In part (1) of SPINS Lab #2, you measured the probabilities of all the possible spin components for each of the unknown initial states $\left|\psi_i\right\rangle(i=1,2,3,4)$. Using your data from that lab, find the unknown states $\left|\psi_1\right\rangle,\left|\psi_2\right\rangle,\left|\psi_3\right\rangle$, and $\left|\psi_4\right\rangle$. Express each of the unknown states as a linear superposition of the $S_z$ basis states $|+\rangle$ and $|-\rangle$. For each state, use your result to calculate the theoretical values of the probabilities for each component measurement and compare these theoretical predictions with your experimental results.

Compute forward and backward difference approximations of O(h) and

$$O(h^2),$$

and central difference approximations of

$$O(h^2)$$

and

$$O(h^4)$$

for the first derivative of y = cos x at

$$x = \pi/4$$

using a value of

$$h = \pi/12.$$

. Estimate the true percent relative error

$$\boldsymbol { \varepsilon } _ { t }$$

for each approximation.

Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval. x ln x-1=0; (1, e)

Ben has a piece of plywood that measures 1 yard on each side. He cuts the wood into 6 equal pieces. Each piece is $\frac{1}{3}$ yard wide and $\frac{1}{2}$ yard long. What is the area ofeach piece that Ben cuts? You can use a model to help you understand the problem. Start by drawing a square that is 1 yard on each side and then divide it as described in the problem. Use your model to solve the problem. area of one piece $=\frac{1}{2} \times \frac{1}{3}=\frac{1 \times 1}{2 \times 3}=\frac{1}{6}$. The area ofeach piece of plywood is $\frac{1}{6}$ square yard. Explain how the product $\frac{1}{2} \times \frac{1}{3}$ relates to the model.

The spin components of a beam of atoms prepared in the state $\left|\psi_{\text {in }}\right\rangle$ are measured and the following experimental probabilities are obtained:

$$\begin{array}{lll} \mathcal{P}_{+}=\frac{1}{2} & \mathcal{P}_{+x}=\frac{3}{4} & \mathcal{P}_{+y}=0.067 \\ \mathcal{P}_{-}=\frac{1}{2} & \mathcal{P}_{-x}=\frac{1}{4} & \mathcal{P}_{-y}=0.933 \end{array}$$

From the experimental data, determine the input state.

Consider a quantum system described by a basis $\left|a_1\right\rangle,\left|a_2\right\rangle$, and $\left|a_3\right\rangle$. The system is initially in a state

$$\left|\psi_i\right\rangle=\frac{i}{\sqrt{3}}\left|a_1\right\rangle+\sqrt{\frac{2}{3}}\left|a_2\right\rangle$$

Find the probability that the system is measured to be in the final state

$$\left|\psi_f\right\rangle=\frac{1+i}{\sqrt{3}}\left|a_1\right\rangle+\frac{1}{\sqrt{6}}\left|a_2\right\rangle+\frac{1}{\sqrt{6}}\left|a_3\right\rangle .$$

Find the derivative of the function. s = 4/(3pi) sin 3t + 4/(5pi) cos 5t

Consider a quantum system in which the energy $E$ is measured and there are four possible measurement results: $2 \mathrm{eV}, 4 \mathrm{eV}, 7 \mathrm{eV}$, and $9 \mathrm{eV}$. The system is prepared in the state

$$|\psi\rangle=\frac{1}{\sqrt{39}}\left\{3|2 \mathrm{eV}\rangle-i|4 \mathrm{eV}\rangle+2 e^{i \pi / 7}|7 \mathrm{eV}\rangle+5|9 \mathrm{eV}\rangle\right\}$$

Calculate the probabilities of all possible measurement results of the energy E. Plot a histogram of the predicted measurement results.

If S := { $\frac{1}{n} - \frac{1}{m} : n, m ∈ ℕ$}, find inf S and sup S.

Evaluate the integral by making an appropriate change of variables. double integral cos(y-x/y+x)dA, where is the trapezoidal region with vertices (1, 0), (2, 0), (0, 2), and (0, 1)

Consider a quantum system with an observable $A$ that has three possible measurement results: $a_1, a_2$, and $a_3$. a) Write down the three kets $\left|a_1\right\rangle,\left|a_2\right\rangle$, and $\left|a_3\right\rangle$ corresponding to these possible results using matrix notation. b) The system is prepared in the state

$$|\psi\rangle=1\left|a_1\right\rangle-2\left|a_2\right\rangle+5\left|a_3\right\rangle .$$

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. c) In a different experiment, the system is prepared in the state

$$|\psi\rangle=2\left|a_1\right\rangle+3 i\left|a_2\right\rangle .$$

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.

Find the degree and a basis for the given field extension. Be prepared to justify your answers. ℚ(∛2, √3) over ℚ

Find the degree and a basis for the given field extension. Be prepared to justify your answers. ℚ(√2, ∛5) over ℚ

Consider a quantum system with an observable $A$ that has three possible measurement results: $a_1, a_2$, and $a_3$. Write down the orthogonality, normalization, and completeness relations for the three kets comprising the basis corresponding to the possible results of the A measurement.