Quantum One: Lecture 13 1. 2 More about Linear Operators 3.

1

Quantum One: Lecture 13

3

More about Linear Operators

In the last lecture, we stated and explored the consequences of the 2nd 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

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

6






We will see later how this actually happens.

7







8







9







10







11







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



That is to say, or


i.e., so that

13



That is to say, or


i.e., so that

14



That is to say, or


i.e., so that

15

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




where

and where

Thus,

17




where

and where

Thus,

18




where

and where

Thus,

19




where

and where

Thus,

20


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




22




23




24


Interchanging the order of integration this takes the form


25




26




27




28




29


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


or



K|k =k|k ⟩ ⟩


31


or



K|k =k|k ⟩ ⟩


32


or



K|k =k|k ⟩ ⟩


33


or



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


or



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


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




P= K and


P|k = k|k k|P|⟩ ⟩ ⟨ ψ =⟩ k ψ(k)

37




P= K


P|k = k|k k|P|⟩ ⟩ ⟨ ψ =⟩ k ψ(k)

38




P= K


P|k = k|k k|P|⟩ ⟩ ⟨ ψ =⟩ k ψ(k)

39




P= K


P|k = k|k k|P|⟩ ⟩ ⟨ ψ =⟩ k ψ(k)

40




P= K and


P|k = k|k k|P|⟩ ⟩ ⟨ ψ =⟩ k ψ(k)

41

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




where

43




where

44




where

45




where

46




where

47




where

48

So in this lecture, we introduced differential operators, whose action on the basis vectors of a particular representation somehow serve to replace the wave function in that representation with a derivative of some kind (sometimes multiplied by other constants).

In particular we introduced the wavevector operator that acts as a differential operator in the position representation, but which we saw also acts as a multiplicative operator in the momentum or wavevector representation.

We also defined and determined the action of the momentum and kinetic energy operators in both the position and the momentum representation.

In the next lecture, we explore even MORE (incredibly useful) ways of defining linear operators on quantum state spaces.

49





50





51





52

Quantum One: Lecture 13 1. 2 More about Linear Operators 3.

Documents

Transcript of Quantum One: Lecture 13 1. 2 More about Linear Operators 3.