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Quantum One: Lecture 17 1

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Ket-Bra Expansions and Integral Representations of Operators 3

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In the last lecture, 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, and 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 this lecture, we extend some of these ideas to continuously indexed bases sets, and develop integral representations of linear operators. 4

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Continuous Ket-Bra Expansion of Operators : Let form a continuous ONB for the space and let A be an operator acting in the space. Then from the trivial identity we can write 5

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Continuous Ket-Bra Expansion of Operators : Let form a continuous ONB for the space and let A be an operator acting in the space. Then from the trivial identity we can write 6

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Continuous Ket-Bra Expansion of Operators : Let form a continuous ONB for the space and let A be an operator acting in the space. Then from the trivial identity we can write 7

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Continuous Ket-Bra Expansion of Operators : Let form a continuous ONB for the space and let A be an operator acting in the space. Then from the trivial identity we can write 8

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Continuous Ket-Bra Expansion of Operators : Let form a continuous ONB for the space 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′ 〉. 9

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Continuous Ket-Bra Expansion of Operators : Let form a continuous ONB for the space 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, and completely specifies the linear operator A in terms of its matrix elements taken between the basis states of this representation. 10

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Continuous Ket-Bra Expansion of Operators : Let form a continuous ONB for the space 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, and completely specifies the linear operator A in terms of its matrix elements taken between the basis states of this representation. 11

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Integral Representation of Operators Thus in the wave function representation induced by any continuous ONB, an operator A is naturally represented by an integral kernel, which is a function of two continuous indices, or arguments, the values of which that are just the matrix elements of A connecting the different members of the basis states defining that continuous representation. Like the matrices associated with discrete representations, knowledge of the kernel facilitates computing quantities related to A itself. 12

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Integral Representation of Operators Thus in the wave function representation induced by any continuous ONB, an operator A is naturally represented by an integral kernel, which is a function of two continuous indices, or arguments, the values of which that are just the matrix elements of A connecting the different members of the basis states defining that continuous representation. Like the matrices associated with discrete representations, knowledge of the kernel facilitates computing quantities related to A itself. 13

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Integral Representation of Operators Thus in the wave function representation induced by any continuous ONB, an operator A is naturally represented by an integral kernel, which is a function of two continuous indices, or arguments, the values of which are just the matrix elements of A connecting the different members of the basis states defining that continuous representation. Like the matrices associated with discrete representations, knowledge of the kernel facilitates computing quantities related to A itself. 14

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Integral Representation of Operators Thus in the wave function representation induced by any continuous ONB, an operator A is naturally represented by an integral kernel, which is a function of two continuous indices, or arguments, the values of which are just the matrix elements of A connecting the different members of the basis states defining that continuous representation. Like the matrices associated with discrete representations, knowledge of the kernel facilitates computing quantities related to A itself. 15

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Integral Representation of Operators Thus, suppose that for some states and. The expansion coefficients for the states and are then clearly related. Note that if then which can be written, rather like a continuous matrix operation 16

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Integral Representation of Operators Thus, suppose that for some states and. The expansion coefficients for the states and are then clearly related. Note that if then which can be written, rather like a continuous matrix operation 17

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Integral Representation of Operators Thus, suppose that for some states and. The expansion coefficients for the states and are then clearly related. Note that if then which can be written, rather like a continuous matrix operation 18

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Integral Representation of Operators Thus, suppose that for some states and. The expansion coefficients for the states and are then clearly related. Note that if then which can be written, rather like a continuous matrix operation 19

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Integral Representation of Operators Thus, suppose that for some states and. The expansion coefficients for the states and are then clearly related. Note that if then which can be written, rather like a continuous matrix operation 20

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Integral Representation of Operators Thus, suppose that for some states and. The expansion coefficients for the states and are then clearly related. Note that if then which can be written, rather like a continuous matrix operation 21

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Integral Representation of Operators Thus, suppose that for some states and. The expansion coefficients for the states and are then clearly related. Note that if then which can be written, rather like a continuous matrix operation 22

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Integral Representation of Operators Consider the matrix element of A between arbitrary states and Inserting our expansion for A this becomes where identifying the wave functions for the two states involved we can write which is the continuous version of product of a row-vector, a square matrix, and a column-vector. 23

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Integral Representation of Operators Consider the matrix element of A between arbitrary states and Inserting our expansion for A this becomes where identifying the wave functions for the two states involved we can write which is the continuous version of product of a row-vector, a square matrix, and a column-vector. 24

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Integral Representation of Operators Consider the matrix element of A between arbitrary states and Inserting our expansion for A this becomes where identifying the wave functions for the two states involved we can write which is the continuous version of product of a row-vector, a square matrix, and a column-vector. 25

