linear operators and their algebra

8/10/2019 Linear Operators and Their Algebra

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Consequences of linear operator alge

Because of the mathematical equivalence ofmatrices and linear operators

the algebra for such operatorsis identical to that of matrices

In particularoperators do not in general commute

is not in general equal tofor any arbitrary

Whether or not operators commute

is very important in quantum mechanics

ˆ ˆ

AB f ˆˆ

BA f f



Generalization to expansion coefficien

We discussed operators

for the case of functions of position (e.g.,

but we can also use expansioncoefficients on basis sets

We expanded and

We could have followed a similar argumentrequiring each expansion coefficient d idepends linearly on all the expansion

coefficients cn

n n

n

f x c x g x



Generalization to expansion coefficien

By similar argumentswe would deduce the most general linea

between the vectors of expansion coefcould be represented as a matrix

The bra-ket statement of the relation betweand remains unchanged as

1 11 12 13 1

2 21 22 23 2

3 31 32 33 3

d A A A cd A A A c

d A A A c

ˆ ˆg A f



Evaluating the matrix elements of an o

Now we will find out how we can write somoperator

as a matrixThat is, we will deduce how to calculat

all the elements of the matrixif we know the operator

Suppose we choose our function

to be the jth basis function

so or equivalently

f x j x

j f x x f




Then, in the expansion

we are choosing

with all the other c’s being 0

Now we operate on this with

into get

Suppose specificallywe want to know the resulting coefficient

in the expansion

n n

n

f x c x 1

jc

f ˆ A

ˆg A f g

n nn

g x d x




From the matrix form of

with our choice and all other c’s 0 twe would have

1 jc

1 11 12 13 1

2 21 22 23 2

3 31 32 33 3

d A A A c

d A A A c

d A A A c

ˆg A f

i ijd A




For example, for

that is, and all other c’s 0 then

so in this example

2 j

2 1c

1 12 11 12 13

2 22 21 22 23

3 32 31 32 33

0

1

0

d A A A A

d A A A A

d A A A A

3 32

d A




But, from the expansions for and

for the specific case of

To extract d i from this expressionwe multiply by on both sides to obta

But we already concluded for this case that

So

f g

j f

ˆ ˆn n j

n

g d A f A

i

ˆi i jd A

ˆij i j A A



Evaluating the matrix elements of an oper

But our choices of i and j here were arbitrary

So quite generally

when writing an operator as a matrixwhen using a basis set

the matrix elements of that operator a

We can now turn any linear operator into a matri

For example, for a simple one-dimensional spa

ˆij i j

A A

n

ˆ A

ˆij i j

x A x dx



Visualization of a matrix element

Operator

acting on the unit vector

generates the vector

with generally a new length anddirection

The matrix element

is the projection of

onto the axis

j

axis j

axisk

ˆ

i

A

ˆ j

A

ˆ A

ˆ j

A

j

ˆi j

A

ˆ j A

i

linear operators and their algebra

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