User contributions for StephanieReynolds
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19 June 2011
- 23:0423:04, 19 June 2011 diff hist +3,393 N Free Particle in Spherical Coordinates New page: A free particle is a specific case when <math>V_0=0\!</math> of the motion in a uniform potential <math>V(r)=V_0\!</math>. So it's more useful to consider a particle moving in a uniform p...
- 23:0323:03, 19 June 2011 diff hist +3,927 N General Formalism New page: A central potential does not depend on time, but rather depends only on the absolute value of the distance away from the potential's center. A central potential is rotationally invariant, ...
- 23:0123:01, 19 June 2011 diff hist +210 Template:Quantum Mechanics A No edit summary
- 22:3522:35, 19 June 2011 diff hist +483 N Problems on Angular Momentum New page: Problem 1: A plane rotator (i.e. a particle confined to move on a unit circle) is in a state with a wavefunction <math>\psi(\phi) = Asin2\phi</math> where <math>\psi</math> is the azimuth...
- 22:3422:34, 19 June 2011 diff hist +2,988 N Orbital Angular Momentum Eigenfunctions New page: [http://wiki.physics.fsu.edu/wiki/index.php/Phy5645/AngularMomentumProblem Worked Problem] about angular momentum. Now we construct our eigenfunctions of the orbital angular momentum expl...
- 13:2813:28, 19 June 2011 diff hist +4,298 N Eigenvalue Quantization New page: The motivation for exploring eigenvalue quantization comes form wanting to solve the energy eigenvalue problem. It is not possible, in general, to specify and measure more than one compon...
- 13:2713:27, 19 June 2011 diff hist +2,676 N Spherical Coordinates New page: Since angular momentum can be represented as a generator of rotations you can use the equation for an infinitesmial rotation to construct the coordinates of angular momentum in spherical c...
- 13:2713:27, 19 June 2011 diff hist +2,563 N Angular Momentum as a Generator of Rotations in 3D New page: Let us consider an infinitesimal rotation <math> \mathbf{\alpha} \!</math> directed along the axis about which the rotation takes place.We then have :<math> \mathbf{w}' = \mathbf{w} + \mat...
- 13:2613:26, 19 June 2011 diff hist +3,393 N Commutation Relations New page: Multidimensional problems entail the possibility of having rotation as a part of solution. Just like in classical mechanics where we can calculate the angular momentum using vector cross p...
- 13:2613:26, 19 June 2011 diff hist +38 N Angular Momentum New page: Introduction----------------------etc.
- 13:2513:25, 19 June 2011 diff hist −515 Template:Quantum Mechanics A No edit summary
- 13:2213:22, 19 June 2011 diff hist −44 WKB Approximation No edit summary
- 13:2113:21, 19 June 2011 diff hist +1 WKB Approximation No edit summary
- 13:2013:20, 19 June 2011 diff hist +2 WKB Approximation No edit summary
- 13:2013:20, 19 June 2011 diff hist −1 WKB Approximation No edit summary
- 13:1913:19, 19 June 2011 diff hist +12,073 N WKB Approximation New page: WKB method (Wentzel-Kramers-Brillouin method) is a technique for finding approximations to certain differential equations, including the one dimensional Schrodinger equation. It was deve...
- 13:1713:17, 19 June 2011 diff hist +7,414 N Charged Particles in an Electromagnetic Field New page: ==Gauge== Gauge theory is a type of field theory in which the Lagrangian is invariant under a certain continuous group of local transformations. Given a distribution of charges and curren...
- 13:1613:16, 19 June 2011 diff hist +2,870 N Harmonic Oscillator: Integration Over Fluctuations New page: Now, let's evaluate the path integral: <math>A=A(t)=\int_{y(0)=0}^{y(t)=0}D[y(t')]e^{\frac{i}{\hbar}\int_{0}^{t}(\frac{1}{2}my'^2-\frac{1}{2}ky^2)dt'}</math> ...