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Integral Representation of Operators Consider the matrix element of A between arbitrary states and Inserting our expansion for A this becomes where identifying the wave functions for the two states involved we can write which is the continuous version of product of a row-vector, a square matrix, and a column-vector. 26

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Integral Representation of Operators As another example, consider the operator product of and The operator product has a similar expansion, i.e., where which gives the continuous analog of a matrix multiplication, i.e., 27

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Integral Representation of Operators As another example, consider the operator product of and The operator product has a similar expansion, i.e., where which gives the continuous analog of a matrix multiplication, i.e., 28

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Integral Representation of Operators As another example, consider the operator product of and The operator product has a similar expansion, i.e., where which gives the continuous analog of a matrix multiplication, i.e., 29

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Integral Representation of Operators As another example, consider the operator product of and The operator product has a similar expansion, i.e., where which gives the continuous analog of a matrix multiplication, i.e., 30

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Integral Representation of Operators As another example, consider the operator product of and The operator product has a similar expansion, i.e., where which gives the continuous analog of a matrix multiplication, i.e., 31

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Integral Representation of Operators As another example, consider the operator product of and The operator product has a similar expansion, i.e., where which gives the continuous analog of a matrix multiplication, i.e., 32

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Integral Representation of Operators As another example, consider the operator product of and The operator product has a similar expansion, i.e., where which gives the continuous analog of a matrix multiplication, i.e., 33

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Integral Representation of Operators As another example, consider the operator product of and The operator product has a similar expansion, i.e., where which gives the continuous analog of a matrix multiplication, i.e., 34

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Integral Representation of Operators So if we know the kernels and representing A and B, we can compute the kernel representing C = AB through the integral relation 35

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Integral Representation of Operators: As a final example, consider the integral kernel 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 prime on the integration variables, and reorder, to find that from which we deduce that 36

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Integral Representation of Operators: As a final example, consider the integral kernel 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 prime on the integration variables, and reorder, to find that from which we deduce that 37

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Integral Representation of Operators: As a final example, consider the integral kernel 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 prime on the integration variables, and reorder, to find that from which we deduce that 38

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Integral Representation of Operators: As a final example, consider the integral kernel 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 prime on the integration variables, and reorder, to find that from which we deduce that 39

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Integral Representation of Operators: As a final example, consider the integral kernel 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 prime on the integration variables, and reorder, to find that from which we deduce that 40

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Integral Representation of Operators: As a final example, consider the integral kernel 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 prime on the integration variables, and reorder, to find that from which we deduce that 41

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This, obviously, is just the continuous analog of the complex-conjugate transpose of a matrix A Hermitian operator is equal to its adjoint, so that the integral kernels representing Hermitian operators obey the relation 42

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This, obviously, is just the continuous analog of the complex-conjugate transpose of a matrix A Hermitian operator is equal to its adjoint, so that the integral kernels representing Hermitian operators obey the relation 43

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Examples: As an example, in 3D, the operator X has as its matrix elements in the position representation This allows us to construct the expansion for this operator where the double integral has been reduced to a single integral because of the delta function. The operator X is said to be diagonal in the position representation, because it has no nonzero elements connecting different basis states. 44

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Examples: As an example, in 3D, the operator X has as its matrix elements in the position representation This allows us to construct the expansion for this operator where the double integral has been reduced to a single integral because of the delta function. The operator X is said to be diagonal in the position representation, because it has no nonzero elements connecting different basis states. 45

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Examples: As an example, in 3D, the operator X has as its matrix elements in the position representation This allows us to construct the expansion for this operator where the double integral has been reduced to a single integral because of the delta function. The operator X is said to be diagonal in the position representation, because it has no nonzero elements connecting different basis states. 46

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Examples: As an example, in 3D, the operator X has as its matrix elements in the position representation This allows us to construct the expansion for this operator where the double integral has been reduced to a single integral because of the delta function. The operator X is said to be diagonal in the position representation, because it has no nonzero elements connecting different basis states. 47

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Examples: As an example, in 3D, the operator X has as its matrix elements in the position representation This allows us to construct the expansion for this operator where the double integral has been reduced to a single integral because of the delta function. The operator X is said to be diagonal in the position representation, because it has no nonzero elements connecting different basis states. 48

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Examples: As an example, in 3D, the operator X has as its matrix elements in the position representation This allows us to construct the expansion for this operator where the double integral has been reduced to a single integral because of the delta function. The operator X is said to be diagonal in the position representation, because it has no nonzero elements connecting different basis states. 49

