Determing the composition of a state

[Sekonda]

Hey,

I have a question on determing the composition of a state of a system of composed of only two eigenvectors, the question is displayed below:

I initially assumed that the ket v was given by:

$$|v>=a|\omega_{1}>+b|\omega_{2}>$$

Where 'a' and 'b' are constants which will determine the probability of either state. So we know the probability of the eigenvalues of ket v are given by the coefficients a and b in the equation:

$$\mid <v|\hat{\Omega}|v>\mid^{2}$$

Where for the probability of attaining the eigenvalue ω(1) we have equation:

$$\frac{1}{4}=a^{4}$$

Though I'm not sure this is correct, it implies

$$a=\frac{1}{\sqrt{2}}\: ,\: b=\sqrt{\frac{\sqrt{3}}{2}}$$

I think I have made a mistake on my third equation...

Thanks for any help,
SK

 

[cosmic dust]

P(ω₁) = |<ω₁|v>|² , so: a = 1/2 and b will be (3/4)^1/2 multiplied by an arbitary phase factor.

 

[Sekonda]

Right okay, this is the other way I done it but wasn't sure which way was correct... though this answer definitely makes more sense!... obviously.

Thanks cosmic dust!
SK