Lesson 17 Operators and quantum mechanicsCommuting operators and sets of eigenfunctions Suppose that...

6.3 Operators and quantum mechanics

Slides: Video 6.3.1 Hermitian operators in quantum mechanics

Text reference: Quantum Mechanics for Scientists and Engineers

Section 5.1

Operators and quantum mechanics

Hermitian operators in quantum mechanics

Quantum mechanics for scientists and engineers David Miller

Commutation of Hermitian operators

For Hermitian operators and representing physical variables

it is very important to know if they commute

i.e., is ? Remember that

because these linear operators obey the same algebra as matrices

in general operators do not commute

ˆ ˆˆ ˆAB BA

A B

Commutator

For quantum mechanics, we formally define an entity

This entity is called the commutatorAn equivalent statement to saying

is then

Strictly, this should be writtenwhere is the identity operator

but this is usually omitted

ˆ ˆ ˆˆ ˆ ˆ,A B AB BA

ˆ ˆˆ ˆAB BAˆ ˆ, 0A B

ˆ ˆ ˆ, 0A B I I

Commutation of operators

If the operators do not commute then does not hold

and in general we can choose to write

where is sometimes referred to as the remainder of commutation or

the commutation rest

ˆ ˆ, 0A B

ˆ ˆˆ,A B iC C

Commuting operators and sets of eigenfunctions

Operators that commute share the same set of eigenfunctions

and

operators that share the same set of eigenfunctions commute

We will now prove both of these statements


Suppose that operators and commute and suppose the are the eigenfunctions of

with eigenvalues Then

i.e.,

But this means that the vector is also the eigenvector or is proportional to it

i.e., for some number Bi

A Bn A

iAˆ ˆˆ ˆ

i iAB BA

ˆ ˆ î i iA B A B

îB

i

î i iB B

î iBA ˆ

i iA B


This kind of relation holds for all the eigenfunctions

so these eigenfunctions are also the eigenfunctions of the operator

with associated eigenvalues BiHence we have proved the first statement that

operators that commute share the same set of eigenfunctions

Note that the eigenvalues Ai and Bi are not in general equal to one another

ˆi i iB B

i

B


Now we consider the statementoperators that share the same set of eigenfunctions commute

Suppose that the Hermitian operators and share the same complete set of eigenfunctions

with associated sets of eigenvalues An and Bnrespectively

Then

and similarly

A Bn

ˆ ˆˆi i i i i iAB AB A B

ˆˆ ˆi i i i i iBA BA B A


Hence, for any function which can always be expanded in this complete set of functions i.e.,

we have

Since we have proved this for an arbitrary function we have proved that the operators commute

hence proving the statementoperators that share the same set of

eigenfunctions commute

f

n i ii

f c

ˆ ˆˆ ˆi i i i i i i i

i i

AB f c A B c B A BA f


Slides: Video 6.3.3 General form of the uncertainty principle


Section 5.2 (up to “Position-momentum uncertainty principle”)


General form of the uncertainty principle



First, we need to set up the concepts of the mean and variance of an expectation value

Using to denote the mean value of a quantity Awe have, in the bra-ket notation

for a measurable quantity associated with the Hermitian operator

when the state of the system is

A

Af

ˆA A f A f


Let us define a new operator associated with the difference between the measured value of A and its average value

Strictly, we should writebut we take such an identity operator to be

understoodNote that this operator is also Hermitian

A

ˆ ˆA A A ˆ ˆ ˆA A AI


Variance in statistics is the “mean square” deviation from the average

To examine the variance of the quantity Awe examine the expectation value of the operator

Expanding the arbitrary functionon the basis of the eigenfunctions of

i.e.,

we can formally evaluate the expectation value of when the system is in state

2ˆ( )Af

i Ai i

i

f c

2ˆ( )A f


We have

2 2ˆ ˆ( ) ( )A f A f 2ˆi i j j

i j

c A A c

ˆi i j j j

i j

c A A c A A

2

i i j j ji j

c c A A 22

i ii

c A A


Because the are the probabilities that the system is found, on measurement, to be in the state

and for that state simply represents the squared deviation of the value of the quantity A

from its average value then by definition

is the mean squared deviation for the quantity Aon repeatedly measuring the system prepared in state

