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.
- 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.
The system is prepared in the state:
\[\left|{\psi_b}\right\rangle = N\left(1\left|{a_1}\right\rangle -2\left|{a_2}\right\rangle +5\left|{a_3}\right\rangle \right)\]
- Staying in bra-ket notation, find the normalization constant.
- Calculate the probabilities of all possible measurement results of the observable \(A\). Check “beasts.”
- Use a New Representation: Plot a histogram of the predicted measurement results.
In a different experiment, the system is prepared in the state:
\[\left|{\psi_c}\right\rangle = N\left(2\left|{a_1}\right\rangle +3i\left|{a_2}\right\rangle \right)\]
- Write this state in matrix notation and find the normalization constant.
- Calculate the probabilities of all possible measurement results of the observable \(A\). Check “beasts”.
- Use a New Representation: Plot a histogram of the predicted measurement results.