Quantum One: Lecture 12 1. 2 Postulate II 3 Observables of Quantum Mechanical Systems 4

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Quantum One: Lecture 12

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Postulate II

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Postulate IIObservables of Quantum Mechanical Systems

In this lecture, we state and explore the consequences of the 2nd postulate of the general formulation of quantum mechanics that we are in the process of developing. As before we begin with a simple statement of the postulate.

Postulate II – Each observable A of a quantum mechanical system is associated with a linear Hermitian operator A whose eigenstates form a complete orthonormal basis for the quantum mechanical state space S.

We are led, therefore, to investigate the nature of linear operators defined on linear vector spaces.

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We are led, therefore, to investigate the nature of linear operators defined on linear vector spaces.

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We are led, therefore, to investigate the nature of linear operators defined on complex linear vector spaces.

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Operators and Their Properties

An operator A associated with a linear vector space S acts on the elements |χ in ⟩S and maps them onto (possibly) other elements |χA of the same space. We ⟩express this mapping of one vector onto another in the form

A|χ = |χ⟩ A . ⟩

An operator A is linear if, for arbitrary states |χ ,|ψ , and arbitrary scalars λ and ⟩ ⟩μ, it obeys the following linearity condition

A(λ|χ + μ|ψ ) = λA|χ +μA|ψ = λ|χ⟩ ⟩ ⟩ ⟩ A +μ|ψ⟩ A ⟩

In what follows we assume, unless otherwise stated, that all operators under consideration are linear operators.

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A|χ = |χ⟩ A . ⟩




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A|χ = |χ⟩ A . ⟩




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One of the useful properties of linear operators is that their action on arbitrary states is completely determined once their action on the elements of any ONB is specified.

To see this, let be an arbitrary ONB, and let the action

|φi =A|i⟩ ⟩

of A on each of these basis states be known.

Then, when an arbitrary state is acted upon by A

the resulting vector is completely determined.

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|φi =A|i⟩ ⟩




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|φi =A|i⟩ ⟩




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|φi =A|i⟩ ⟩




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Properties of Linear Operators

The sum and difference of two operators is defined through vector addition

(A + B)|ψ = A|ψ + B|ψ = |ψ⟩ ⟩ ⟩ A +|ψ⟩ B ⟩

(A - B)|ψ = A|ψ - B|ψ = |ψ⟩ ⟩ ⟩ A - |ψ⟩ B . ⟩

The product of operators is defined through the combined action of each. If C = AB,

then C|ψ = AB|ψ = A|ψ⟩ ⟩ B .⟩

In general the operator product is not commutative, since reversing the order can give a different result, i.e., the vector

BA|ψ = B|ψ⟩ A ⟩

need not have any relation to the vector A|ψB . ⟩

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(A + B)|ψ = A|ψ + B|ψ = |ψ⟩ ⟩ ⟩ A +|ψ⟩ B ⟩

(A - B)|ψ = A|ψ - B|ψ = |ψ⟩ ⟩ ⟩ A - |ψ⟩ B . ⟩






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(A + B)|ψ = A|ψ + B|ψ = |ψ⟩ ⟩ ⟩ A +|ψ⟩ B ⟩

(A - B)|ψ = A|ψ - B|ψ = |ψ⟩ ⟩ ⟩ A - |ψ⟩ B . ⟩






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It is useful define the commutator

[A,B] = AB - BA = - [B,A]

of two operators, which is also (generally) an operator.

If

[A,B]=0, then AB = BA,

and the two operators commute.

From the definition of the commutator it is straightforward to prove the following useful relations

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[A,B] = AB - BA = - [B,A]


If




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[A,B] = AB - BA = - [B,A]


If




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Properties of The Commutator

[A, A] = 0

[A, B+C] = [A, B] + [A, C]

[A+B, C] = [A, C] + [B, C]

[A, BC] = B[A, C] + [A, B]C

[AB, C] = A[B,C]+[A,C]B

[A, [B, C]] + [C, [A, B]] + [B, [C, A]] = 0.

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[A, A] = 0

[A, B+C] = [A, B] + [A, C]

[A+B, C] = [A, C] + [B, C]

[A, BC] = B[A, C] + [A, B]C

[AB, C] = A[B,C]+[A,C]B

[A, [B, C]] + [C, [A, B]] + [B, [C, A]] = 0.

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[A, A] = 0

[A, B+C] = [A, B] + [A, C]

[A+B, C] = [A, C] + [B, C]

[A, BC] = B[A, C] + [A, B]C

[AB, C] = A[B,C]+[A,C]B

[A, [B, C]] + [C, [A, B]] + [B, [C, A]] = 0.

