How can I visualize a complex wave function of quantum mechanics?
484329
see this link\nhttp://www.colorado.edu/physics/2000/quantumzone/index.html
46678
The most popular atomic geometries are Linear, Trigonal Planar, Octahedral and Tetrahedra.\n\nI've added a link to a visualization program, though I couldn't get the plug-in to work.
693063
Here are some good topics for your Physics Project:\n\nI. Algebraic Quantum Field Theory \nII. Quantum Information Theory \nIII. Phase, the Operator Conjugate to the Number Operator \nIV. IAMP-approved problems \nV. Skyrmions \nVI. Yang-Mills-Shaw Theory \n\nRefer to this website for more info:\n\nhttp://www.mth.kcl.ac.uk/~streater/regainedcauses.html
376295
For those who understand mathematics, and dare to imagine beyond the usual, quantum mechanics could easily be the most beautiful branch of physics.\n\nQuantum mechanics is a fundamental branch of theoretical physics that replaces Newtonian mechanics and classical electromagnetism at the atomic and subatomic levels. It is the underlying framework of many fields of physics and chemistry, including condensed matter physics, quantum chemistry, and particle physics. Along with general relativity, it is one of the pillars of modern physics.\n\nQuantum mechanics was developed in the first part of the 20th century and was highly successful in explaining the behavior of atoms, molecules and nuclei. Developed between 1900 and 1930 and later combined with the general and special theory of relativity, it revolutionized the concepts of physics. The new concepts, which were the particle properties of radiation, the wave properties of matter, quantization of physical properties and the idea that one can no longer know exactly where a single particle such as an electron is at any one instance were necessary to explain all of the new experimental evidence that was available at the time. For example, quantum mechanics explains the behavior of semiconductors which are used to make the myriad of devices we use every day.\n\nFollowing are the important contributors to the foundation of quantum mechanics and the principles they uncovered.\n\nYear  Researcher    Quantum Mechanics Concept\n1901  Planck        Blackbody radiation\n1905  Einstein      Photoelectric effect\n1913  Bohr          Quantum theory of spectra\n1922  Compton       Scattering of photons off electrons\n1924  Pauli         Exclusion principle\n1925  de Broglie    Matter waves\n1926  Schroedinger  Wave equation\n1927  Heisenberg    Uncertainty principle\n1927  Davison and Germer  Wave properties of electrons\n1927  Born          Interpretation of the wavefunction\n\n[The answer given by the first user (theswarm666x) is foolish. Although, one gives such examples in order to explain students of quantum mechanics what is actually meant by the terms like 'tunneling', you could not and should not use quantum mechanics in such a situation. Quantum mechanics talks about the wave aspect of objects (which is significant only for very small bodies) and waves can travel through walls.]
112900
you must use it when you looking for free particle in quantum mechanic to normalize wave function
240384
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
914601
http://mathworld.wolfram.com/PoincareConjecture.html
1069671
yep...scalars multiply vectors component by component just like you have shown.  you can also show it by plotting out the original vectors, scaling them appropriately, adding them, and finding the resukltant components.
86731
The splitting and shift of a spectral line into several components in the presence of an electric field.
1015775
I think a high school student needs to be familiar with the traditional display. But if you want to emphasize certain properties it's fun to rearrage things. I always preferred the 3D cylidrical display with the Lanthanides and Actinides looping out from a single spot - makes the inner electron orbitals visually obvious. Some of the animated table are interesting but they never show the proper orbital shapes. It all depends on what you want to emphasize - I think using a variety will show how chemistry and physics are intertwined.
729152
Try  this site it looks good!\n\nhttp://www.dositey.com/mathk2.htm
578062
use the properties of equalities and inequalities.\n(subraction,additon,multiplication,division).
1067262
This is covered in most quantum mechanics textbooks. I'm looking at the appendix to Albert Messiah's Quantum Mechanics Vol I. The wavefunction for n=2,l=0 is:\n\n(1/(a^3/2)) * 1/sqrt(8pi) * (1-r/2a) * exp(-r/2a)\n\nwhere a is the Bohr radius. Let's use units in which a=1. The probability density is:\n\nP(x) = 1/(8pi) * (1-r/2)^2 * exp(-r)\n\nThe radial probability density is the above times the area of the sphere:\n\nR(r) = 4pi * r^2 * P(x) = 1/2 * r^2(1-r/2)^2 * exp(-r)\n\nIf you want the radius of the location in space at which P(x) is maximal solve:\n\ndP(r)/dr = 0 (a quadratic equation)\n\nIf you want the radius at which the radial probability density is maximal solve:\n\ndR(r)/dr = 0 (a cubic equation)\n\nR(r) has two relative maxima. I think the one farther out is bigger. The expectation value of the radius <r> is:\n\n<r> = &#92;int_0^oo R(r) r dr \n\nstraightforward using &#92;int_0^oo r^n e^(-r) dr = n! etc.. The answer <r> should be 6a.
558518
The problem is that there are not two wavefunctions for entangled particles. There is only one, but that one wave function describes *both* of the particles. So the observation collapses that one wave function as fixes the state of the system. This is one difficulty of the 'particle' view of things. There is too much of a tendency to think of a two particle state as two one particle states. This is wrong.
8585
Hi,\n\nMagnetic Quantum Number "m" in relation to Azimuthal Quantum Number "l" can be depicted by- \n\nm = 2l + 1\n\nSo, for any subshell (defined by the value of L), there are (2L + 1) values of M possible given by-\n\nM = -L, -(L-1), ......, 0, ....., (L-1), L\n\nSo, from your given options, I think the second one- -L, 0, L  is the closest.
834583
You can use the Rydberg formula, which relates the reciprocal of the wavelength of the photon absorbed or emitted to the squares of the reciprocals of the principal quantum numbers.\n\n1/wavelength = R(1/n_l^2 - 1/n_u^2)\n\nSo, for the listed values of n, \n\nA.  1/w = R(1/ 2^2 - 1/4^2) = 0.1875R\nB.  1/w = -0.75R\nC.  1/w = 0.21R\nD.  1/w = -0.04861R\n\nSince energy and frequency are proportional to 1/wavelength,\nlooks like the order is D, A, C, B.
745993
There is a great animation that explains the experiments in detail (with pictures) on the Mendel site at http://www.mendel-museum.org/eng/1online/experiment.htm