diff options
| -rw-r--r-- | MIT Courses | 9 | ||||
| -rw-r--r-- | physics/quantum/fermigas | 51 | ||||
| -rw-r--r-- | physics/units | 53 | ||||
| -rw-r--r-- | software/freebsd-tricks | 5 | ||||
| -rw-r--r-- | software/gimp | 19 | ||||
| -rw-r--r-- | software/scheme | 6 |
6 files changed, 138 insertions, 5 deletions
diff --git a/MIT Courses b/MIT Courses index 31ebaf1..4c70d7f 100644 --- a/MIT Courses +++ b/MIT Courses @@ -45,8 +45,8 @@ humanities concentration in philosophy. * Spring 2008
+ **8.04** - Quantum Mechanics I (last time!)
+ **8.14** - Experimental Physics II
- + **21F.052** - French Film Classics (spring, in english)
- + **18.100b** - Analysis
+ + **21F.052** - French Film Classics
+ + **18.100b** - Analysis I
* Fall 2008
+ **8.05** - Quantum Mechanics II
@@ -56,10 +56,8 @@ humanities concentration in philosophy. * Spring 2009 (planned)
+ 8.06 - Quantum Mechanics III
- + 8.962 - General Relativity
+ 6.945 - Adventures in Symbolic Programming
+ 4.602 - Modern Art and Mass Culture
- + 8.ThU - Thesis
Other interesting courses:
+ 24.215 - Topics in the Philosophy of Science (fall)
@@ -81,5 +79,6 @@ Other interesting courses: + 8.371J - Quantum Information Science (grad, spring)
+ 8.594J - Introduction to Neural Networks (grad, fall)
+ 8.284 - Modern Astrophysics (spring)
- + 8.942 - Cosmology (grad, spring)
+ + 8.942 - Cosmology (grad, fall)
+ + 8.962 - General Relativity (grad, spring)
diff --git a/physics/quantum/fermigas b/physics/quantum/fermigas new file mode 100644 index 0000000..0114b43 --- /dev/null +++ b/physics/quantum/fermigas @@ -0,0 +1,51 @@ +=============== +Fermi Gas +=============== + +Derivation of the Fermi Energy +--------------------------------- +Consider a crystal lattice with an electron gas as a 3 dimensional infinite +square well with dimensions :m:`$l_{x}, l_{y}, l_z$`. The wavefunctions of +individual fermions (pretending they are non-interacting) can be seperated +as :m:`$\psi(x,y)=\psi_{x}(x)\psi_{y}(y)\psi_{z}(z)$`. The solutions will be +the usual ones to the Schrodinger equation: + +:m:`$$\frac{-\hbar^2}{2m}\frac{d^2 \psi_x}{dx}=E_x \psi_x$$` + +with the usual wave numbers :m:`$k_x=\frac{\sqrt{2mE_x}}{\hbar}$`, and quantum +numbers satisfying the boundry conditions :m:`$k_x l_x = n_x \pi$`. The full +wavefunction for each particle will be: + +:m:`$$\psi_{n_{x}n_{y}n_{z}}(x,y,z)=\sqrt{\frac{4}{l_{x}l_{y}}}\sin\left(\frac{n_{x}\pi}{l_{x}}x\right)\sin\left(\frac{n_{y}\pi}{l_{y}}y\right)\sin\left(\frac{n_{z}\pi}{l_{z}}z\right)$$` + +and the associated energies (with :m:`$E = E_x + E_y + E_z$`): + +:m:`$$E_{n_{x}n_{y}n_z}=\frac{\hbar^{2}\pi^{2}}{2m}\left(\frac{n_{x}^{2}}{l_{x}^{2}}+\frac{n_{y}^{2}}{l_{y}^{2}}+\frac{n_{z}^{2}}{l_{z}^{2}}\right)=\frac{\hbar^2|\vec{k}|^2}{2m}$$` + +where :m:`$|\vec{k}|^2$` is the magnitude of the particle's k-vector in k-space. +This k-space can be imagined as a grid of blocks, each representing a possible +particle state (with a double degeneracy for spin). Positions on this grid have +coordinates :m:`$(k_{x},k_{y},k_z)$` corresponding to the positive integer +quantum numbers. These blocks will be filled +from the lowest energy upwards: for large numbers of occupying particles, +the filling pattern can be approximated as an expanding spherical shell with +radius :m:`$|\vec{k_F}|^2$`. + +Note that we're "over counting" the number of occupied states because the +"sides" of the quarter sphere in k-space (where one of the associated quantum +numbers is zero) do not represent valid states. These surfaces can be ignored +for very large N because the surface area to volume ratio is so low, but the +correction can be important. There will then be a second correction due to +removing the states along the individual axes twice (once for each +side-surface), u.s.w. + +The surface of this shell is called the Fermi surface and represents the most +excited states in the gas. The radius can be derived by calculating the total +volume enclosed: each block has volume :m:`$\frac{\pi^3}{l_x l_y +l_z}=\frac{\pi^3}{V}$` and there are N/2 blocks occupied by N fermions, so: + +:m:`$$\frac{1}{8}(\frac{4\pi}{3} |k_{F}|^{3})&=&\frac{Nq}{2}(\frac{\pi^{3}}{V})\\|k_{F}|&=&\sqrt{\frac{3Nq\pi^2}{V}}^3=\sqrt{3\pi^2\rho}^3$$` + +:m:`$\rho$` is the "free fermion density". The corresponding energy is: + +:m:`$$E_{F}=\frac{\hbar^{2}}{2m}|k_{F}|^{2}=\frac{\hbar^{2}}{2m}(3\rho \pi)^{2/3}$$` diff --git a/physics/units