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[A, A] = 0

[A, B+C] = [A, B] + [A, C]

[A+B, C] = [A, C] + [B, C]

[A, BC] = B[A, C] + [A, B]C

[AB, C] = A[B,C]+[A,C]B

[A, [B, C]] + [C, [A, B]] + [B, [C, A]] = 0.

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[A, A] = 0

[A, B+C] = [A, B] + [A, C]

[A+B, C] = [A, C] + [B, C]

[A, BC] = B[A, C] + [A, B]C

[AB, C] = A[B,C]+[A,C]B

[A, [B, C]] + [C, [A, B]] + [B, [C, A]] = 0.

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[A, A] = 0

[A, B+C] = [A, B] + [A, C]

[A+B, C] = [A, C] + [B, C]

[A, BC] = B[A, C] + [A, B]C

[AB, C] = A[B,C]+[A,C]B

[A, [B, C]] + [C, [A, B]] + [B, [C, A]] = 0.

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Properties and Examples of Linear Operators

Two very important operators are

1. The null operator 0, which maps each vector in the space onto the null vector,

0|ψ = 0. ⟩

2. The identity operator 1, which maps each vector in the space onto itself, i.e.,

1|ψ = |ψ . ⟩ ⟩

The inverse of an operator A, if it exists, is denoted A⁻¹ and obeys the property

AA⁻¹ = A⁻¹A = 1.

Note: There exist numerous examples of operators A and B for which

AB = 1 but BA ≠ 1

in which case A and B are not inverses.

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0|ψ = 0. ⟩


1|ψ = |ψ . ⟩ ⟩


AA⁻¹ = A⁻¹A = 1.


AB = 1 but BA ≠ 1


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0|ψ = 0. ⟩


1|ψ = |ψ . ⟩ ⟩


AA⁻¹ = A⁻¹A = 1.


AB = 1 but BA ≠ 1


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0|ψ = 0. ⟩


1|ψ = |ψ . ⟩ ⟩


AA⁻¹ = A⁻¹A = 1.


AB = 1 but BA ≠ 1


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0|ψ = 0. ⟩


1|ψ = |ψ . ⟩ ⟩


AA⁻¹ = A⁻¹A = 1.


AB = 1 but BA ≠ 1


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A nonzero vector |χ is said to be an ⟩ eigenvector of the operator A with eigenvalue a (where generally, a ∈ C) if it satisfies the eigenvalue equation

A|χ = a|χ⟩ ⟩

The set of eigenvalues {a} for which solutions to this equation exist is referred to as the spectrum of A.

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A nonzero vector |χ is said to be an ⟩ eigenvector of the operator A with eigenvalue a (where generally, a ∈ C) if it satisfies the eigenvalue equation

A|χ = a|χ⟩ ⟩

The set of eigenvalues {a} for which solutions to this equation exist is referred to as the spectrum of A.

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Functions of Linear Operators

It is also possible to define operators that are, themselves, functions of other operators. This can be done in a number of ways.

For example, from the product rule given above, it is clear that in general the n-fold product of an operator A with itself is well-defined.

Aⁿ |ψ = A A . . . A |ψ⟩ ⟩

n terms

Thus, we may always speak of positive integer powers Aⁿ of an operator.

If the inverse A⁻¹ of an operator is also defined, then we can define negative powers through the relation

A⁻ⁿ ψ = A⁻¹ . . . A⁻¹ |ψ⟩ ⟩

n terms

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Aⁿ |ψ = A A . . . A |ψ⟩ ⟩

n terms



A⁻ⁿ ψ = A⁻¹ . . . A⁻¹ |ψ⟩ ⟩

n terms

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Aⁿ |ψ = A A . . . A |ψ⟩ ⟩

n terms



A⁻ⁿ ψ = A⁻¹ . . . A⁻¹ |ψ⟩ ⟩

n terms

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Aⁿ |ψ = A A . . . A |ψ⟩ ⟩

n terms



A⁻ⁿ |ψ = A⁻¹ . . . A⁻¹ |ψ⟩ ⟩

n terms

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Then, if

is any power series expandable function with a suitable radius of convergence, we can define the operator valued function

of the operator A.

Thus,

are often well defined

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Then, if


of the operator A.

Thus,


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Then, if


of the operator A.

Thus,


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Then, if


of the operator A.

Thus,


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Examples of Linear Operators

We now consider a variety of different operators.

We begin by noting that if we multiply any state |ψ in the space by a scalar λ we ⟩generate a new vector

|ψλ = λ|ψ . ⟩ ⟩

Thus, each scalar defines a very simple type of linear operator.

