# Quantum state of system after measurement

> Operator $$\hat{A}$$ has two normalized eigenstates $$\psi_1,\psi_2$$ with
> eigenvalues $$\alpha_1,\alpha_2$$. Operator $$\hat{B}$$ has also two
> normalized eigenstates $$\phi_1,\phi_2$$ with eigenvalues
> $$\beta_1,\beta_2$$. Eigenstates satisfy:

> $$\psi_1=(\phi_1+2\phi_2)/\sqrt{5}$$
> $$\psi_2=(2\phi_1-\phi_2)/\sqrt{5}$$
>
> We measure the quantity $$A$$ and we get the value $$\alpha_1$$. What's
> the state of the system after the measurement?
>
> What are the possible outcomes after the measurement of the quantity
> $$B$$? What is the probability of getting each one of them?

To begin with, I assume that, since the eigenfunction corresponding to the measured eigenvalue is $\psi_1$, this is also the state of the system after the measurement.

As for the possible outcomes of the measurement of $B$, I'm thinking of expressing $\psi_1,\psi_2$ as functions of $\phi_1,\phi_2$ and using them as eigenstates of $B$. Any help?

mfb
Mentor
I moved the thread to our homework section.
To begin with, I assume that, since the eigenfunction corresponding to the measured eigenvalue is $\psi_1$, this is also the state of the system after the measurement.
Right.
As for the possible outcomes of the measurement of $B$, I'm thinking of expressing $\psi_1,\psi_2$ as functions of $\phi_1,\phi_2$ and using them as eigenstates of $B$.
Why that direction? While possible, I don't think it is necessary.

By the way: You can make inline LaTeX with double #.

Sorry about the wrong syntax. After reexamining it, I see that the only possible outcomes after the measurement of ##B## is either ##\beta_1## or ##\beta_2##, but how are the given expressions useful for the calculation of the probability of getting each state?

nrqed