- 13:1513:15, 19 June 2011 diff hist +2,501 N Propagator for the Harmonic Oscillator New page: The classical action <math>S</math> can be evaluated as follows: <math>S=\int_{0}^{t}(KE-PE)dt </math> Where <math>KE\!</math> is the kinetic engergy and <math>PE\!</math> is the potent...
- 13:1413:14, 19 June 2011 diff hist +4,005 N The Free-Particle Propagator New page: Although our heuristic analysis yielded an exact free-particle propagator, we will now repeat the calculation without any approximation to illustrate the partial integration. Consider <mat...
- 13:1313:13, 19 June 2011 diff hist −494 Template:Quantum Mechanics A No edit summary
- 13:1113:11, 19 June 2011 diff hist +368 N Feynman Path Integral Evaluation of the Propagator New page: The propagator for harmonic oscillator can be evaluated as follows: <math><x|\hat{U}(t,0)|x_0>=e^{\frac{i}{\hbar}S}\int_{y(0)=0}^{y(t)=0}D[y(t')]e^{\frac{i}{\hbar}\int_{0}^{t}(\frac{1}{2...
- 13:1013:10, 19 June 2011 diff hist +3,905 N Coherent States New page: The general states of an harmonic oscillator can be expressed as a superpostion of the energy eigenstates <math>|n\rangle\!</math>. A class of states that is of particular importance consi...
- 13:1013:10, 19 June 2011 diff hist +3,180 N Analytical Method for Solving the Simple Harmonic Oscillator New page: In contrast to the elegant method described above to solve the harmonic oscillator, there is another "brute force" method to find out the eigenvalues and eigenfunctions. This method uses e...
- 13:0913:09, 19 June 2011 diff hist +7,814 N Harmonic Oscillator Spectrum and Eigenstates New page: thumb|650px| 1-D harmonic oscillator is a particle moving under the harmonic oscillator potential with the form: <math>V(x)=\frac{1}{2}k x^2</math> where <m...
- 10:4310:43, 19 June 2011 diff hist +216 Template:PhysicsNavigation No edit summary
- 10:4110:41, 19 June 2011 diff hist −18 Template:PhysicsNavigation No edit summary
- 10:3910:39, 19 June 2011 diff hist +73 Template:Quantum Mechanics A No edit summary
- 10:3710:37, 19 June 2011 diff hist +22 N Discrete Eigenvalues and Bound States - The Harmonic Oscillator and the WKB Approximation New page: Introduction------etc.
- 10:3610:36, 19 June 2011 diff hist +297 N Problems New page: 4.1) Prove that there is a unitary operator <math>\tilde{U}(a)</math>, which is a function of <math>\hat p =\frac{\hbar}{i}\frac{d}{dx}</math>, such that for some wavefunction <math>\psi(x...
- 10:3610:36, 19 June 2011 diff hist +4,641 N Feynman Path Integrals New page: thumb|650px| The path integral formulation was developed in 1948 by Richard Feynman. The path integral formulation of quantum mechanics is a description of q...
- 10:3510:35, 19 June 2011 diff hist +938 N The Virial Theorem New page: Consider <math> \begin{align} &\frac{d}{dt}<xp>\\ &=\frac{1}{i\hbar}<[xp,H]> \\ &=\frac{ 2<p^{2}> }{2m}+\frac{1}{i\hbar}<xpV-xVp>\\ &=\frac{2<p^{2}>}{2m}+ \frac{1}{i\hbar}\int_{-\infty}...
- 10:3510:35, 19 June 2011 diff hist +1,842 N The Interaction Picture New page: The interaction picture (or Dirac picture) is a hybrid between the Schrödinger and Heisenberg pictures. In this picture both the operators and the state kets are time dependent. The tim...
- 10:3410:34, 19 June 2011 diff hist +6,264 N The Heisenberg Picture: Equations of Motion for Operators New page: There are several ways to mathematically approach the change of a quantum mechanical system with time. The Schrodinger picture considers wave functions which change with time while the Hei...
