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Quantum One: Lecture 13 1

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More about Linear Operators 3

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In the last lecture, we stated and explored the consequences of the 2 nd postulate of the general formulation of quantum mechanics, which associates observables with linear Hermitian operators whose eigenstates form a complete orthonormal basis for the state space S. We then defined operators, linear operators, and explored a number of properties associated with linear operators. We also introduced multiplicative operators, whose action on the basis vectors of a particular representation is to simply multiply each basis vector by a function of the index which identifies them. In this lecture we continue to explore different kinds of linear operators. We begin with what we will refer to as differential operators. 4

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Differential Operators: We can illustrate what we mean by differential operators by again working in the position representation, i.e., Define operators in such a way that if|ψ 〉 =∫d³rψ(r)|r 〉, then Thus, D_ acts on this expansion to somehow replace the wave function in the position representation by its partial derivative with respect to x. Similar actions are implicitly defined for D and, but it is important to note that in this expression does not act on the wave function, but on the basis kets to replace them with a linear combination that somehow gives this effect. We will see later how this actually happens. 5

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Differential Operators: We can illustrate what we mean by differential operators by again working in the position representation, i.e., Define operators in such a way that if|ψ 〉 =∫d³rψ(r)|r 〉, then Thus, D_ acts on this expansion to somehow replace the wave function in the position representation by its partial derivative with respect to x. Similar actions are implicitly defined for D and, but it is important to note that in this expression does not act on the wave function, but on the basis kets to replace them with a linear combination that somehow gives this effect. We will see later how this actually happens. 6

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Differential Operators: We can illustrate what we mean by differential operators by again working in the position representation, i.e., Define operators in such a way that if|ψ 〉 =∫d³rψ(r)|r 〉, then Thus, D_ acts on this expansion to somehow replace the wave function in the position representation by its partial derivative with respect to x. Similar actions are implicitly defined for D and, but it is important to note that in this expression does not act on the wave function, but on the basis kets to replace them with a linear combination that somehow gives this effect. We will see later how this actually happens. 7

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Differential Operators: We can illustrate what we mean by differential operators by again working in the position representation, i.e., Define operators in such a way that if|ψ 〉 =∫d³rψ(r)|r 〉, then Thus, D_ acts on this expansion to somehow replace the wave function in the position representation by its partial derivative with respect to x. Similar actions are implicitly defined for D and, but it is important to note that in this expression does not act on the wave function, but on the basis kets to replace them with a linear combination that somehow gives this effect. We will see later how this actually happens. 8

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Differential Operators: We can illustrate what we mean by differential operators by again working in the position representation, i.e., Define operators in such a way that if|ψ 〉 =∫d³rψ(r)|r 〉, then Thus, D_ acts on this expansion to somehow replace the wave function in the position representation by its partial derivative with respect to x. Similar actions are implicitly defined for D and, but it is important to note that in this expression does not act on the wave function, but on the basis kets to replace them with a linear combination that somehow gives this effect. We will see later how this actually happens. 9

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Differential Operators: We can illustrate what we mean by differential operators by again working in the position representation, i.e., Define operators in such a way that if|ψ 〉 =∫d³rψ(r)|r 〉, then Thus, D_ acts on this expansion to somehow replace the wave function in the position representation by its partial derivative with respect to x. Similar actions are implicitly defined for D and, but it is important to note that in this expression does not act on the wave function, but on the basis kets to replace them with a linear combination that somehow gives this effect. We will see later how this actually happens. 10

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Differential Operators: We can illustrate what we mean by differential operators by again working in the position representation, i.e., Define operators in such a way that if|ψ 〉 =∫d³rψ(r)|r 〉, then Thus, D_ acts on this expansion to somehow replace the wave function in the position representation by its partial derivative with respect to x. Similar actions are implicitly defined for D and, but it is important to note that in this expression does not act on the wave function, but on the basis kets to replace them with a linear combination that somehow gives this effect. We will see later how this actually happens. 11

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Differential Operators These three operators form the components of the vector operator which "takes the gradient in this representation“: That is to say, or It turns out that this operator is not Hermitian (a term we have not yet defined) and is useful to trade it in for something called the wavevector operator which is defined so that i.e., so that 12

