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6.2 Unitary and Hermitian operators
Slides: Video 6.2.1 Using unitaryoperators
Text reference: Quantum Mechanicsfor Scientists and Engineers
Section 4.10 (starting from“Changing the representation ofvectors”)
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Unitary and Hermitian operato
Using unitary operators
Quantum mechanics for scientists and engineers David M
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Unitary operators to change representations of vecto
Suppose that we have a vector (function)that is represented
when expressed as an expansion onthe functions
as the mathematical column vectorThese numbers c1, c2, c3, …
are the projections ofon the orthogonal coordinate axesin the vector space
labeled with , , …
old f
n
old f
old f
1 2 3
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Unitary operators to change representations of vecto
Suppose we want to represent this vector on a new setof orthogonal axes
which we will label , , …Changing the axes
which is equivalent to changing the basis set offunctions
does not change the vector we are representingbut it does changethe column of numbers used to represent th
vector
1 2 3
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Unitary operators to change representations of vecto
For example, suppose the original vectorwas actually the first basis vector in the old basis
Then in this new representationthe elements in the column of numbers
would be the projections of this vectoron the various new coordinate axes
each of which is simplySo under this coordinate transformationor change of basis
old f
1m 1
0
0
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Unitary operators to change representations of vecto
Writing similar transformations for each basis vectorwe get the correct transformation
if we define a matrix
where
and we define our new column of numbers
11 12 13
21 22 23
31 32 33ˆ
u u u
u u uU u u u
ij i ju
ˆnew old f U f
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Unitary operators to change representations of vecto
Note incidentally that hereand are the same vector in the vector sp
Only the representationthe coordinate axes
and, consequentlythe column of numbers
that have changednot the vector itself
old f new f
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Unitary operators to change representations of vecto
Now we can prove that is unitaryWriting the matrix multiplication in its sum form
sohence is unitary
since its Hermitian transpose is therefore itinverse
U
†ˆ ˆmi mj
ijm
U U u u
†
ˆ ˆ ˆU U I U
m i m jm
i m
i m m jm
ˆ
i j I i j
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Unitary operators to change representations of vecto
Hence any change in basiscan be implemented with a unitary operator
We can also say thatany such change in representation to a new
orthonormal basisis a unitary transform
Note also, incidentally, that
so the mathematical order of this multiplicationmakes no difference
†
† † †ˆ ˆ ˆ ˆ ˆUU U U I I
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Unitary operators to change representations of opera
Consider a number such aswhere vectors and and operator are arbitr
This result should not depend on the coordinate systemso the result in an “old” coordinate system
should be the same in a “new” coordinate systemthat is, we should have
Note the subscripts “new” and “old” refer to representnot the vectors (or operators) themselveswhich are not changed by change of representati
Only the numbers that represent them are chan
ˆg A f f g ˆ A
og
ˆ new new new g A f
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Unitary operators to change representations of opera
With unitary operator to go from “old” to “new” sywe can write
Since we believe also that
then we identify
or since
then
†ˆ ˆnew new new new new newg A f g A f
U
ˆ new new new old g A f g †ˆ ˆˆ ˆ
old newU A U
† † †ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆold new newUA U UU A UU A
†ˆ ˆˆ ˆnew old A UA U
† ˆˆ ˆ
old new old U g A U f old g U
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Unitary operators that change the state vector
For example, if the quantum mechanical state
is expanded on the basis to givethen
and if the particle is to be conservedthen this sum is retained as the quantum
mechanical system evolves in timeBut this is just the square of the vector l
Hence a unitary operator, which conserves lengthdescribes changes that conserve the particle
n n
2
1nn
a
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