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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