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Differential Operators These three operators form the components of the vector operator which "takes the gradient in this representation“: That is to say, or It turns out that this operator is not Hermitian (a term we have not yet defined) and is useful to trade it in for something called the wavevector operator which is defined so that i.e., so that 13

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Differential Operators These three operators form the components of the vector operator which "takes the gradient in this representation“: That is to say, or It turns out that this operator is not Hermitian (a term we have not yet defined) and is useful to trade it in for something called the wavevector operator which is defined so that i.e., so that 14

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Differential Operators These three operators form the components of the vector operator which "takes the gradient in this representation“: That is to say, or It turns out that this operator is not Hermitian (a term we have not yet defined) and is useful to trade it in for something called the wavevector operator which is defined so that i.e., so that 15

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The Wavevector Operator It is instructive to consider the action of this “differential” operator in the momentum representation. By assumption, an arbitrary state can be expanded in this representation in the form where Thus, 16

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The Wavevector Operator It is instructive to consider the action of this “differential” operator in the momentum representation. By assumption, an arbitrary state can be expanded in this representation in the form where and where Thus, 17

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The Wavevector Operator It is instructive to consider the action of this “differential” operator in the momentum representation. By assumption, an arbitrary state can be expanded in this representation in the form where and where Thus, 18

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The Wavevector Operator It is instructive to consider the action of this “differential” operator in the momentum representation. By assumption, an arbitrary state can be expanded in this representation in the form where and where Thus, 19

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The Wavevector Operator It is instructive to consider the action of this “differential” operator in the momentum representation. By assumption, an arbitrary state can be expanded in this representation in the form where and where Thus, 20

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The Wavevector Operator The gradient operator, which acts only on the position variables, now just "pulls down the wavevector" from the exponential, i.e., Thus, we deduce that 21

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The Wavevector Operator The gradient operator, which acts only on the position variables, now just "pulls down the wavevector" from the exponential, i.e., Thus, we deduce that 22

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The Wavevector Operator The gradient operator, which acts only on the position variables, now just "pulls down the wavevector" from the exponential, i.e., Thus, we deduce that 23

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The Wavevector Operator The gradient operator, which acts only on the position variables, now just "pulls down the wavevector" from the exponential, i.e., Thus, we deduce that 24

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The Wavevector Operator Interchanging the order of integration this takes the form Thus, we deduce that 25

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The Wavevector Operator Interchanging the order of integration this takes the form Thus, we deduce that 26

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The Wavevector Operator Interchanging the order of integration this takes the form Thus, we deduce that 27

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The Wavevector Operator Interchanging the order of integration this takes the form Thus, we deduce that 28

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The Wavevector Operator Interchanging the order of integration this takes the form Thus, we deduce that 29

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The Wavevector Operator or Thus K acts "in the k representation" to multiply the wave function in that representation by k.. Since K really acts only on the kets |k 〉,, we deduce the action K|k 〉 =k|k 〉 Thus, the operator K plays the same role in the k representation that the operator R plays in the r representation, i.e., it simply multiplies the basis vectors by the value of the parameter k that labels them. You can think of this as the operator "pulling out" the label. 30

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The Wavevector Operator or Thus K acts "in the k representation" to multiply the wave function in that representation by k.. Since K really acts only on the kets |k 〉,, we deduce the action K|k 〉 =k|k 〉 Thus, the operator K plays the same role in the k representation that the operator R plays in the r representation, i.e., it simply multiplies the basis vectors by the value of the parameter k that labels them. You can think of this as the operator "pulling out" the label. 31

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The Wavevector Operator or Thus K acts "in the k representation" to multiply the wave function in that representation by k.. Since K really acts only on the kets |k 〉,, we deduce the action K|k 〉 =k|k 〉 Thus, the operator K plays the same role in the k representation that the operator R plays in the r representation, i.e., it simply multiplies the basis vectors by the value of the parameter k that labels them. You can think of this as the operator "pulling out" the label. 32

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The Wavevector Operator or Thus K acts "in the k representation" to multiply the wave function in that representation by k.. Since K really acts only on the kets |k 〉,, we deduce the action K|k 〉 =k|k 〉 Thus, the operator K plays the same role in the k representation that the operator R plays in the r representation, i.e., it simply multiplies the basis vectors by the value of the parameter k that labels them. You can think of this as the operator "pulling out" the label. 33

