Eigenfunctions and hermitian operators

[baldywaldy]

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]

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 \psi is an eigenfunction of the hermitian operator A_1, what does this mean? Can you write the eigenvalue problem?

 

[baldywaldy]

is it.

phi|A>= a|A> ?

 

[Hootenanny]

is it.

phi|A>= a|A> ?

Yes. So, we have (preserving the index):

A_1 \psi = a_1\psi.

Suppose we now have a second hermitian operator, A_2. Can you write a similar equation?

 

[baldywaldy]

Yes. So, we have (preserving the index):

A_1 \psi = a_1\psi.

Suppose we now have a second hermitian operator, A_2. Can you write a similar equation?

A_2 \psi = a_2\psi.

 

[Hootenanny]

A_2 \psi = a_2\psi.

Excellent!

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

 

[baldywaldy]

Excellent!

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

A_2A_1\psi = A_2(A_1\psi) =A_2(a1\psi)=a_1A_2\psi= a_1a_2\psi

A_1A_2\psi=a_2a_1\psi

Therefore they commute! XD

 

[Hootenanny]

A_2A_1\psi = A_2(A_1\psi) =A_2(a1\psi)=a_1A_2\psi= a_1a_2\psi

A_1A_2\psi=a_2a_1\psi

Therefore they commute! XD

Indeed they do!

 

[baldywaldy]

Indeed they do!

Thanks! :Danother 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

A_1\psi=a_1\psi

A_1θ=a_1θ

 

[dextercioby]

Shouldn't there be a₂ for the second line ?