- 10:3310:33, 19 June 2011 diff hist +16,714 N Time Evolution of Expectation Values and Ehrenfest's Theorem New page: It is reasonable to expect the motion of a wave packet to agree with the motion of the corresponding classical particle whenever the potential energy changes by a negligible amount over th...
- 10:3010:30, 19 June 2011 diff hist +2,686 N Transformations of Operators and Symmetry New page: Symmetry of any quantum mechanical state is determined by how the state transforms under certain mathematical transformations, examples being translation and rotation. A symmetry transform...
- 10:2610:26, 19 June 2011 diff hist +11,907 N Commutation Relations and Simultaneous Eigenvalues New page: ==Commutator=== The commutator of two operators A and B is defined as follows: <math>[A,B]=AB-BA\,\!.</math> When 2 operators <math>A</math> and <math>B</math> commute, then <math>\left...
- 10:2410:24, 19 June 2011 diff hist +13,583 N Linear Vector Spaces and Operators New page: Quantum Mechanics can be conveniently formulated in the language of abstract state vectors, from which the various representations (wave mechanics, matrix mechanics, Schrodinger, Heisenber...
- 10:2110:21, 19 June 2011 diff hist +206 Template:Quantum Mechanics A No edit summary
- 10:1810:18, 19 June 2011 diff hist +24 N Operators, Eigenfunctions, and Symmetry New page: Introduction--------etc.
- 10:1610:16, 19 June 2011 diff hist +1,439 N Summary of One-Dimensional Systems New page: '''1. ''Zero'' Potential''' Physical Example: Proton in beam from cyclotron Significant Feature: Results used for other systems '''2. ''Step'' Potential (energy below top)''' Physical ...
- 10:1510:15, 19 June 2011 diff hist +9,145 N Motion in a Periodic Potential New page: An example of a periodic potential is given in Figure 1 which consists of a series of continuous repeating form of potentials. In other words, the potential is translational symmetric over...
- 10:1410:14, 19 June 2011 diff hist +10,533 N Scattering States, Transmission and Reflection New page: ''' The step potential ''' Let's consider one dimensional potential step with an energy <math> E > V_0 \!</math>. That is, we have a potential :<math> V(x) = \begin{cases} 0, & x < 0, ...
- 10:1310:13, 19 June 2011 diff hist +1,791 N Oscillation Theorem New page: Let us concentrate on the bound states of a set of wavefunctions. Let <math> \frac{\hbar^2}{2m}=1</math>. Let <math>\psi_1\!</math> be an eigenstate with energy <math>E_1\!</math> and <mat...
- 10:1210:12, 19 June 2011 diff hist +711 N Scattering States New page: The scattering states are those not bound, where the energy spectrum is a continuous band. Unlike the bound case, the wave-function does not have to vanish at infinity, though a particle ...
- 10:1110:11, 19 June 2011 diff hist +3,647 N The Dirac Delta Function Potential New page: A delta potential, eg. <math>V_0\delta(x-a)\!</math>, is a special case of the finite square well, where the width of the well goes to zero and the depth of the well goes to infinity, whil...
- 10:1010:10, 19 June 2011 diff hist +1,945 One-Dimensional Bound States No edit summary
- 10:0810:08, 19 June 2011 diff hist −1,938 One-Dimensional Bound States No edit summary
- 10:0710:07, 19 June 2011 diff hist +10,241 N One-Dimensional Bound States New page: When the energy of a particle is less than the potential at both positive and negative infinity, the particle is '''trapped''' in a well and it goes back and forth between the turning poin...
- 10:0510:05, 19 June 2011 diff hist +2,036 N Motion in One Dimension New page: We study the one dimensional problems in quantum theory, not only because the interest of study the simplest cases to learn about the general properties. Actually there are many cases in t...
