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Eigenfunctions and hermitian operators

  • Thread starter baldywaldy
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Hi. I'm just a bit stuck on this question:

Write down two equations to represent the fact that a given wavefunction is simultaneously an eiigenfunction of two different hermitian operators. what conclusion can be drawn about these operators?

Im not quite sure how to start it.

Thanks!
 

Hootenanny

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Hi. I'm just a bit stuck on this question:

Write down two equations to represent the fact that a given wavefunction is simultaneously an eiigenfunction of two different hermitian operators. what conclusion can be drawn about these operators?

Im not quite sure how to start it.

Thanks!
If [itex]\psi[/itex] is an eigenfunction of the hermitian operator [itex]A_1[/itex], what does this mean? Can you write the eigenvalue problem?
 
is it.

phi|A>= a|A> ?
 

Hootenanny

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is it.

phi|A>= a|A> ?
Yes. So, we have (preserving the index):

[tex]A_1 \psi = a_1\psi.[/tex]

Suppose we now have a second hermitian operator, [itex]A_2[/itex]. Can you write a similar equation?
 
Yes. So, we have (preserving the index):

[tex]A_1 \psi = a_1\psi.[/tex]

Suppose we now have a second hermitian operator, [itex]A_2[/itex]. Can you write a similar equation?
[tex]A_2 \psi = a_2\psi.[/tex]
 

Hootenanny

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[tex]A_2 \psi = a_2\psi.[/tex]
Excellent!

So what happens if you first operate on [itex]\psi[/itex] with [itex]A_1[/itex], followed by [itex]A_2[/itex]? Compare this with what happens when you do it the other way round.
 
Excellent!

So what happens if you first operate on [itex]\psi[/itex] with [itex]A_1[/itex], followed by [itex]A_2[/itex]? Compare this with what happens when you do it the other way round.
[itex]A_2[/itex][itex]A_1[/itex][itex]\psi[/itex] = [itex]A_2[/itex]([itex]A_1[/itex][itex]\psi[/itex]) =[itex]A_2[/itex]([itex]a1[/itex][itex]\psi[/itex])=[itex]a_1[/itex][itex]A_2[/itex][itex]\psi[/itex]= [itex]a_1[/itex][itex]a_2[/itex][itex]\psi[/itex]

[itex]A_1[/itex][itex]A_2[/itex][itex]\psi[/itex]=[itex]a_2[/itex][itex]a_1[/itex][itex]\psi[/itex]

Therefore they commute!! XD
 

Hootenanny

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[itex]A_2[/itex][itex]A_1[/itex][itex]\psi[/itex] = [itex]A_2[/itex]([itex]A_1[/itex][itex]\psi[/itex]) =[itex]A_2[/itex]([itex]a1[/itex][itex]\psi[/itex])=[itex]a_1[/itex][itex]A_2[/itex][itex]\psi[/itex]= [itex]a_1[/itex][itex]a_2[/itex][itex]\psi[/itex]

[itex]A_1[/itex][itex]A_2[/itex][itex]\psi[/itex]=[itex]a_2[/itex][itex]a_1[/itex][itex]\psi[/itex]

Therefore they commute!! XD
Indeed they do! :approve:
 
Indeed they do! :approve:
Thanks! :D


another similar question to that one I have is :

write down two equations to represent the fact that two different wavefunctions are simultaneously eigenfunctions of the same hermation operator, with different eigenvalues. what conclusion can be drawn about these wavefunctions. So far I have

[itex]A_1[/itex][itex]\psi[/itex]=[itex]a_1[/itex][itex]\psi[/itex]

[itex]A_1[/itex]θ=[itex]a_1[/itex]θ
 

dextercioby

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Shouldn't there be a2 for the second line ?
 

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