Quantum mechanics

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How was Schrodinger equation perceived pre-Born Schrodinger at first believed it was the density of a wave associated with the electron, and considered it like a classical wave equation, describing the motion of a wave in space. He considered the square of psi to be the electron density, and the phase to be the electron field momentum. This is the reason he gives a local expression for the momentum density in the field, what is now called the probability current in quantum mechanics books, so that the square of psi is conserved with a current, not just conserved globally. What makes it interesting is that Schrodinger's interpretation is completely correct for a different kind of Schrodinger equation, the classical Schrodinger equation, the kind that describes Bose-Einstein condensates. How was Schrodinger equation perceived pre-Born
Ab initio quantum chemistry methods are computational chemistry methods based on quantum chemistry.[1] The term ab initio was first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene.[2][3] The background is described by Parr.[4] In its modern meaning ('from first principles of quantum mechanics') the term was used by Chen[5] (when quoting an unpublished 1955 MIT report by Allen and Nesbet), by Roothaan[6] and, in the title of an article, by Allen and Karo,[7] who also clearly define it. Almost always the basis set (which is usually built from the LCAO ansatz) used to solve the Schrödinger equation is not complete, and does not span the Hilbert space associated with ionization and scattering processes (see continuous spectrum for more details). Ab initio quantum chemistry methods Ab initio quantum chemistry methods
DFT has been very popular for calculations in solid-state physics since the 1970s. However, DFT was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions. In many cases the results of DFT calculations for solid-state systems agree quite satisfactorily with experimental data. Density functional theory Density functional theory
Spin (physics) In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.[1][2] Spin is a solely quantum-mechanical phenomenon; it does not have a counterpart in classical mechanics (despite the term spin being reminiscent of classical phenomena such as a planet spinning on its axis).[2] Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. Orbital angular momentum is the quantum-mechanical counterpart to the classical notion of angular momentum: it arises when a particle executes a rotating or twisting trajectory (such as when an electron orbits a nucleus).[3][4] The existence of spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which particles are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone.[5] Spin (physics)
Fermi hole Fermi hole Neglecting the spin-orbit interaction, the wavefunction for the two electrons can be written as , where we have split the wavefunction into spatial and spin parts. As mentioned above, needs to be antisymmetric, and so the antisymmetry can arise either from the spin part or the spatial part. There are 4 possible spin states for this system:
Fermi holes and Fermi heaps, Fall 2002, CH352 Physical Chemistry You may have learned the "rule" that no more than two electrons can be in the same orbital. If you have, you may also have puzzled about why such a rule is so. If you have decided, like many people who have been presented with just the rule without any explanation, that it has to do with electrical repulsion—that it reflects the electrons repelling one another due to their electrical charge—then you are in for a neat surprise. The "rule" instead traces to a deep algebraic property of nature that has nothing whatsoever to do with the charge on electrons! Perhaps you, like me, will find it fascinating that such a crucial aspect of the world has such a subtle origin. The essence is that many-electron wavefunctions must change sign when the labels on any two electrons are interchanged. Fermi holes and Fermi heaps, Fall 2002, CH352 Physical Chemistry
A vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer)—and ergodic theory, which forms the mathematical underpinning of thermodynamics. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. Hilbert space Hilbert space
In high school science class, you probably saw a picture of an atom that looked like this: The picture shows a stylized nucleus with red protons and blue neutrons, surrounded by three grey electrons. It’s an attractive and iconic image. Ethan Hein's answer to Particle Physics: Why don't electrons crash into the nucleus Ethan Hein's answer to Particle Physics: Why don't electrons crash into the nucleus