b/physics/units new file mode 100644 index 0000000..b1968f4 --- /dev/null +++ b/physics/units @@ -0,0 +1,53 @@ +====================== +Units +====================== + +.. contents:: + +SI Units +-------------------- +The SI system uses meters-kilograms-seconds. It also defines the Coulomb as +a unit for measuring electric charge, which introduces redundant conversions +between mass-length-time units and the electric charge. + +cgs Units +-------------------- +The cgs system uses centimeters-grams-seconds, and also defines electric charge +in terms of the fundamental quantities of mass, length, and time. The unit of +charge is "esu" or electrostatic unit. + +Natural Units +-------------------- +Natural units are a system of units which replace (or re-scale) the usual mass, +length, and time bases with quantities which have "natural" (physical) +constants associated with them. The two constants usually chosen are the speed +of light (c) and Plank's constant (:m:`$\hbar$`); the gravitational constant +(G) is a possibility for the third constant/unit, but energy (in +electron-volts: eV) is often used instead because it gives more useful +relations and because there is no accepted theory of quantum gravity to unite +these three constants. See _`Plank Units` for more on using G as a unit. + +Working with natural units simplifies physical relations and equations because +many conversion factors drop out. + +Given the relations between cgs units (gm, cm, sec) and natural units (c, +:m:`$\hbar$` , eV), we can find the natural units of an arbitrary quantity +:m:`$[Q]=[gm]^{a}[cm]^{b}[sec]^{c}=[c]^{\alpha}[\hbar]^{\beta}[eV]^{\gamma}$`: + +:m:`$$(\alpha,\beta,\gamma)=\left(\begin{array}{ccc} -2 & 1 & 0\\ 0 & 1 & 1\\ 1 & -1 & -1\end{array}\right)\left(\begin{array}{c} a\\ b\\ c\end{array}\right)=(-2a+b,b+c,a-b-c)$$` + +or in reverse: + +:m:`$$(a,b,c)=\left(\begin{array}{ccc} 0 & 1 & 1\\ 1 & 2 & 2\\ -1 & -1 & -2\end{array}\right)\left(\begin{array}{c} \alpha\\ \beta\\ \gamma\end{array}\right)=(\beta+\gamma,\alpha+2\beta+\gamma,-\alpha-\beta-2\gamma)$$` + +Plank Units +---------------- +Plank units (defined by Plank soon after defining his constant :m:`$\hbar$`) are a version of _`Natural Units` using the gravitational constant G as the the +third unit (instead of the common measure of energy). When converted back into +mass-length-time units we get three quantities which define the "Plank Scale", +which may provide estimation of the domain where quantum gravity effects become +important (similar to how the speed of light and Plank's constant provide +estimation of when special relativistic and quantum mechanical effects become +important). + + diff --git a/software/freebsd-tricks b/software/freebsd-tricks index 84010f6..0344abd 100644 --- a/software/freebsd-tricks +++ b/software/freebsd-tricks @@ -7,3 +7,8 @@ Burn an ISO image To burn a CD-R, use ``burncd`` like so:: $ burncd -f /dev/acd0 data FILENAME.iso fixate + +Network Tools +---------------------- + +``iftop`` is great. diff --git a/software/gimp b/software/gimp new file mode 100644 index 0000000..97d7393 --- /dev/null +++ b/software/gimp @@ -0,0 +1,19 @@ +====================== +The Gimp +====================== + +How to Sharpen Photos +---------------------------- +First decompose the image into HSV: Colors/Compose.../Decompose. Select +"HSV" from the list and "Decompose to Layers". This will make a new image +(close the original). + +Select the "Value" layer for editing (so that colors aren't affected). The +"Unsharped Mask" filter is under Filters/Enhance. + +Radius is how many pixels to select, amount is the strength of the filter, +and threshold allows parts of the image to be ignored. + +After applying the filter go back to Colors/Compose.../Compose and select +HSV again to regenerate the color image. + diff --git a/software/scheme b/software/scheme index c64d000..dda68f1 100644 --- a/software/scheme +++ b/software/scheme @@ -77,3 +77,9 @@ image can be specified with the ``--band`` option at runtime or with ``(disk-restore filename)`` from within the interpreter. Bands are also called worlds. +"First Class" +------------------ +"Procedures as first class objects" is one of the features commonly attributed +to scheme. What does that mean? SICP describes first class objects as those +that can be: named by variables, passed as arguments, returned as results, +and included in data structures. |