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The Wavevector Operator or Thus K acts "in the k representation" to multiply the wave function in that representation by k.. Since K really acts only on the kets |k 〉,, we deduce the action K|k 〉 =k|k 〉 Thus, the operator K plays the same role in the k representation that the operator R plays in the r representation, i.e., it simply multiplies the basis vectors by the value of the parameter k that labels them. You can think of this as the operator "pulling out" the label. 34

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The Wavevector Operator or Thus K acts "in the k representation" to multiply the wave function in that representation by k.. Since K really acts only on the kets |k 〉,, we deduce the action K|k 〉 =k|k 〉 Thus, the operator K plays the same role in the k representation that the operator R plays in the r representation, i.e., it simply multiplies the basis vectors by the value of the parameter k that labels them. You can think of this as the operator just "pulling out" the label. 35

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The Wavevector Operator Clearly the wavevector operator is a differential operator in the position representation, but a multiplicative operator in the wavevector representation Similarly the momentum and kinetic energy operators P=K and are multiplicative operators in the wavevector representation P|k 〉 =k|k 〉 〈 k|P|ψ 〉 =k ψ(k) 36

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The Wavevector Operator Clearly the wavevector operator is a differential operator in the position representation, but a multiplicative operator in the wavevector representation Similarly the momentum and kinetic energy operators P=K and are multiplicative operators in the wavevector representation P|k 〉 =k|k 〉 〈 k|P|ψ 〉 =k ψ(k) 37

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The Wavevector Operator Clearly the wavevector operator is a differential operator in the position representation, but a multiplicative operator in the wavevector representation Similarly the momentum and kinetic energy operators P=K are multiplicative operators in the wavevector representation P|k 〉 =k|k 〉 〈 k|P|ψ 〉 =k ψ(k) 38

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The Wavevector Operator Clearly the wavevector operator is a differential operator in the position representation, but a multiplicative operator in the wavevector representation Similarly the momentum and kinetic energy operators P=K are multiplicative operators in the wavevector representation P|k 〉 =k|k 〉 〈 k|P|ψ 〉 =k ψ(k) 39

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The Wavevector Operator Clearly the wavevector operator is a differential operator in the position representation, but a multiplicative operator in the wavevector representation Similarly the momentum and kinetic energy operators P=K are multiplicative operators in the wavevector representation P|k 〉 =k|k 〉 〈 k|P|ψ 〉 =k ψ(k) 40

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The Wavevector Operator Clearly the wavevector operator is a differential operator in the position representation, but a multiplicative operator in the wavevector representation Similarly the momentum and kinetic energy operators P=K and are multiplicative operators in the wavevector representation P|k 〉 =k|k 〉 〈 k|P|ψ 〉 =k ψ(k) 41

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but are differential operators in the position representation Thus, whether an operator is a multiplicative operator or a differential operator is very much a representation-dependent statement. It is left as an exercise to show that in the wavevector representation the position operator acts as a differential operator, i.e., that where 42

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but are differential operators in the position representation Thus, whether an operator is a multiplicative operator or a differential operator is very much a representation-dependent statement. It is left as an exercise to show that in the wavevector representation the position operator acts as a differential operator, i.e., that where 43

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but are differential operators in the position representation Thus, whether an operator is a multiplicative operator or a differential operator is very much a representation-dependent statement. It is left as an exercise to show that in the wavevector representation the position operator acts as a differential operator, i.e., that where 44

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but are differential operators in the position representation Thus, whether an operator is a multiplicative operator or a differential operator is very much a representation-dependent statement. It is left as an exercise to show that in the wavevector representation the position operator acts as a differential operator, i.e., that where 45

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but are differential operators in the position representation Thus, whether an operator is a multiplicative operator or a differential operator is very much a representation-dependent statement. It is left as an exercise to show that in the wavevector representation the position operator acts as a differential operator, i.e., that where 46

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but are differential operators in the position representation Thus, whether an operator is a multiplicative operator or a differential operator is very much a representation-dependent statement. It is left as an exercise to show that in the wavevector representation the position operator acts as a differential operator, i.e., that where 47

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but are differential operators in the position representation Thus, whether an operator is a multiplicative operator or a differential operator is very much a representation-dependent statement. It is left as an exercise to show that in the wavevector representation the position operator acts as a differential operator, i.e., that where 48

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