the identity operator

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6.1 Types of linear operators

Slides: Video 6.1.3 The identity

operatorText reference: Quantum Mechanics

for Scientists and EngineersSection 4.8

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Types of linear operator

The identity operator

Quantum mechanics for scientists and engineers Da

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Identity operator

The identity operator is the operator that

when it operates on a vector (function)

leaves it unchanged

In matrix form, the identity operator is

In bra-ket form

the identity operator can be written

where the

form a complete basis for the space

ˆ I 

1 0

ˆ

0

 I 

ˆ

i

 I   i 

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Identity operator - proof 

For an arbitrary function we know

so

Now, with our proposed form

then

But is just a number

and so it can be moved in the productHence

and hence, using ,

ˆi i

i

 I     

i i

i

 f c      c

i i

i

 f f   

ˆ

i ii

 I f f   

i

  f  

ˆi i

i

 I f f    i i

i

 f f      ˆ I f f 

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Identity operator

The statement

is trivial if is the basis used to represent th

Then

so that

ˆi i

i

 I     

i 

1

1

0

0 

1 1

1

01 0 0 0

0

0

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Identity operator

Similarly

so

2 2

0 0 0

0 1 0

0 0 0  

3 3

0

0

0  

ˆi i

i

 I     

1 0 0

0 1 0

0 0 1

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Identity operator

Note, however, that

even if the basis being used is not the set

Then some specific

is not a vector with an ith element of 1 an

other elements 0

and the matrix in general has p

all of its elements non-zeroNonetheless, the sum of all matrices

still gives the identity matrixWe can use any convenient complete basis to wr

ˆi i

i

 I     

i

i 

i i  

i i  ˆ

 I 

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Identity operator

The expression has a simple vecto

In the expressionis just the projection of onto the

so multiplying by

that is,

gives the vector component of on

Provided the form a complete set

adding these components up just reconstructs

ˆi i

i

 I     

i ii f f   

i i i i f f   

i  f  

i  f 

i  i   f  

 f 

i 

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Identity matrix in formal proofs

Since the identity matrix is the identity matr

no matter what complete orthonormal

basis we use to represent itwe can use the following tricks

First, we “insert” the identity matrixin some basis

into an expressionThen, we rearrange the expression

Then, we find an identity matrix wcan take out of the result

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Proof that the trace is independent of the

Consider the sum, S 

of the diagonal elements of an operator

on some complete orthonormal basis

Now suppose we have some other completeorthonormal basis

We can therefore also write the identity opera

ˆ A

i 

ˆi i

i

S A  

m 

ˆm m

m

 I     

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Proof that the trace is independent of the

In

we can insert an identity operator just beforewhich makes no difference to the result

since

so we have

ˆi i

i

S A  

 A

ˆ ˆˆ IA A

ˆˆi i

iS IA  

i m mi m  A

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Proof that the trace is independent of the

Rearranging

reordering the sums

moving the number

moving a sum and associating

recognizing

ˆ ˆi i i m m

i i m

S IA

i m

m i

S

i m  

ˆ i i

i

 I     

ˆm

m i

 A

ˆm

m i

 A

ˆ ˆm m

m

 AI   

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Proof that the trace is independent of the

So, with now

the final step is to note that

so

Hence the trace of an operator

the sum of the diagonal elements

is independent of the basis used to represent toperator

which is why the trace is a useful operator p

ˆ ˆ ˆi i m m

i m

S A AI    

ˆ ˆˆ AI Aˆ ˆ

i i m m

i m

S A A  

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