Homework due February 11 — MTH 496

Homework due February 11.

A PDF version of these exercises is available here.

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)<M} 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}}.

2 thoughts on “Homework due February 11 — MTH 496

  1. Pingback: Corrections to HW 1 — MTH 496 | Jeffrey Schenker

  2. Pingback: Additional correction to HW1 — MTH 496 | Jeffrey Schenker

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