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Examples: As an example, in 3D, the operator X has as its matrix elements in the position representation This allows us to construct the expansion for this operator where the double integral has been reduced to a single integral because of the delta function. The operator X is said to be diagonal in the position representation, because it has no nonzero elements connecting different basis states. 50

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Examples: As an example, in 3D, the operator X has as its matrix elements in the position representation This allows us to construct the expansion for this operator where the double integral has been reduced to a single integral because of the delta function. The operator X is said to be diagonal in the position representation, because it has no nonzero elements connecting different basis states. 51

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This concept of diagonality extends to arbitrary representations. An operator A is said to be diagonal in the discrete representation if so that which only has one summation index, in contrast to the general form which requires two. 52

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This concept of diagonality extends to arbitrary representations. An operator A is said to be diagonal in the discrete representation if so that which only has one summation index, in contrast to the general form which requires two. 53

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This concept of diagonality extends to arbitrary representations. An operator A is said to be diagonal in the discrete representation if so that which only has one summation index, in contrast to the general form which requires two. 54

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This concept of diagonality extends to arbitrary representations. An operator A is said to be diagonal in the discrete representation if so that which only has one summation index, in contrast to the general form which requires two. 55

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In a discrete representation, an operator that is diagonal in that representation is represented by a diagonal matrix, i.e., if then 56

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In a discrete representation, an operator that is diagonal in that representation is represented by a diagonal matrix, i.e., if then 57

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Similarly, in a continuous representation an operator A is diagonal if so that which ends up with only one integration variable, i.e., in contrast to the general form which requires two. 58

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Similarly, in a continuous representation an operator A is diagonal if so that which ends up with only one integration variable, i.e., in contrast to the general form which requires two. 59

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Similarly, in a continuous representation an operator A is diagonal if so that which ends up with only one integration variable, i.e., in contrast to the general form which requires two. 60

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It is easy to show that in any basis in which an operator is diagonal, it is what we referred to earlier as a multiplicative operator in that representation. That is, if is diagonal in the representation, and if then which shows that a diagonal operator G acts in the representation to multiply the wave function by. 61

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It is easy to show that in any basis in which an operator is diagonal, it is what we referred to earlier as a multiplicative operator in that representation. That is, if is diagonal in the representation, and if then which shows that a diagonal operator G acts in the representation to multiply the wave function by. 62

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It is easy to show that in any basis in which an operator is diagonal, it is what we referred to earlier as a multiplicative operator in that representation. That is, if is diagonal in the representation, and if then which shows that a diagonal operator G acts in the representation to multiply the wave function by. 63

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It is easy to show that in any basis in which an operator is diagonal, it is what we referred to earlier as a multiplicative operator in that representation. That is, if is diagonal in the representation, and if then which shows that a diagonal operator G acts in the representation to multiply the wave function by. 64

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It is easy to show that in any basis in which an operator is diagonal, it is what we referred to earlier as a multiplicative operator in that representation. That is, if is diagonal in the representation, and if then which shows that a diagonal operator G acts in the representation to multiply the wave function by. 65

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It is easy to show that in any basis in which an operator is diagonal, it is what we referred to earlier as a multiplicative operator in that representation. That is, if is diagonal in the representation, and if then which shows that a diagonal operator G acts in the representation to multiply the wave function by. 66

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We list below ket-bra expansions and matrix elements of important operators. The position operator The potential energy operator The wavevector operator The momentum operator The kinetic energy operator 67

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We list below ket-bra expansions and matrix elements of important operators. The position operator The potential energy operator The wavevector operator The momentum operator The kinetic energy operator 68

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We list below ket-bra expansions and matrix elements of important operators. The position operator The potential energy operator The wavevector operator The momentum operator The kinetic energy operator 69

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We list below ket-bra expansions and matrix elements of important operators. The position operator The potential energy operator The wavevector operator The momentum operator The kinetic energy operator 70

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We list below ket-bra expansions and matrix elements of important operators. The position operator The potential energy operator The wavevector operator The momentum operator The kinetic energy operator 71

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Another nice thing about ket-bra expansions of this sort, particularly in a representation in which the operator is diagonal, is that it is very easy to determine whether or not the operator is Hermitian. For example, we can take the Hermitian adjoint of the position operator by replacing each term in this continuous summation with its adjoint. Thus we easily see that so the position operator (and each of its components) is clearly Hermitian. 72

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Another nice thing about ket-bra expansions of this sort, particularly in a representation in which the operator is diagonal, is that it is very easy to determine whether or not the operator is Hermitian. For example, we can take the Hermitian adjoint of the position operator by replacing each term in this continuous summation with its adjoint. Thus we easily see that so the position operator (and each of its components) is clearly Hermitian. 73

