# 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?

## Answers and Replies

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
Science Advisor
Homework Helper
Gold Member
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?
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?
You know the wave function after the measurement of A, you just need to express it in terms of the eigenstates of B.