# 1.5A: Wave Mechanics of Electrons

- Page ID
- 2615

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)All matter has intrinsic wave properties. These are described mathematically by the Schrödinger Equations and it's solutions. The wavenature of electrons and other fundamental principles (eg charge and momentum) together produce the wave mechanics of electron. The effects of electron wave mechanics are far reaching, responsible for such phenomena as electricity, emission and absorption, and bonding and hybridization.

## Background

### Wave-Nature of Matter

Accurate explanations of atomic natural physical chemical phenomena are dependent on energy quantization. The realization of this fundamental characteristic of matter was developed through treatment of a couple well known experiments, notably Max Planck's explanation of black body radiation and Einstein's explanation of the photoelectric effect. The conclusions of energy quantization were consolidated by Louis De Broglie as

\[\Lambda = \dfrac{h}{p} \nonumber \]

and, by rearrangement:

\[f= \dfrac{E}{h} \nonumber \]

### On Waves

Quantum mechanically, an electron can be described by a wave function oscillating in space and time that has mean values equal to the expectation values of observables corresponding to given operators. According to the Born interpretation of quantum mechanics, the complex conjugate of this wavefunction correlates to the electron's positional probability density.

Electrons are fermions. They are charged particles. When they are confined by a potential to a limited space they display harmonics analogous to those of other wavelike phenomena. This occurs most notable in atoms and molecules. The hydrogen atom proves the most simple atomic example. The three dimensional harmonics of an electron bound within the potential energy well of a proton results in what are conventionally called orbitals. Orbitals are commonly depicted as contours of some percent of the complex conjugate of the electron's approximate wave function, though realistically without external potentials they diffuse infinitely.

A bound electron occupies higher harmonics of the bound state with increasing energy. Energy can only be increased in specific quanta as demanded for the wave function to exist. The discrete energy levels of higher harmonics correspond to higher orbitals.

Electrons can gain energy to exist in a higher orbital. When this process occurs via interaction with electromagnetic radiation, it is referred to as absorption. Similarly, the regression of an electron into a lower energy orbital results in the release of electromagnetic radiation, and is referred to as emission. Because of the quantized energy levels demanded by a bound system, electrons in a molecule or atom can only absorb or emit light at specific frequencies, which depend on the properties of the system.

Certain materials have energy level spacing such that excitation by an energy source can produce a greater number of electrons in an excited state than in the ground state. This is known as population inversion. When this happens for a transition which releases light upon relaxation, light of a specific nature is produced that has great practical importance. This light is monochromatic, and can be channeled back and forth through the medium (gain medium) and allowed only to disperse through a very narrow slit to produce monochromatic, directional, coherent light source. The apparatus is called a LASER, which is an acronym for light amplification by stimulated emission of radiation.

The inherent charge of electron incites movement of the particle in accordance to the forces described by coulomb's law. Rotational motion of a charged particle produces an electric field. The potential of an electron attraction to positive charge can be used to store energy in chemical form. The motion of electrons by batteries or other sources through conductive media, such as copper wire can be facilitated to do work. Computer. Light.

Electronic absorption and emission within roughly 350 to 750 nm produces radiation that is in the visible spectrum. the sky is blue because of the interaction of light from the sun with electrons in atoms in the atmosphere in what's known as scattering. Scattering of this type occurs to the inverse cube of wavelength so light with shorter wavelength (blue in the visible) is scattered much more than other wavelengths. The other light passes through the atmosphere or something. maybe i have this backwards.

electrons can tunnel due to their wave nature. quantum mechanical tunneling is where a particle goes somewhere that is classically impossible, meaning that it simply did not have enough energy to to past a potential barrier, ut it did. We experts in science call this quantum weirdness. There are a lot of electrons, but perhaps not more than there are stupid people in the world. This page needs revision.

Most basic wave to satisfy boundary conditions (which are...) is

\[A \sin(n\pi x/L) \nonumber \]

The superposition principle allows for the fourier theorem which allows an infinite number of such waves to be combined to form any any curve that obeys the requirements of a bound system.

Such a wave provides an accurate (nonrelativistic) description of the electron.

## References

- D.A.McQuarrieandJ.D.Simon,PhysicalChemistry:AMolecularApproach (1997)
- P. W. Atkin and R. S. Friedman, Molecular Quantum Mechanics, 4th Ed. Oxford University Press, 2005.