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Another nice thing about ket-bra expansions of this sort, particularly in a representation in which the operator is diagonal, is that it is very easy to determine whether or not the operator is Hermitian. For example, we can take the Hermitian adjoint of the position operator by replacing each term in this continuous summation with its adjoint. Thus we easily see that so the position operator (and each of its components) is clearly Hermitian. 74

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Another nice thing about ket-bra expansions of this sort, particularly in a representation in which the operator is diagonal, is that it is very easy to determine whether or not the operator is Hermitian. For example, we can take the Hermitian adjoint of the position operator by replacing each term in this continuous summation with its adjoint. Thus we easily see that so the position operator (and each of its components) is clearly Hermitian. 75

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Another nice thing about ket-bra expansions of this sort, particularly in a representation in which the operator is diagonal, is that it is very easy to determine whether or not the operator is Hermitian. For example, we can take the Hermitian adjoint of the position operator by replacing each term in this continuous summation with its adjoint. Thus we easily see that so the position operator (and each of its components) is clearly Hermitian. 76

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You should verify to yourself that each of the basic operators whose diagonal ket- bra expansion we previously displayed is also Hermitian. It follows that the wavevector operator is also Hermitian, and so the operator D=iK, satisfies the relationD⁺=-iK⁺=-iK=-D Thus we see that the operator D, which takes the gradient in the position representation, is actually an anti-Hermitian operator, That’s why we traded it in for the wavevector operator. 77

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You should verify to yourself that each of the basic operators whose diagonal ket- bra expansion we previously displayed is also Hermitian. It follows that the wavevector operator is also Hermitian, and so the operator D=iK, satisfies the relationD⁺=-iK⁺=-iK=-D Thus we see that the operator D, which takes the gradient in the position representation, is actually an anti-Hermitian operator, That’s why we traded it in for the wavevector operator. 78

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You should verify to yourself that each of the basic operators whose diagonal ket- bra expansion we previously displayed is also Hermitian. It follows that the wavevector operator is also Hermitian, and so the operator D=iK, satisfies the relationD⁺=-iK⁺=-iK=-D Thus we see that the operator D, which takes the gradient in the position representation, is actually an anti-Hermitian operator, That’s why we traded it in for the wavevector operator. 79

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You should verify to yourself that each of the basic operators whose diagonal ket- bra expansion we previously displayed is also Hermitian. It follows that the wavevector operator is also Hermitian, and so the operator D=iK, satisfies the relationD⁺=-iK⁺=-iK=-D Thus we see that the operator D, which takes the gradient in the position representation, is actually an anti-Hermitian operator, That’s why we traded it in for the wavevector operator. 80

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You should verify to yourself that each of the basic operators whose diagonal ket- bra expansion we previously displayed is also Hermitian. It follows that the wavevector operator is also Hermitian, and so the operator D=iK, satisfies the relationD⁺=-iK⁺=-iK=-D Thus we see that the operator D, which takes the gradient in the position representation, is actually an anti-Hermitian operator, That’s why we traded it in for the wavevector operator. 81

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You should verify to yourself that each of the basic operators whose diagonal ket- bra expansion we previously displayed is also Hermitian. It follows that the wavevector operator is also Hermitian, and so the operator D=iK, satisfies the relationD⁺=-iK⁺=-iK=-D Thus we see that the operator D, which takes the gradient in the position representation, is actually an anti-Hermitian operator, That’s why we traded it in for the wavevector operator. 82

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You should verify to yourself that each of the basic operators whose diagonal ket- bra expansion we previously displayed is also Hermitian. It follows that the wavevector operator is also Hermitian, and so the operator D=iK, satisfies the relationD⁺=-iK⁺=-iK=-D Thus we see that the operator D, which takes the gradient in the position representation, is actually an anti-Hermitian operator, That’s why we traded it in for the wavevector operator. 83

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You should verify to yourself that each of the basic operators whose diagonal ket- bra expansion we previously displayed is also Hermitian. It follows that the wavevector operator is also Hermitian, and so the operator D=iK, satisfies the relationD⁺=-iK⁺=-iK=-D Thus we see that the operator D, which takes the gradient in the position representation, is actually an anti-Hermitian operator, That’s why we traded it in for the wavevector operator. 84

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As an additional example, we work out below the matrix elements of the wavevector operator K in the position representation. Recall that for any state |ψ 〉 for which the state has a position wave function given by the expression But we we can also write where The right hand side of this last expression seems to be the wave function in the position representation for the state. 85

