Homework for Spins
- (General State)
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.
- (YBasis)
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.
- (Orthogonal) This problem displays the benefits of using bra-ket notation as opposed to matrix notation. Part a) involves practice using orthogonality conditions to find unknown states, and use of the inner product is common.
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 $j=1$, $2$, $3$.
- (MeasurementProbabilities)
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?
- (Histogram) This is designed to be a relatively easy problem that tests students' knowledge of the fourth postulate of quantum mechanics. The problem also introduces histograms in a simple setting.
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 $(b)$.
- (InOut) The introduction of states that do not have the basis $\vert + \rangle$ or $\vert - \rangle$ may throw some students off, but using the fourth postulate makes this problem straightforward.
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.”
- (Phase) This problem shows how relative phases can affect probability calculations. Practice with the inner product and the fourth postulate of quantum mechanics is a main focus as well.
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.
