the identity operator
TRANSCRIPT
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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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