## General State

• Quantum Fundamentals 2021

Use a New Representation: Consider a quantum system with an observable $A$ that has three possible measurement results: $a_1$, $a_2$, and $a_3$. States $\left|{a_1}\right\rangle$, $\left|{a_2}\right\rangle$, and $\left|{a_3}\right\rangle$ are eigenstates of the operator $\hat{A}$ corresponding to these possible measurement results.

1. Using matrix notation, express the states $\left|{a_1}\right\rangle$, $\left|{a_2}\right\rangle$, and $\left|{a_3}\right\rangle$ in the basis formed by these three eigenstates themselves.
2. The system is prepared in the state:

$\left|{\psi_b}\right\rangle = N\left(1\left|{a_1}\right\rangle -2\left|{a_2}\right\rangle +5\left|{a_3}\right\rangle \right)$

1. Staying in bra-ket notation, find the normalization constant.
2. Calculate the probabilities of all possible measurement results of the observable $A$. Check “beasts.”
3. Use a New Representation: Plot a histogram of the predicted measurement results.

3. In a different experiment, the system is prepared in the state:

$\left|{\psi_c}\right\rangle = N\left(2\left|{a_1}\right\rangle +3i\left|{a_2}\right\rangle \right)$

1. Write this state in matrix notation and find the normalization constant.
2. Calculate the probabilities of all possible measurement results of the observable $A$. Check “beasts”.
3. Use a New Representation: Plot a histogram of the predicted measurement results.