2ic

i 2

iA A

2 2 22 ˆ ˆ ˆA A A A f A A f

f


In statistical language

the quantity is called the variance and the square root of the variance

which we can write asis the standard deviation

In statisticsthe standard deviation gives a

well-defined measure of the width of a distribution

2A

2A A


We can also consider some other quantity Bassociated with the Hermitian operator

and, with similar definitions

So we have ways of calculating the uncertainty in the measurements of the quantities A and B

when the system is in a state to use in a general proof of the uncertainty principle

BˆB B f B f

2 2 22 ˆ ˆ ˆB B B B f B B f

f


Suppose two Hermitian operators and do not commute and have a commutation rest

as defined above in Consider, for some arbitrary real number , the number

By we mean the vector

written this way to emphasize it is simply a vector so it must have an inner product with itself

that is greater than or equal to zero

A BC

ˆ ˆˆ,A B iC

ˆ ˆˆ ˆ 0G A i B f A i B f

ˆ ˆA i B f ˆ ˆA i B f


So

By Hermiticity

†ˆ ˆˆ ˆG f A i B A i B f

ˆ ˆˆ ˆf A i B A i B f

0

† †ˆ ˆˆ ˆf A i B A i B f

2 22 ˆ ˆ ˆˆ ˆ ˆf A B i A B B A f

2 22 ˆ ˆˆ ˆ,f A B i A B f

2 22 ˆ ˆˆf A B C f


So

where

By a simple (though not obvious) rearrangement

2 22 ˆ ˆˆG f A B C f

2 22 2

2 20

2 4

CCG A BA A

0

ˆC C f C f

2 22 A B C


But

must be true for arbitrary

so it is true for

which sets the first term equal to zero

so or

2 22 2

2 20

2 4

CCG A BA A

22

C

A

22 2

4C

A B 2C

A B


So, for two operators andcorresponding to measurable quantities A and B

for which in some state for which

we have the uncertainty principle

where A an B are the standard deviations of the values of A and B we would measure

2C

A B

A B

ˆ ˆˆ,A B iC f ˆC C f C f


Only if the operators and commute

i.e., (or, strictly, )

or if they do not commute, i.e.,

but we are in a state for which

is it possible for both A and B simultaneously to have exact measurable values

A B

ˆ ˆ, 0A B ˆ ˆ ˆ, 0A B I

f

ˆ ˆˆ,A B iC ˆ 0f C f


Slides: Video 6.3.5 Specific uncertainty principles


Section 5.2 (starting from “Position-momentum uncertainty principle”)


Specific uncertainty principles


Position-momentum uncertainty principle

We now formally derive the position-momentum relationConsider the commutator of and x

(We treat the function x as the operator for position) To be sure we are taking derivatives correctly

we have the commutator operate on an arbitrary function

ˆ xp

ˆ ,xd dp x f i x x fdx dx

d di x f x f

dx dx

d di f x f x fdx dx

i f

Position-momentum uncertainty principle

Insince is arbitrary

we can writeand the commutation rest operator

is simply the numberHence

and so, from

we have

ˆ ,xp x f i f f

ˆ ,xp x i C

C C

/ 2A B C

2xp x

Energy-time uncertainty principle

The energy operator is the Hamiltonianand from Schrödinger’s equation

so we use If we take the time operator to be just t

then using essentially identical algebra as used for the momentum-position uncertainty principle

so, similarly we have

HH i

t

ˆ /H i t

ˆ ,H t i t t it t

2E t

Frequency-time uncertainty principle

We can relate this result mathematically to the frequency-time uncertainty principle

that occurs in Fourier analysis Noting that in quantum mechanics

we haveE

12

t

Lesson 17 Operators and quantum mechanicsCommuting operators and sets of eigenfunctions Suppose that...

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