note, moreover that

54. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary. Note, moreover that

55. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary. Note, moreover that

56. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary. Note, moreover that

57. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary. Note, moreover that

58. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary. Note, moreover that

59. It follows that if
then
so that
So the operators U and U⁺ that connect these two sets of basis states are unitary. Note, moreover that
so that and

60. In addition, we note that
are just the expansion coefficients for the basis states |ψ_{j}〉 in terms of the basis states |φ_{i}〉 . So we have the related pair of relation involving the matrix elements of U and U⁺ that
The practical use of these matrix elements come when we wish to transform from one representation to another.
Suppose, for example, that you and I have solved a given problem using different representations, and we want to compare our results.
Then I need to transform my expressions derived in my representation to yours, or vice versa.

61. In addition, we note that
are just the expansion coefficients for the basis states |ψ_{j}〉 in terms of the basis states |φ_{i}〉 . So we have a related pair of relations involving the matrix elements of U and U⁺ , i.e.,
The practical use of these matrix elements come when we wish to transform from one representation to another.
Suppose, for example, that you and I have solved a given problem using different representations, and we want to compare our results.
Then I need to transform my expressions derived in my representation to yours, or vice versa.

62. In addition, we note that
are just the expansion coefficients for the basis states |ψ_{j}〉 in terms of the basis states |φ_{i}〉 . So we have a related pair of relations involving the matrix elements of U and U⁺ , i.e.,
The practical use of these matrix elements come when we wish to transform from one representation to another.
Suppose, for example, that you and I have solved a given problem using different representations, and we want to compare our results.
Then I need to transform my expressions derived in my representation to yours, or vice versa.

63. In addition, we note that
are just the expansion coefficients for the basis states |ψ_{j}〉 in terms of the basis states |φ_{i}〉 . So we have a related pair of relations involving the matrix elements of U and U⁺ , i.e.,
The practical use of these matrix elements come when we wish to transform from one representation to another.
Suppose, for example, that you and I have both solved a given problem using different representations, and we want to compare our results.
Then I need to transform my expressions derived in my representation to yours, or vice versa.

64. In addition, we note that
are just the expansion coefficients for the basis states |ψ_{j}〉 in terms of the basis states |φ_{i}〉 . So we have a related pair of relations involving the matrix elements of U and U⁺ , i.e.,
The practical use of these matrix elements come when we wish to transform from one representation to another.
Suppose, for example, that you and I have both solved a given problem using different representations, and we want to compare our results.
Then I need to transform expressions derived in my representation to those derived in yours, and vice versa.

65. Transformation of Kets - Let |χ〉 be an arbitrary ket in the space.
It can be expanded in either of the two bases considered above, i.e., or
Question: how are the expansion coefficients in these two different representations related to one another?
To find out we use an appropriate decomposition of unity

66. Transformation of Kets - Let |χ〉 be an arbitrary ket in the space.
It can be expanded in either of the two bases considered above, i.e., or
Question: how are the expansion coefficients in these two different representations related to one another?
To find out we use an appropriate decomposition of unity

67. Transformation of Kets - Let |χ〉 be an arbitrary ket in the space.
It can be expanded in either of the two bases considered above, i.e., or
Question: how are the expansion coefficients in these two different representations related to one another?
To find out we use an appropriate decomposition of unity

68. Transformation of Kets - Let |χ〉 be an arbitrary ket in the space.
It can be expanded in either of the two bases considered above, i.e., or
Question: how are the expansion coefficients in these two different representations related to one another?
To find out we use an appropriate decomposition of unity

69. Transformation of Kets - Let |χ〉 be an arbitrary ket in the space.
It can be expanded in either of the two bases considered above, i.e., or
Question: how are the expansion coefficients in these two different representations related to one another?
To find out we use an appropriate decomposition of unity

70. Transformation of Kets - Let |χ〉 be an arbitrary ket in the space.
It can be expanded in either of the two bases considered above, i.e., or
Question: how are the expansion coefficients in these two different representations related to one another?
To find out we use an appropriate decomposition of unity

