Abstract
The Schrödinger equation, first obtained in 19261, was an extension of de Broglie’s hypothesis, proposed two years earlier,2 that each material particle has associated with it a wavelength λ related to the linear momentum p of the particle by the equation
where h is Planck’ s constant. Since any sinusoidally varying wave motion of amplitude ψ and λ wavelength X satisfies the differential equation*
matter waves would obey the equation
In particular, a particle of mast m with no forces acting on it has energy EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWGfbGaeyypa0ZaaSaaaeaacaaIXaaa % baGaaGOmaaaacaWGTbGaamODamaaCaaaleqabaGaaGOmaaaakiabg2 % da9maalyaabaGaamiCamaaCaaaleqabaGaaGOmaaaaaOqaaiaaikda % caWGTbaaaaaa!4A08!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$E = \frac{1}{2}m{v^2} = {{{p^2}} \mathord{\left/ {\vphantom {{{p^2}} {2m}}} \right. \kern-\nulldelimiterspace} {2m}}$$, or p 2 = 2m(E-V). Equation (3) may thus be written
This is the Schrödinger equation for a free particle. Of greater importance is the equation for a bound particle for which the binding force can be related to a potential energy V. In this case, the total energy of the particle is equal to the sum of the kinetic and potential energies, i.e., E = p 2 /2m + V, or p 2 = 2m (E−V). Equation (3) thus becomes
This is known as the time-independent Schr??dinger equation. Its great utility lies in the fact that it enables one to calculate energy levels, or eigenvalues, of the energy E (see QUANTUM THEORY). The remarkable agreement of the results of the equation with experimental fact
has led to its being regarded as one of the fundamental equations of physics.
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General
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On the Forces Which Hold Matter Together
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On Thermodynamics
Pippard, A, A., “Elements of Classical Thermodynamics,” Cambridge, Cambridge Univ. Press, 1957.
Porter, A. W., “Thermodynamics,” London, Methuen, and New York, John Wiley and Sons, 1951.
Loeb, Leonard B., “Static Electrification,” Berlin, Goettingen, and Heidelberg, Springer-Verlag, 1958. Covers solids, liquids, and gases.
Klinkenberg, A., and Van der Minne, J. L., “Electrostatics in the Petroleum Industry,” Amsterdam and New York, Elsevier Publishing Company, 1958. An excellent coverage of liquid-solid contacts.
Montgomery, D. J., “Static Electrification of Solids,” Solid State Physics 9, 139 (1959). An introduction into solid-solid contacts.
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Fowles, G.R. et al. (1990). S. In: Besançon, R.M. (eds) The Encyclopedia of Physics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-6902-2_18
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