Matrix representing projection operators

[SouthQuantum]

Hey everyone! I have a question regarding the matrix representation of a projection operator. Specifically, does the wavefunction have to be normalized before determining the projection operator? For example:

|Ψ1> = 1/3|u1> + i/3|u2> + 1/3|u3>

|Ψ2> = 1/3|u1> + i/3|u3>

Ψ1 is obviously normalized and Ψ2 isnt

Now to calculate the matrix that represents the projection operator, just make row and column matrices and multiply out. For Ψ1:

| 1/3|u1> |
| i/3|u2> | * (| 1/3|u1> i/3|u2> 1/3|u3> |)
| 1/3|u3> |

Which gives a 3x3 Hermitian matrix.

My question is do I have to normalize Ψ2 in order to do the same process? The result is a Hermitian operator either way, I believe, but I just want to know if 'technically' it needs to be normalized first.

thanks for any input!

Timmy

 

[SouthQuantum]

ps: I realize this is a little 'homeworky' and if it is too much so, I apologize for posting it in the wrong place.

 

[RedX]

yes if you want to build a projection operator out of a wavefunction then the wavefunction has to be normalized, but it's also true that it's Hermitian either way. A projection operator P should obey PP=P, so |$$\psi$$><$$\psi$$|$$\psi$$><$$\psi$$|=|$$\psi$$><$$\psi$$| requires <$$\psi$$|$$\psi$$>=1

 

[SouthQuantum]

Alright! Thanks for the help--I appreciate it!

Timmy