How Do You Normalize an Angular Momentum State and Determine Lz Probabilities?

[kcasali]

Homework Statement

A particle is in an angular momentum state

Ψ(θ,φ) = |l=1,m=1> + 2|1,0> + 3|1,-1>

Normalize this state and find the probabilities for finding the system with its third component Lz with values hbar, 0, -hbar.

Homework Equations

The Attempt at a Solution

I know that to normalize the kets, I just have to multiply them by their bras, but I'm not sure if I multiply the entire equation by the same bra or how to multiply it by three different bras. (I'm also not sure if I'm even on the right track.)

 

[xcrunner2414]

Homework Statement

A particle is in an angular momentum state

Ψ(θ,φ) = |l=1,m=1> + 2|1,0> + 3|1,-1>

Normalize this state and find the probabilities for finding the system with its third component Lz with values hbar, 0, -hbar.

- the coefficients 1, 2, and 3 are probability amplitudes. you know that if you square the probability amplitude, you get a probability. Right now, your probability adds up to 1^2 + 2^2 + 3^2. You want to do something to these coefficients so that their squares add up to 1... instead of 1^2 + 2^2 + 3^2. But you have to do the same thing to each amplitude so that their relative sizes don't change.

 

[kcasali]

- the coefficients 1, 2, and 3 are probability amplitudes. you know that if you square the probability amplitude, you get a probability. Right now, your probability adds up to 1^2 + 2^2 + 3^2. You want to do something to these coefficients so that their squares add up to 1... instead of 1^2 + 2^2 + 3^2. But you have to do the same thing to each amplitude so that their relative sizes don't change.

That makes sense, but where am I supposed to get the hbar and -hbar from? Am I missing something really obvious?

 

[xcrunner2414]

That makes sense, but where am I supposed to get the hbar and -hbar from? Am I missing something really obvious?

- If you measure the particle to have z-component spin of hbar, that means it's in the |1,1> state. If its z-component is 0, it's in the |1,0> state. If its z-component is -hbar, it's in the |1,-1> state.

 

[kcasali]

- If you measure the particle to have z-component spin of hbar, that means it's in the |1,1> state. If its z-component is 0, it's in the |1,0> state. If its z-component is -hbar, it's in the |1,-1> state.

Ahhh, thank you! Am I on the right track to normalize the state initially?

 

[kcasali]

Nevermind, I got it. Thank you!

 

[Theory4G]

Had the same problem quasite a while ago.
Did the same mistake.