Quantum One: Lecture 18 1

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Canonical Commutation Relations 3

In the last lecture, 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 lecture by deriving what are referred to as canonical commutation relations. 4

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 5

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 6

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 7

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 8

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 9

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. 10

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. 11

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. 12

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. 13

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. 14

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 15

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 16

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 17

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 18

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 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 Subtracting these two terms, we find that in the position representation 23

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., 24

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., 25

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., 26

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., 27

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., 28

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Changing Representations: Unitary Operators 31

In an earlier lecture 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 32

In an earlier lecture 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 33

In an earlier lecture 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 34

In an earlier lecture 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 35

In an earlier lecture 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 36

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 37

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 38

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 39

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 40

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 41

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 42

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 43

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 44

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 45

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 46

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 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 so 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 so 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 so 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 so 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 so 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 so that 53

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. 54

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. 55

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. 56

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. 57

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. 58

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 59

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 60

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 61

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 62

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 63

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 64

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 65

Transformation of Kets This last relation, thus takes the form which is equivalent to a matrix-coulumn vector multiplication, i.e., 66

Transformation of Kets This last relation, thus takes the form which is equivalent to a matrix-column vector multiplication, i.e., 67

Transformation of Kets This last relation, thus takes the form which is equivalent to a matrix-column vector multiplication, i.e., 68

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 69

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 70

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: 71

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: 72

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: 73

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: 74

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: 75

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: 76

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 77

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 78

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 79

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…) 80

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…) 81

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…) 82

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…) 83

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 84

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 85

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 86

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. 87

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. 88

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. 89

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. 90

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., 91

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., 92

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., 93

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., 94

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., 95

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., 96

In this lecture, 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 lecture 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. 97

In this lecture, 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 lecture 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. 98

In this lecture, 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 lecture 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. 99

In this lecture, 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 lecture we learn about a number of properties that the matrices that represent a given linear operator share, i.e., representation independent properties. 100

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