71. Transformation of Kets - Let |χ〉 be an arbitrary ket in the space.
It can be expanded in either of the two bases considered above, i.e., or
Question: how are the expansion coefficients in these two different representations related to one another?
To find out we use an appropriate decomposition of unity

72. Transformation of Kets
This last relation, thus takes the form
which is equivalent to a matrix-coulumn vector multiplication, i.e.,

73. Transformation of Kets
This last relation, thus takes the form
which is equivalent to a matrix-column vector multiplication, i.e.,

74. Transformation of Kets
This last relation, thus takes the form
which is equivalent to a matrix-column vector multiplication, i.e.,

75. Transformation of Kets
By a similar approach it can be shown that the reverse transformation is effected by the matrix representing U⁺. Thus, we have the relation
which is equivalent to

76. Transformation of Kets
By a similar approach it can be shown that the reverse transformation is effected by the matrix representing U⁺. Thus, we have the relation
which is equivalent to

77. Transformation of Matrices - If A is an operator it has matrix elements in the two bases considered above of the form
To find the relationship between the matrices representing this operator in these two different bases we write
and insert decompositions of unity in the basis:

78. Transformation of Matrices - If A is an operator it has matrix elements in the two bases considered above of the form
and
To find the relationship between the matrices representing this operator in these two different bases we write
and insert decompositions of unity in the basis:

79. Transformation of Matrices - If A is an operator it has matrix elements in the two bases considered above of the form
To find the relationship between the matrices representing this operator in these two different bases we write
and insert decompositions of unity in the basis:

80. Transformation of Matrices - If A is an operator it has matrix elements in the two bases considered above of the form
To find the relationship between the matrices representing this operator in these two different bases we write
and insert decompositions of unity in the basis:

81. Transformation of Matrices - If A is an operator it has matrix elements in the two bases considered above of the form
To find the relationship between the matrices representing this operator in these two different bases we write
and insert decompositions of unity in the basis:

82. Transformation of Matrices - If A is an operator it has matrix elements in the two bases considered above of the form
To find the relationship between the matrices representing this operator in these two different bases we write
and insert decompositions of unity in the basis:

83. Transformation of Matrices
This last relation, thus takes the form
which can be expressed as a threefold matrix product
The reverse transformation is found in the same way, and yields the result

84. Transformation of Matrices
This last relation, thus takes the form
which can be expressed as a threefold matrix product
The reverse transformation is found in the same way, and yields the result

85. Transformation of Matrices
This last relation, thus takes the form
which can be expressed as a threefold matrix product
The reverse transformation is found in the same way, and yields the result

86. Transformation of Representations: Extension to Continuous Representations
Let |ψ〉 be an arbitrary vector in the space of a quantum particle in three dimensions, i.e., the space spanned by the vectors of the position representation and by the vectors of the wavevector representation.
We can expand the ket |ψ〉 in either of these two bases, i.e.,
and
Question: How are the expansion coefficients related to the expansion coefficients ?
(Let’s pretend we didn’t already know…)

87. Transformation of Representations: Extension to Continuous Representations
Let |ψ〉 be an arbitrary vector in the space of a quantum particle in three dimensions, i.e., the space spanned by the vectors of the position representation and by the vectors of the wavevector representation.
We can expand the ket |ψ〉 in either of these two bases, i.e.,
and
Question: How are the expansion coefficients related to the expansion coefficients ?
(Let’s pretend we didn’t already know…)

88. Transformation of Representations: Extension to Continuous Representations
Let |ψ〉 be an arbitrary vector in the space of a quantum particle in three dimensions, i.e., the space spanned by the vectors of the position representation and by the vectors of the wavevector representation.
We can expand the ket |ψ〉 in either of these two bases, i.e.,
and
Question: How are the expansion coefficients related to the expansion coefficients ?
(Let’s pretend we didn’t already know…)
Transformation of Representations: Extension to Continuous Representations
Let |ψ〉 be an arbitrary vector in the space of a quantum particle in three dimensions, i.e., the space spanned by the vectors of the position representation and by the vectors of the wavevector representation.
We can expand the ket |ψ〉 in either of these two bases, i.e.,
and
Question: How are the expansion coefficients related to the expansion coefficients ?
(Let’s pretend we didn’t already know…)

