- #1
maxverywell
- 197
- 2
Let's suppose that we have an entangled state of two systems ##A## and ##B##:
$$
\frac{1}{2}\left(|\psi_1 \phi_1\rangle+|\psi_2 \phi_2\rangle \right)
$$
where ##|\psi \rangle## and ##|\phi \rangle## are energy eigenstates of ##A## and ##B## respectively. However the eigenstates##|\phi_1\rangle## and ##|\phi_2\rangle## are degenerate:
$$
\hat{H}_B|\phi_1\rangle=E|\phi_1\rangle
$$
$$
\hat{H}_B|\phi_2\rangle=E|\phi_2\rangle
$$
What will be the state of the system ##AB## after measuring the energy of ##B## and finding the value ##E##?
My guess is:
$$
\frac{\left(|\psi_1\rangle+|\psi_2 \rangle \right)}{\sqrt{2}} \frac{\left(|\phi_1\rangle+\phi_2\rangle \right)}{\sqrt{2}}
$$
$$
\frac{1}{2}\left(|\psi_1 \phi_1\rangle+|\psi_2 \phi_2\rangle \right)
$$
where ##|\psi \rangle## and ##|\phi \rangle## are energy eigenstates of ##A## and ##B## respectively. However the eigenstates##|\phi_1\rangle## and ##|\phi_2\rangle## are degenerate:
$$
\hat{H}_B|\phi_1\rangle=E|\phi_1\rangle
$$
$$
\hat{H}_B|\phi_2\rangle=E|\phi_2\rangle
$$
What will be the state of the system ##AB## after measuring the energy of ##B## and finding the value ##E##?
My guess is:
$$
\frac{\left(|\psi_1\rangle+|\psi_2 \rangle \right)}{\sqrt{2}} \frac{\left(|\phi_1\rangle+\phi_2\rangle \right)}{\sqrt{2}}
$$
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