Editing Matter wave (section)

=== Schrödinger's (matter) wave equation ===

Following up on de Broglie's ideas, physicist [[Peter Debye]] made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, [[Erwin Schrödinger]] decided to find a proper three-dimensional wave equation for the electron. He was guided by [[William Rowan Hamilton]]'s analogy between mechanics and optics (see [[Hamilton's optico-mechanical analogy]]), encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system – the trajectories of [[light rays]] become sharp tracks that obey [[Fermat's principle]], an analog of the [[principle of least action]].<ref>{{Cite book | last=Schrödinger | first=E. | year=1984 | title=Collected papers | publisher=Friedrich Vieweg und Sohn | isbn=978-3-7001-0573-2}} See the introduction to first 1926 paper.</ref>

In 1926, Schrödinger published the [[Schrödinger equation|wave equation that now bears his name]]<ref name="Schroedinger">{{Cite journal |last=Schrödinger |first=E. |date=1926 |title=An Undulatory Theory of the Mechanics of Atoms and Molecules |url=https://link.aps.org/doi/10.1103/PhysRev.28.1049 |journal=Physical Review |language=en |volume=28 |issue=6 |pages=1049–1070 |doi=10.1103/PhysRev.28.1049 |bibcode=1926PhRv...28.1049S |issn=0031-899X|url-access=subscription }}</ref> – the matter wave analogue of [[Maxwell's equations]] – and used it to derive the [[Emission spectrum|energy spectrum]] of [[hydrogen]]. Frequencies of solutions of the non-relativistic Schrödinger equation differ from de Broglie waves by the [[Compton wavelength|Compton frequency]] since the energy corresponding to the [[Invariant mass|rest mass]] of a particle is not part of the non-relativistic Schrödinger equation. The Schrödinger equation describes the time evolution of a [[wavefunction]], a function that assigns a [[complex number]] to each point in space. Schrödinger tried to interpret the [[modulus squared]] of the wavefunction as a charge density. This approach was, however, unsuccessful.<ref name=Moore1992>{{cite book | last=Moore | first=W. J. | year=1992 | title=Schrödinger: Life and Thought | publisher=[[Cambridge University Press]] | isbn=978-0-521-43767-7|pages=219–220}}</ref><ref name="jammer1974">{{cite book | last=Jammer | first=Max | author-link=Max Jammer | title=Philosophy of Quantum Mechanics: The interpretations of quantum mechanics in historical perspective | url=https://archive.org/details/philosophyofquan0000jamm | url-access=registration | year=1974 | publisher=Wiley-Interscience | isbn=978-0-471-43958-5 |pages=24–25}}</ref><ref>{{Cite journal|last=Karam|first=Ricardo|date=June 2020| title=Schrödinger's original struggles with a complex wave function|url=http://aapt.scitation.org/doi/10.1119/10.0000852| journal=[[American Journal of Physics]] | language=en| volume=88| issue=6| pages=433–438| doi=10.1119/10.0000852| bibcode=2020AmJPh..88..433K |s2cid=219513834 |issn=0002-9505| url-access=subscription}}</ref> [[Max Born]] proposed that the modulus squared of the wavefunction is instead a [[Probability density function|probability density]], a successful proposal now known as the [[Born rule]].<ref name=Moore1992/>

[[File:Guassian Dispersion.gif|180 px|thumb|right|Position space probability density of an initially Gaussian state moving in one dimension at minimally uncertain, constant momentum in free space]]
The following year, 1927, [[Charles Galton Darwin|C. G. Darwin]] (grandson of the [[Charles Darwin|famous biologist]]) explored [[Schrödinger equation|Schrödinger's equation]] in several idealized scenarios.<ref>Darwin, Charles Galton. "Free motion in the wave mechanics." Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 117.776 (1927): 258–293.</ref> For an unbound electron in free space he worked out the propagation of the wave, assuming an initial [[Wave packet#Gaussian wave packets in quantum mechanics|Gaussian wave packet]]. Darwin showed that at time <math>t</math> later the position <math>x</math> of the packet traveling at velocity <math>v</math> would be
<math display=block>x_0 + vt \pm \sqrt{\sigma^2 + (ht/2\pi\sigma m)^2}</math>
where <math>\sigma</math> is the uncertainty in the initial position. This position uncertainty creates uncertainty in velocity (the extra second term in the square root) consistent with [[Heisenberg]]'s [[uncertainty relation]] The wave packet spreads out as show in the figure.