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.
Show that the kets $\vert + \rangle_y$ and $\vert - \rangle_y$ defined by $$\vert + \rangle_y\doteq \frac{1}{\sqrt{2}}\pmatrix{1\cr i}$$ $$\vert - \rangle_y\doteq \frac{1}{\sqrt{2}}\pmatrix{1\cr -i}$$ form an orthonormal basis for the vector space of two-component complex vectors, i.e.
Show that $\vert + \rangle_y$ and $\vert - \rangle_y$ are normalized.
Show that $\vert + \rangle_y$ and $\vert - \rangle_y$ are orthogonal.
Show that $\vert + \rangle_y$ and $\vert - \rangle_y$ are complete, i.e. that any vector in the vector space can be written as a linear combination of these two vectors.
Consider the three quantum states:
$$\left\vert \psi_1\right\rangle = \frac{1}{\sqrt{3}}\left\vert +\right\rangle + i\frac{\sqrt{2}}{\sqrt{3}} \left\vert -\right\rangle$$ $$\left\vert \psi_2\right\rangle = \frac{1}{\sqrt{5}}\left\vert +\right\rangle - \frac{2}{\sqrt{5}} \left\vert -\right\rangle$$ $$\left\vert \psi_3\right\rangle = \frac{1}{\sqrt{2}}\left\vert +\right\rangle + i\frac{e^{\frac{i\pi}{4}}}{\sqrt{2}} \left\vert -\right\rangle$$
For each of the $\vert \psi_i\rangle$ above, find the normalized vector $\vert \phi_i\rangle$ that is orthogonal to it.
Calculate the inner products $\langle \psi_i\vert \psi_j\rangle$ for $i$ and $j=1$, $2$, $3$.
A beam of spin-$\frac{1}{2}$ particles is prepared in the initial state $$ \left\vert \psi\right\rangle_x = \sqrt{2/5} |+\rangle_x - \sqrt{3/5} |-\rangle_x $$(Note the x-subscript on the kets!)
What are the possible results of a measurement of $S_x$, with what probabilities?
Repeat part A for measurements of $S_z$.
Suppose we start with this , first measure $S_x$, and happen to get $+\hbar /2$ . We then measure $S_z$. What are the possible results and with what probabilities?
A beam of spin-$\frac{1}{2}$ particles is prepared in the state:
$$\left\vert \psi\right\rangle = \frac{2}{\sqrt{13}}\left\vert +\right\rangle + i\frac{3}{\sqrt{13}} \left\vert -\right\rangle$$
What are the possible results of a measurement of the spin component $S_z$, and with what probabilities would they occur? Check Beasts: Check that you have the right “beast.”
What are the possible results of a measurement of the spin component $S_x$, and with what probabilities would they occur? Check Beasts: Check that you have the right “beast.”
Use Another Representation: Plot histograms of the predicted measurement results from parts $(a)$ and $(b)$.
Consider a quantum system described by an orthonormal basis $\ket{a_1}$, $\ket{a_2}$, and $\ket{a_3}$. The system is initially in a state:
$$\ket{\psi_{\hbox{in}}} = \frac{i}{\sqrt{3}}\ket{a_1}+\sqrt{\frac{2}{3}}\ket{a_2}$$ Find the probability that the system is measured to be in the final state: $$\ket{\psi_{\hbox{out}}} = \frac{1+i}{\sqrt{3}}\ket{a_1} +\frac{1}{\sqrt{6}}\ket{a_2} +\frac{1}{\sqrt{6}}\ket{a_3}$$
Check Beasts: Check that you have the right “beast.”
Consider the three quantum states:
$$\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle + i\frac{3}{5} \left\vert -\right\rangle$$ $$\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle - i\frac{3}{5} \left\vert -\right\rangle$$ $$\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle + i\frac{3}{5} \left\vert -\right\rangle$$
For each of the $\ket{\psi_i}$ above, calculate the probabilities of spin component measurements along the $x$, $y$, and $z$-axes.
Look For a Pattern (and Generalize): Use your results from $(a)$ to comment on the importance of the overall phase and of the relative phases of the quantum state vector.