The Scientific – Problems & Solutions Blog

Quantum Information

Posted by Oscar Riveros on April 15, 2012 – 2:52 am

Filed under Hilbert Spaces, Quantum Information

Given the orthonormal basis

$\vert\psi_{1}>=(\begin{array}{c} e^{i\phi}\cos\theta\\ \sin\theta \end{array}),\vert\psi_{2}>=(\begin{array}{c} -\sin\theta\\ e^{-i\phi}\cos\theta \end{array})$

in the Hilbert space $C^{2}$ . Use this basis to find a basis in $C^{4}$ .

Classical Electrodynamics

Posted by Oscar Riveros on April 13, 2012 – 2:19 pm

Filed under Classical Electrodynamics

The time-averaged potential of a neutral hydrogen atom is given by

$\Phi=\frac{q}{4\pi\epsilon_{0}}\frac{e^{-\alpha r}}{r}(1+\frac{\alpha r}{2})$

Where $q$ is the magnitude of the electronic charge, and $\alpha^{-1}=\frac{a_{0}}{2}$ , $a_{0}$ being the Bohr radius. Find the distribution of charge (both continuous and discrete) that will give this potential and interpret your result physically.

Classical Mechanics

Posted by Oscar Riveros on April 13, 2012 – 12:33 am

Filed under Classical Mechanics, Lax Representations

Let

$\frac{d\mathbf{u}}{dt}=\mathbf{V}(\mathbf{u}),u\equiv(u_{1},u_{2},...,u_{m})^{^{T}}$

Be an autonomous system of first-order ordinary differential equations. Assume that the functions $V_{k}:\mathbb{R}^{m}\rightarrow\mathbb{R}$ are smooth. Assume that this system can be written in the form (the so-called $Lax\, representation$ )

$\frac{d\mathbf{L}}{dt}=[A,L](t)\equiv[A(t),L(t)]$

where $A$ and $L$ are $n\times n$ matrices and $[A,L]\equiv AL-LA$ . The $n\times n$ matrices $L$ and $A$ are called a $\mathit{Lax\,}pair$ .

(i) Show that

$\frac{dL^{k}}{dt}=[A,L^{k}](t)$

(ii) Show that $tr(L^{k})(k=1,2,...)$ are first integrals, where $tr(\cdot)$ denotes the trace.

(iii) Assume that $L^{-1}$ exists. Show that $tr(L^{-1})$ is first integral.

(iv) Show that the solution of the matrix differential Eq. $\frac{d\mathbf{L}}{dt}=[A,L](t)\equiv[A(t),L(t)]$ is given by $L(t)=e^{t}Le^{-tA}$ (where $L=L(t=0)$ ).

Mathematical Physics

Posted by Oscar Riveros on April 12, 2012 – 10:50 pm

Filed under Mathematical Physics, Vector Calculus

If $\phi=\frac{1}{r}$ , where $r=(x^{2}+y^{2}+z^{2})^{^{\frac{1}{2}}}$ , show that $\nabla\phi=\frac{r}{r^{3}}$ .

Category Theory

Posted by Oscar Riveros on April 12, 2012 – 2:42 pm

Filed under Category Theory

Let $S$ and $T$ be sets, and let $Hom(S,T)$ denote the set of all functions with domain $S$ and codomain $T$ . Let $f:T\rightarrow V$ be a function; define the function