89. Transformation of Representations: Extension to Continuous Representations
Let |ψ〉 be an arbitrary vector in the space of a quantum particle in three dimensions, i.e., the space spanned by the vectors of the position representation and by the vectors of the wavevector representation.
We can expand the ket |ψ〉 in either of these two bases, i.e.,
and
Question: How are the expansion coefficients related to the expansion coefficients ?
(Let’s pretend we didn’t already know…)
Transformation of Representations: Extension to Continuous Representations
Let |ψ〉 be an arbitrary vector in the space of a quantum particle in three dimensions, i.e., the space spanned by the vectors of the position representation and by the vectors of the wavevector representation.
We can expand the ket |ψ〉 in either of these two bases, i.e.,
and
Question: How are the expansion coefficients related to the expansion coefficients ?
(Let’s pretend we didn’t already know…)

90. Transformation of Representations: Extension to Continuous Representations
Let |ψ〉 be an arbitrary vector in the space of a quantum particle in three dimensions, i.e., the space spanned by the vectors of the position representation and by the vectors of the wavevector representation.
We can expand the ket |ψ〉 in either of these two bases, i.e.,
and
Question: How are the expansion coefficients related to the expansion coefficients ?
(Let’s pretend we didn’t already know…)
Transformation of Representations: Extension to Continuous Representations
Let |ψ〉 be an arbitrary vector in the space of a quantum particle in three dimensions, i.e., the space spanned by the vectors of the position representation and by the vectors of the wavevector representation.
We can expand the ket |ψ〉 in either of these two bases, i.e.,
and
Question: How are the expansion coefficients related to the expansion coefficients ?
(Let’s pretend we didn’t already know…)

91. Transformation of Representations: Extension to Continuous Representations
We can find out in the same way as we just did for the discrete case, i.e., we write
which we (suggestively) write as
where the (continuous) matrix elements of the unitary operator connecting these two bases are

92. Transformation of Representations: Extension to Continuous Representations
We can find out in the same way as we just did for the discrete case, i.e., we write
which we (suggestively) write as
where the (continuous) matrix elements of the unitary operator connecting these two bases are

93. Transformation of Representations: Extension to Continuous Representations
We can find out in the same way as we just did for the discrete case, i.e., we write
which we (suggestively) write as
where the (continuous) matrix elements of the unitary operator connecting these two bases are

94. Transformation of Representations: Extension to Continuous Representations
Thus, we find that
which, of course, we already knew. Similarly, we find that
Thus, the Fourier transform is just an example of a unitary transformation from one continuous basis to another.
It is also possible to use the unitary transformation represented by the Fourier transform to derive the matrix elements of some of the operators already encountered.

95. Transformation of Representations: Extension to Continuous Representations
Thus, we find that
which, of course, we already knew. Similarly, we find that
Thus, the Fourier transform is just an example of a unitary transformation from one continuous basis to another.
It is also possible to use the unitary transformation represented by the Fourier transform to derive the matrix elements of some of the operators already encountered.

96. Transformation of Representations: Extension to Continuous Representations
Thus, we find that
which, of course, we already knew. Similarly, we find that
Thus, the Fourier transform is just an example of a unitary transformation from one continuous basis to another.
It is also possible to use the unitary transformation represented by the Fourier transform to derive the matrix elements of some of the operators already encountered.

97. Transformation of Representations: Extension to Continuous Representations
Thus, we find that
which, of course, we already knew. Similarly, we find that
Thus, the Fourier transform is just an example of a unitary transformation from one continuous basis to another.
It is also possible to use the unitary transformation represented by the Fourier transform to derive the matrix elements of some of the operators we have already encountered.

