Linear Algebra and Quantum Mechanics Crash Course (50 pts.)

1. Quantum Mechanics (9 pts.) In your own words, define the following:
   1. Wave-particle duality (3pts.)
   2. Superposition (3pts.)
   3. Entanglement (3pts.)
2. Statistics (13 pts.) Consider a group of 10 rooms in an old building. Three of the rooms have no spiders, six of the rooms have one spider, and one room has 5 spiders.
   • (1 pt.) Is the number of spiders in a room a discrete or continuous variable? Explain.
   • (2 pts.) What is the probability that you randomly select a room with 1 spider? Show your work.
   • (4 pts.) Show that the total probability of this data set is 1.
   • (2 pts.) What is the average number of spiders in a room?
   • (4 pts.) If s is the number of spiders in a room, then the number of flies in the room, f, can be described by the equation \(f(s) = 10-2s\). What is the expectation value for the number of flies in the building?
3. Wavefunctions (4pts.). Consider a quantum mechanical state that can be in one of three states, \(|1\rangle\), \(|2\rangle\), and \(|3\rangle\). State \(|1\rangle\) has an energy of a, state \(|2\rangle\) has an energy of b, and state \(|3\rangle\) has an energy of c. The quantum mechanical state being studied is described by the following wavefunction. \[|\psi\rangle = \frac{2i}{\sqrt{30}}|1\rangle -\frac{5}{\sqrt{30}}|2\rangle +\frac{1}{\sqrt{30}}|3\rangle\] If you measure the energy of state \(|\psi\rangle\), what is the most likely result? Explain your answer.
4. Linear Algebra for Quantum Mechanics (24 pts.)

• (6 pts.) Consider the two vectors \(|x\rangle\) and \(|y\rangle\) below. Show that \(\langle x | y \rangle ^* = \langle y | x \rangle\), where the \(*\) denotes the complex conjugate. Note that this is a general rule that we will be using many times throughout the course. \[|x\rangle = \begin{bmatrix}-1 \\ 2i \\ 1 \end{bmatrix}\] \[|y\rangle = \begin{bmatrix} 1 \\ 0 \\ i \end{bmatrix}\]
• (6 pts.) Given below are the three Pauli matrices which we will be using many times in this course. Show that \(\sigma_x\sigma_x = \sigma_y\sigma_y = \sigma_z\sigma_z = \textbf{I}\) (the identity matrix) and that \([\sigma_x, \sigma_y] = 2i\sigma_z\). \[\sigma_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\] \[\sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}\] \[\sigma_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\]
• (6 pts.) Given N vectors of length N, these vectors are called a basis if any vector of length N can be constructed as a weighted linear sum of the original vectors (i.e. the original vectors multiplied by scalars and added or subtracted from each other). Show that \(|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\) and \(|1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\) for a basis for all vectors of length 2.
• (6 pts.) Given \(|0\rangle\) and \(|1\rangle\) defined in the previous problem, show that \(\langle 0 | 0 \rangle = \langle 1 | 1 \rangle = 1\) and that \(\langle 0 | 1 \rangle = \langle 1 | 0 \rangle = 0\). Any basis which has these properties (\(\langle i | j \rangle = \delta_{ij}\), where \(\delta\) is the Kronecker delta) is called an orthonormal basis (it is both normalized and orthogonal).