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phyzmatix
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Homework Statement
The components of the initial state [tex]|\psi_i>[/tex] of a quantum system are given in a complete and orthonormal basis of three states [tex]|\phi_1>, |\phi_2>, |\phi_3>[/tex] by
[tex]<\phi_1|\psi_i>=\frac{i}{\sqrt{3}}[/tex]
[tex]<\phi_2|\psi_i>=\sqrt{\frac{2}{3}}[/tex]
[tex]<\phi_3|\psi_i>=0[/tex]
Calculate the probability of finding the system in a state [tex]|\psi_f>[/tex] whose components are given in the same basis by
[tex]<\phi_1|\psi_f>=\frac{1+i}{\sqrt{3}}[/tex]
[tex]<\phi_2|\psi_f>=\frac{1}{\sqrt{6}}[/tex]
[tex]<\phi_3|\psi_f>=\frac{1}{\sqrt{6}}[/tex]
The Attempt at a Solution
Actually, I must admit that I don't really know what I have to do to answer this question. However, while experimenting with possible approaches to a solution I got to
[tex]P_1=|<\phi_1|\psi_i>|^2=|\frac{i}{\sqrt{3}}|^2=-\frac{1}{3}[/tex]
[tex]P_2=|<\phi_2|\psi_i>|^2=|\sqrt{\frac{2}{3}}|^2=\frac{2}{3}[/tex]
[tex]P_3=|<\phi_3|\psi_i>|^2=|0|^2=0[/tex]
But is it possible to get a negative probability?
Also, since the three states are orthonormal, shouldn't they automatically be normalized, and the total probability [tex]\sum{P_i}=1[/tex]?
Any help here will be greatly appreciated.
phyz