To avoid any cumbersome notation we simply will denote by λ that operator which multiplies a vector by λ.

This allows us, e.g., to form operators of the form λ + A, where A is an arbitrary linear operator and λ is a scalar, so that

(λ + A)|ψ = λ|ψ + A|ψ ⟩ ⟩ ⟩

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|ψλ = λ|ψ . ⟩ ⟩




(λ + A)|ψ = λ|ψ + A|ψ ⟩ ⟩ ⟩

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|ψλ = λ|ψ . ⟩ ⟩




(λ + A)|ψ = λ|ψ + A|ψ ⟩ ⟩ ⟩

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|ψλ = λ|ψ . ⟩ ⟩




(λ + A)|ψ = λ|ψ + A|ψ ⟩ ⟩ ⟩

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Special cases: λ=1 is the identity operator

λ=0 is the null operator.

Multiplication by a scalar is an operation that always commutes with any other linear operator, i.e., scalars always commute with everything. Thus, we can write

[λ,A]=0

The next class of operators acts in a slightly more complicated fashion. It still multiplies states by something, but what it multiplies by depends on what the state is. We call them multiplicative operators.

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[λ, A]=0


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[λ, A]=0


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Multiplicative Operators

In the space of a quantum particle moving in one dimension let us define an operator X by defining its action on the one-dimensional position states {|x } as ⟩follows:

X|x =x|x .⟩ ⟩

Thus, X just multiplies the basis vector |x by its label.⟩

This implies that the basis states |x are all eigenstates of the operator X, and are ⟩in fact labeled by their associated eigenvalues.

With this definition, we find that the action of X on an arbitrary state |ψ is ⟩rather simply expressed in the position representation, i.e.,

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X|x =x|x .⟩ ⟩




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X|x =x|x .⟩ ⟩




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X|x =x|x .⟩ ⟩




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X|x =x|x .⟩ ⟩




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X|x =x|x .⟩ ⟩




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Thus:

This shows that the wave function representing X|ψ is just xψ(x). ⟩

It’s useful at this point to distinguish the operators of Schrödinger’s mechanics from those of our abstract state space.

From this point onward, the corresponding Schrödinger operator will be denoted with a caret so that, for the operator X, which acts on kets, we can define a Schrödinger operator that acts on wave functions such that, in the position representation

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Thus:




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Thus:




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Thus:




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Thus:




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This idea is easily extended to functions of x. For any function f(x) we can define an operator F which has the action

of multiplying each basis vector |x in the position representation by the function ⟩f evaluated at the point x labeling the basis vector.

When F acts on arbitrary vectors it leads to the result

that is, in the position representation,

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For a particle moving in 3D we can thus define on the position states the components

of the (vector) position operator for which , so that in the position representation

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For a particle moving in 3D we can thus define on the position states the components

of the (vector) position operator for which , so that in the position representation

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The potential energy operator is also, clearly, a multiplicative operator in the position representation, i.e., we define such that

so that in the position representation

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The potential energy operator is also, clearly, a multiplicative operator in the position representation, i.e., we define such that

so that in the position representation

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These kinds of multiplicative operators can be defined for any representation.

If {|α } is an ONB for the space, we can define an operator A such that⟩

A|α =α|α⟩ ⟩

for all basis vectors of this representation. Then, for any function g(α) we can define an operator G=G(A) such that

G|α =g(α)|α ,⟩ ⟩

and then

Thus, in the α representation

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A|α =α|α⟩ ⟩


G|α =g(α)|α ,⟩ ⟩

and then


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A|α =α|α⟩ ⟩


G|α =g(α)|α ,⟩ ⟩

and then


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A|α =α|α⟩ ⟩


G|α =g(α)|α ,⟩ ⟩

and then


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A|α =α|α⟩ ⟩


G|α =g(α)|α ,⟩ ⟩

and then


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So in this lecture, we stated the second postulate, which associates observables with linear Hermitian operators, and explored the mathematical properties of linear operators on complex linear vector spaces.

We then studied in some detail a class of multiplicative operators, defined with respect to a particular representation, in which the operator simply multiplies the basis vectors by a function of the label that is used to distinguish them.

Each of the basis vectors of the representation on which the operator is defined, is for this class of operators, an eigenstate of the corresponding observable.

In the next lecture, we explore other ways of defining linear operators on the state spaces of quantum mechanical systems.

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Documents

Quantum One: Lecture 12 1. 2 Postulate II 3 Observables of Quantum Mechanical Systems 4