Use a New Representation: Consider a quantum system with an observable $A$ that has three possible measurement results: $a_1$, $a_2$, and $a_3$.
Write down the three kets $\ket{a_1}$, $\ket{a_2}$, and $\ket{a_3}$, corresponding to these possible results, using matrix notation.
The system is prepared in the state:
$$\ket{\psi} = 1\ket{a_1}-2\ket{a_2}+5\ket{a_3}$$
Staying in bra-ket notation, calculate the probabilities of all possible measurement results of the observable $A$. Plot a histogram of the predicted measurement results.
In a different experiment, the system is prepared in the state:
$$\ket{\psi} = 2\ket{a_1}+3i\ket{a_2}$$
Write this state in matrix notation and calculate the probabilities of all possible measurement results of the observable $A$. Plot a histogram of the predicted measurement results.
Using the Spins simulation, follow the link to switch to the \texttt{Spin-1} case. Set up an experiment for two successive measurements of spin projections.
Measure the probability that a state which starts out with $z$-component of spin equal to $\hbar$ ends up with $z$-component of spin equal to $\hbar$ after the $z$-component of spin is measured. Write your statement in bra-ket language.
Measure the probability that a state which starts out with $z$-component of spin equal to $\hbar$ ends up with $z$-component of spin equal to zero after the $z$-component of spin is measured. Write your statement in bra-ket language. What does this probability tell you about the $z$ basis?
Measure the probability that a state which starts out with $x$-component of spin equal to zero ends up with $z$-component of spin equal to zero after the $z$-component of spin is measured. Write your statement in bra-ket language. What does this probability tell you about the $x$ and $z$ bases?
Use your simulation to find the value of $\vert\braket{1}{-1}_x\vert^2$. State in words what the measured quantity represents. Compare your “measured” value to a theoretical value computed from the Spin Reference Sheet.
If a beam of spin-3/2 particles is input to a Stern-Gerlach analyzer, there are four output beams whose deflections are consistent with magnetic moments arising from spin angular momentum components of $\frac{3}{2}\hbar$, $\frac{1}{2}\hbar$, $-\frac{1}{2}\hbar$, and $-\frac{3}{2}\hbar$. For a spin-3/2 system:
Write down the eigenvalue equations for the $S_z$ operator.
Write down the matrix representation of the $S_z$ eigenstates.
Write down the matrix representation of the $S_z$ operator.
Write down the eigenvalue equations for the $S^2$ operator. (The eigenvalues of the $S^2$ are $\hbar^2s(s+1)$, where $s$ is the spin quantum number, and $S^2$ is proportional to the identify operator.)
Write down the matrix representation of the $S^2$ operator. Check Beasts: Is your operator proportional to the identity operator?