98. Transformation of Representations: Extension to Continuous Representations
As an example, consider the position operator whose matrix elements in the position representation are given by the expression
The matrix elements in the wavevector representation can be obtained from this by a unitary transformation, i.e.,

99. Transformation of Representations: Extension to Continuous Representations
As an example, consider the position operator whose matrix elements in the position representation are given by the expression
The matrix elements in the wavevector representation can be obtained from this by a unitary transformation, i.e.,

100. Transformation of Representations: Extension to Continuous Representations
As an example, consider the position operator whose matrix elements in the position representation are given by the expression
The matrix elements in the wavevector representation can be obtained from this by a unitary transformation, i.e.,

101. Transformation of Representations: Extension to Continuous Representations
As an example, consider the position operator whose matrix elements in the position representation are given by the expression
The matrix elements in the wavevector representation can be obtained from this by a unitary transformation, i.e.,

102. Transformation of Representations: Extension to Continuous Representations
As an example, consider the position operator whose matrix elements in the position representation are given by the expression
The matrix elements in the wavevector representation can be obtained from this by a unitary transformation, i.e.,

103. Transformation of Representations: Extension to Continuous Representations
As an example, consider the position operator whose matrix elements in the position representation are given by the expression
The matrix elements in the wavevector representation can be obtained from this by a unitary transformation, i.e.,

104. Transformation of Representations: Extension to Continuous Representations
As an example, consider the position operator whose matrix elements in the position representation are given by the expression
The matrix elements in the wavevector representation can be obtained from this by a unitary transformation, i.e.,

105. Transformation of Representations: Extension to Continuous Representations
As an example, consider the position operator whose matrix elements in the position representation are given by the expression
The matrix elements in the wavevector representation can be obtained from this by a unitary transformation, i.e.,

106. In this segment, we derived the canonical commutation relations obeyed by the Cartesian components of the position and wavevector (or momentum) operators.
We then began a study of unitary operators and showed that any two sets of basis vectors are connected by a unitary operator and by its adjoints.
We saw how to transform between two discrete representations, using the matrices that represent the unitary operators connecting those two representations, and extend this transformation to continuous representation, noting that the Fourier transform relation between position and momentum actually represents a unitary transformation between those two representations.
In the next segment we learn about a number of properties that the matrices that represent a given lilnear operator share, i.e., they are independent of the representation in which one is working.

107. In this segment, we derived the canonical commutation relations obeyed by the Cartesian components of the position and wavevector (or momentum) operators.
We then began a study of unitary operators and showed that any two sets of basis vectors are connected by a unitary operator and by its adjoints.
We saw how to transform between two discrete representations, using the matrices that represent the unitary operators connecting those two representations, and extend this transformation to continuous representation, noting that the Fourier transform relation between position and momentum actually represents a unitary transformation between those two representations.
In the next segment we learn about a number of properties that the matrices that represent a given lilnear operator share, i.e., they are independent of the representation in which one is working.

108. In this segment, we derived the canonical commutation relations obeyed by the Cartesian components of the position and wavevector (or momentum) operators.
We then began a study of unitary operators and showed that any two sets of basis vectors are connected by a unitary operator and by its adjoints.
We saw how to transform between two discrete representations, using the matrices that represent the unitary operators connecting them, and extended this idea to continuous representation, noting that the Fourier transform relation between position and momentum actually represents a unitary transformation between those two representations.
In the next segment we learn about a number of properties that the matrices that represent a given lilnear operator share, i.e., they are independent of the representation in which one is working.

109. In this segment, we derived the canonical commutation relations obeyed by the Cartesian components of the position and wavevector (or momentum) operators.
We then began a study of unitary operators and showed that any two sets of basis vectors are connected by a unitary operator and by its adjoints.
We saw how to transform between two discrete representations, using the matrices that represent the unitary operators connecting them, and extended this idea to continuous representation, noting that the Fourier transform relation between position and momentum actually represents a unitary transformation between those two representations.
In the next segment we learn about a number of properties that the matrices that represent a given linear operator share, i.e., representation independent properties.