# Homework due February 11 — MTH 496

Homework due February 11.

Exercise 1 Consider the Hamiltonian ${H(x,p)=\frac{1}{2}p^{2}+V(x)}$ on the phase space ${\mathbb{R}^{2}}$, where ${V(x)}$ is bounded from above: ${V(x) for all ${x}$ for some ${M}$. Suppose the initial momentum ${p_{0}>0}$ and that the initial position ${x_{0}}$ is such that ${H(x_{0},p_{0})\ge M}$.

1. Show that the position ${x(t)}$ satisfies $\displaystyle \frac{dx(t)}{dt}=\sqrt{2E-2V(x(t))}$

where ${E=H(x_{0},p_{0})}$.

2. Suppose ${V(x)=-\frac{1}{\left|x\right|^{\alpha}}}$ with ${\alpha>0}$. Find the trajectory ${x(t)}$ provided ${p_{0}>0}$ and ${\frac{1}{2}p_{0}^{2}+V(x_{0})=0}$.

Exercise 2 Let ${V\subset\mathbb{C}^{n}}$ be a subspace.

1. Given ${\mathbf{z}\in\mathbb{C}^{n}}$ prove that there is a unique element ${P_{V}\mathbf{z}\in V}$ such that $\displaystyle \left|\mathbf{z}-P_{V}\mathbf{z}\right|=\min\left\{ \left|\mathbf{z}-\mathbf{w}\right|\middle|\mathbf{w}\in V\right\} .$

2. Prove that ${\mathbf{z}\mapsto P_{V}\mathbf{z}}$ is a linear map and that $\displaystyle \left\langle \mathbf{z}-P_{V}\mathbf{z},\mathbf{w}\right\rangle =0$

for all ${\mathbf{w}\in V}$.

3. Prove that $\displaystyle V^{\perp}=\left\{ \mathbf{z}\in\mathbb{C}^{n}\ \middle|\ \left\langle \mathbf{z},\mathbf{w}\right\rangle =\mathbf{0}\ \text{ for all }\mathbf{w}\in V\right\} ,$

is a subspace and that $\displaystyle P_{V^{\perp}}=I-P_{V}$

where ${I}$ is the identity map.

4. Prove that ${\dim V+\dim V^{\perp}=n}$.

Exercise 3 The Pauli spin matrices are $\displaystyle \sigma_{x}=\begin{pmatrix}0 & 1\\ 1 & 0 \end{pmatrix},\quad\sigma_{y}=\begin{pmatrix}0 & -i\\ i & 0 \end{pmatrix},\quad\sigma_{z}=\begin{pmatrix}1 & 0\\ 0 & -1 \end{pmatrix}.$

1. Prove the commutation relations $\displaystyle \left[\sigma_{x},\sigma_{y}\right]=2i\sigma_{z},\quad\left[\sigma_{z},\sigma_{x}\right]=2i\sigma_{y},\quad\left[\sigma_{y},\sigma_{z}\right]=2i\sigma_{x}.$

2. Let ${\mathbf{x}\in\mathbb{R}^{3}}$ be a unit vector, ${\left|\mathbf{x}\right|=1}$. Let ${\Sigma_{\mathbf{x}}}$ denote the observable $\displaystyle \Sigma_{\mathbf{x}}=x_{1}\sigma_{x}+x_{2}\sigma_{y}+x_{3}\sigma_{z}.$

Show that the spectrum of ${\Sigma_{x}}$ is ${\left\{ -1,1\right\} }$ and find its eigenvectors.

3. Compute the commutator $\displaystyle \left[\Sigma_{\mathbf{x}},\Sigma_{\mathbf{y}}\right]$

for distinct unit vectors ${\mathbf{x},\mathbf{y}\in\mathbb{R}^{3}}$.

4. Consider the state ${\omega}$ with density matrix ${\begin{pmatrix}1 & 0\\ 0 & 0 \end{pmatrix}}$. Compute the expectation and variance of ${\Sigma_{\mathbf{x}}}$ in this state for any ${\mathbf{x}}$.