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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.

How was Schrodinger equation perceived pre-Born

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. But it is false for the fundamental kind of Schrodinger equation basic to quantum mechanics. Schrodinger seems to have initially guessed that electron waves would stay little blobs in space as they move, so that they stay particles. Ab initio quantum chemistry methods.

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In its modern meaning ('from first principles of quantum mechanics') the term was used by Chen[5] (when quoting an unpublished 1955 MIT report by Allen and Nesbet), by Roothaan[6] and, in the title of an article, by Allen and Karo,[7] who also clearly define it.

Ab initio quantum chemistry methods

Accuracy and scaling[edit] Density functional theory. DFT has been very popular for calculations in solid-state physics since the 1970s.

Density functional theory

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. Computational costs are relatively low when compared to traditional methods, such as Hartree–Fock theory and its descendants based on the complex many-electron wavefunction. Overview of method[edit] Although density functional theory has its conceptual roots in the Thomas–Fermi model, DFT was put on a firm theoretical footing by the two Hohenberg–Kohn theorems (H–K).[9] The original H–K theorems held only for non-degenerate ground states in the absence of a magnetic field, although they have since been generalized to encompass these.[10][11] where, for the and . . . .

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 (physics)

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.

Fermi hole

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: However, only the first two are symmetric or anti-symmetric to electron exchange (which corresponds to exchanging 1 and 2). 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.

Fermi holes and Fermi heaps, Fall 2002, CH352 Physical Chemistry

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. Hilbert space. The state of a vibrating string can be modeled as a point in a Hilbert space.

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. Ethan Hein's answer to Particle Physics: Why don't electrons crash into the nucleus. 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.

Ethan Hein's answer to Particle Physics: Why don't electrons crash into the nucleus

It’s an attractive and iconic image. It makes a nice logo. Unfortunately, it’s also totally wrong. There’s an extent to which subatomic particles are like little marbles, but it’s a limited extent. The problem with textbook images like the one above is that they mislead you into thinking of particles as “things.” Protons and electrons pull on each other the way refrigerators and magnets do. You can get a good idea of how particles really behave by looking at television static, which consists of huge numbers of electrons being fired at the screen at random.