Hermitian Operators in Quantum Mechanics
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Transcript of Hermitian Operators in Quantum Mechanics
8/10/2019 Hermitian Operators in Quantum Mechanics
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6.3 Operators and quantummechanics
Slides: Video 6.3.1 Hermitian
operators in quantum mechanicsText reference: Quantum Mechanics
for Scientists and EngineersSection 5.1
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Operators and quantum mec
Hermitian operators in quanmechanics
Quantum mechanics for scientists and engineers Da
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Commutation of Hermitian operators
For Hermitian operators and representinphysical variables
it is very important to know if theycommute
i.e., is ?Remember that
because these linear operators obey thesame algebra as matrices
in general operators do not commute
ˆ ˆˆ ˆ B BA
ˆ ˆ B
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Commutator
For quantum mechanics, we formally definean entity
This entity is called the commutatorAn equivalent statement to saying
is then
Strictly, this should be written
where is the identity operator
but this is usually omitted
ˆ ˆ ˆˆ ˆ ˆ, B AB BA
ˆ ˆˆ ˆ B BAˆ ˆ, 0 A B
ˆ ˆ ˆ, 0 B I ˆ I
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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, 0 A B
ˆ ˆˆ, A B iC C
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Commuting operators and sets of eigenfu
Operators that commute share the same set ofeigenfunctions
and
operators that share the same set of eigenfunctiocommute
We will now prove both of these statements
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Commuting operators and sets of eigenfu
Suppose that operators and commute
and suppose the are the eigenfunctions o
with eigenvalues
Then
i.e.,
But this means that the vectoris also the eigenvector or is proportional t
i.e., for some number Bi
ˆ A ˆ B
n
i A
ˆ ˆˆ ˆi i
AB BA
ˆ ˆ ˆi i i
A B A B ˆ
i B
i
ˆi i i
B B
ˆi i
BA ˆi i
A B
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Commuting operators and sets of eigenfu
This kind of relation
holds for all the eigenfunctions
so these eigenfunctionsare also the eigenfunctions of the operato
with associated eigenvalues Bi
Hence we have proved the first statement that
operators that commute share the same set ofeigenfunctions
Note that the eigenvalues Ai and Bi are not in geequal to one another
ˆi i i
B B
i
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Commuting operators and sets of eigenfu
Now we consider the statement
operators that share the same set of eigenfunc
commuteSuppose that the Hermitian operators and
share the same complete set of eigenfuncwith associated sets of eigenvalues An and B
respectively
Then
and similarly
ˆ A ˆ B
n
ˆ ˆˆi i i i i i
AB AB A B
ˆˆ ˆi i i i i i
BA BA B A
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Commuting operators and sets of eigenfu
Hence, for any function
which can always be expanded in this complet
functions i.e.,we have
Since we have proved this for an arbitrary functio
we have proved that the operators commutehence proving the statement
operators that share the same set ofeigenfunctions commute
f
n
i i
i
f c
ˆ ˆˆ ˆi i i i i i i i
i i
B f c A B c B A BA
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