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As an additional example, we work out below the matrix elements of the wavevector operator K in the position representation. Recall that for any state |ψ 〉 for which the state has a position wave function given by the expression But we we can also write where The right hand side of this last expression seems to be the wave function in the position representation for the state. 86

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As an additional example, we work out below the matrix elements of the wavevector operator K in the position representation. Recall that for any state |ψ 〉 for which the state has a position wave function given by the expression But we we can also write where The right hand side of this last expression seems to be the wave function in the position representation for the state. 87

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As an additional example, we work out below the matrix elements of the wavevector operator K in the position representation. Recall that for any state |ψ 〉 for which the state has a position wave function given by the expression But we we can also write where The right hand side of this last expression seems to be the wave function in the position representation for the state. 88

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As an additional example, we work out below the matrix elements of the wavevector operator K in the position representation. Recall that for any state |ψ 〉 for which the state has a position wave function given by the expression But we we can also write where The right hand side of this last expression seems to be the wave function in the position representation for the state. 89

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Reminding ourselves of the position eigenfunctions we see that, evidently i.e., is -i times the gradient of the delta function. The properties of this not-so-frequently encountered object are reviewed in the appendix at the end of the first chapter. The most important of which is that for any function, 90

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Reminding ourselves of the position eigenfunctions we see that, evidently i.e., is -i times the gradient of the delta function. The properties of this not-so-frequently encountered object are reviewed in the appendix at the end of the first chapter. The most important of which is that for any function, 91

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Reminding ourselves of the position eigenfunctions we see that, evidently i.e., is -i times the gradient of the delta function. The properties of this not-so-frequently encountered object are reviewed in the appendix at the end of the first chapter. The most important of which is that for any function, 92

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Reminding ourselves of the position eigenfunctions we see that, evidently i.e., is -i times the gradient of the delta function. The properties of this not-so-frequently encountered object are reviewed in the appendix at the end of the first chapter. The most important of which is that for any function, 93

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Reminding ourselves of the position eigenfunctions we see that, evidently i.e., is -i times the gradient of the delta function. The properties of this not-so-frequently encountered object are reviewed in the appendix at the end of the first chapter. The most important of which is that for any function, 94

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We deduce, therefore that can be expanded in the position representation in the form so that when we apply this to any state, we obtain consistent with our previous definition. 95

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We deduce, therefore that can be expanded in the position representation in the form so that when we apply this to any state, we obtain consistent with our previous definition. 96

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We deduce, therefore that can be expanded in the position representation in the form so that when we apply this to any state, we obtain consistent with our previous definition. 97

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We deduce, therefore that can be expanded in the position representation in the form so that when we apply this to any state, we obtain consistent with our previous definition. 98

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We deduce, therefore that can be expanded in the position representation in the form so that when we apply this to any state, we obtain consistent with our previous definition. 99

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In a similar fashion, one finds that and that 100

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In a similar fashion, one finds that and that 101

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In this 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 particle in representations in which they are diagonal. Finally, we saw how differential operators can be expressed as ket-bra expansions with integral kernels that involve derivatives the delta function, so that we could understand how a linear operator acting on kets, can somehow end up replacing the wave function with its derivative or gradient. In the next lecture we consider still other, representation independent, properties of linear operators. 102

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In this 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 particle in representations in which they are diagonal. Finally, we saw how differential operators can be expressed as ket-bra expansions with integral kernels that involve derivatives the delta function, so that we could understand how a linear operator acting on kets, can somehow end up replacing the wave function with its derivative or gradient. In the next lecture we consider still other, representation independent, properties of linear operators. 103

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In this 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 particle in representations in which they are diagonal. Finally, we saw how differential operators can be expressed as ket-bra expansions with integral kernels that involve derivatives the delta function, so that we could understand how a linear operator acting on kets, can somehow end up replacing the wave function with its derivative or gradient. In the next lecture we consider still other, representation independent, properties of linear operators. 104

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In this 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 particle in representations in which they are diagonal. Finally, we saw how differential operators can be expressed as ket-bra expansions with integral kernels that involve derivatives of the delta function, so that we could understand how a linear operator acting on kets, can somehow end up replacing the wave function with its derivative or gradient. In the next lecture we consider still other, representation independent, properties of linear operators. 105

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In this 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 particle in representations in which they are diagonal. Finally, we saw how differential operators can be expressed as ket-bra expansions with integral kernels that involve derivatives the delta function, so that we could understand how a linear operator acting on kets, can somehow end up replacing the wave function with its derivative or gradient. In the next lecture we consider still other, representation independent, properties of linear operators. 106

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