Quantum Mechanics Fundamentals

Name	Formula	Related Formulae	Variables/Constants
Planck relation and photoelectric effect	$E=h\nu =\hslash \omega$	$E_m=h\nu -q\Phi$	$E:photonenergy$ $h:Planckconstant(6.63\times 10^{-34}Js)$ $\nu :frequency$ $\hslash :reducedPlanckconstanth/2\pi$ $\omega :angularfrequency$ $\begin{array}{r}{E}_m:maximumkineticenergyofanejected\\ electron\end{array}$ $q:electroniccharge$ $\Phi :workfunctioninvolts$
de Broglie wavelength	$\lambda =\frac{h}{p}=\frac{h}{mv}$	$p=\hslash k,E=cp(\text{photon})$	$\lambda :wavelength$ $p:momentum$ $m:mass$ $v:velocity$ $k:wavenumber2\pi /\lambda$ $c:speedoflight$
Bohr model of hydrogen	$E_n=-\frac{m{q}^4}{2K^2{n}^2{\hslash }^2},K=4\pi {\epsilon }_0$	$r_n=\frac{K{n}^2{\hslash }^2}{m{q}^2},h\nu =E_2-E_1,\nu =cR\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$	$E_n:energyoforbitn$ $n:principalquantumnumber$ $r_n:orbitradius$ ${\epsilon }_0:permittivityoffreespace$ $R:Rydbergconstant$ $n_1:lowerlevel$ $\begin{array}{r}{n}_2:upperlevel;angularmomentummvr=\\ n\hslash \end{array}$
Heisenberg uncertainty principle	$\Delta x\Delta p_x\ge \frac{\hslash }{2}$	$\Delta E\Delta t\ge \frac{\hslash }{2}$	$\Delta x:positionuncertainty$ $\Delta p_x:momentumuncertainty$ $\Delta E:energyuncertainty$ $\Delta t:timeuncertainty$
Probability and expectation value	${\int }_{-\infty }^{\infty }{\Psi }^{\ast }\Psi dx=1$	$⟨Q⟩={\int }_{-\infty }^{\infty }{\Psi }^{\ast }{Q}_{op}\Psi dx$	$\Psi :wavefunction,{\Psi }^{\ast }:complexconjugate$ ${\Psi }^{\ast }\Psi :probabilitydensity$ $\begin{array}{r}{Q}_{op}:operatorforquantityQ,⟨Q⟩:average\\ (expectation)value\end{array}$
Quantum operators	$p_x\to \frac{\hslash }{j}\frac{\partial }{\partial x},E\to -\frac{\hslash }{j}\frac{\partial }{\partial t}$	$\frac{\hslash }{j}=-j\hslash ,-\frac{\hslash }{j}=j\hslash (j=i=\sqrt{-1})$	$p_x:momentumoperator$ $E:energyoperator$ $\begin{array}{r}j:imaginaryunit(engineeringnotation;\\ physicsbookswritei)\end{array}$
Classical energy equation	$\frac{p^2}{2m}+V=E$	$\text{kinetic energy}+\text{potential energy}=\text{total energy}$	$p:momentum$ $m:mass$ $V:potentialenergy$ $E:totalenergy$
Time-dependent Schrodinger equation	$-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\Psi }{\partial x^2}+V\Psi =-\frac{\hslash }{j}\frac{\partial \Psi }{\partial t}$	$-\frac{{\hslash }^2}{2m}{\nabla }^2\Psi +V\Psi =j\hslash \frac{\partial \Psi }{\partial t}$	$\begin{array}{r}\Psi (x,t):wavefunction,V(x):potential\\ energy\end{array}$ ${\nabla }^2:Laplacian$ $m:particlemass$
Time-independent Schrodinger equation	$\left[-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V\left(x\right)\right]\psi =E\psi$	$\Psi \left(x,t\right)={\sum }_n{c}_n{\psi }_n\left(x\right)e^{-j{E}_{n}t/\hslash }$	$\psi :spatialeigenfunction$ $E:energyeigenvalue$ $\begin{array}{r}{c}_n:expansioncoefficient(|c_n{|}^{2}isthe\\ probabilityofmeasuring{E}_n)\end{array}$ $E_n:eigenenergy$
Free particle	$\Psi =A{e}^{j\left(kx-\omega t\right)},E=\frac{{\hslash }^2{k}^2}{2m}$	$⟨p_x⟩=\hslash k,⟨E⟩=\hslash \omega$	$A:amplitude$ $k:wavenumber$ $\begin{array}{r}\omega :angularfrequency;planewaveina\\ regionwithV=0\end{array}$
Infinite potential well	$E_n=\frac{n^2{\pi }^2{\hslash }^2}{2m{L}^2}$	${\psi }_n=\sqrt{\frac{2}{L}}\sin \frac{n\pi x}{L},k=\frac{\sqrt{2mE}}{\hslash }=\frac{n\pi }{L}$	$E_n:energyofleveln$ $n:quantumnumber1,2,3$ $L:wellwidth$ $m:mass(m^{\ast }inasemiconductorwell)$
Harmonic oscillator	$E_n=\left(n+\frac{1}{2}\right)\hslash \omega$	$V\left(x\right)=\frac{1}{2}m{\omega }^2{x}^2$	$n:quantumnumber0,1,2$ $\begin{array}{r}\omega :oscillatorangularfrequency;equally\\ spacedlevels,quantaoflatticevibration\\ arephonons\end{array}$
Hydrogen atom quantum numbers	$n=1,2,3,\dots ;l=0,\dots ,n-1;m=-l,\dots ,+l$	$s=\pm \frac{\hslash }{2},2n^2\text{states in shell }n$	$n:principal$ $l:orbital$ $m:magnetic$ $\begin{array}{r}s:spinangularmomentum;Pauliexclusion\\ principleallowsoneelectronpersetof\\ n,l,m,s\end{array}$
Tunnelling through a rectangular barrier	$T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa W}$	$\kappa =\frac{\sqrt{2m\left(V_0-E\right)}}{\hslash }$	$T:transmissionprobability$ $E:particleenergy(E<V_0)$ $V_0:barrierheight$ $W:barrierthickness$ $\kappa :decayconstantinsidethebarrier$
WKB tunnelling probability	$T\approx \exp \left[-2{\int }_{x_1}^{x_2}\kappa \left(x\right)dx\right]$	$\kappa \left(x\right)=\frac{\sqrt{2m^{\ast }\left[V\left(x\right)-E\right]}}{\hslash }$	$\begin{array}{r}{x}_1,x_2:classicalturningpoints,V(x):\\ barrierprofile,m^{\ast }:effectivemassinthe\\ barrier\end{array}$
Fowler-Nordheim tunnelling	$J_{FN}=A{\mathcal{E}}_{ox}^2\exp \left(-\frac{B}{{\mathcal{E}}_{ox}}\right)$	$A=\frac{q^3}{8\pi h{\varphi }_B},B=\frac{8\pi \sqrt{2m^{\ast }}{\varphi }_B^{3/2}}{3qh}$	$J_{FN}:tunnellingcurrentdensity$ ${\mathcal{E}}_{ox}:oxidefield$ $\begin{array}{r}{\varphi }_B:barrierheightinjoules(3.1eVfor\\ theSi-SiO2conductionband),m^{\ast }:\\ electroneffectivemassintheoxide;\\ triangularbarrierwhenq{V}_{ox}>{\varphi }_B,direct\\ tunnellingthroughatrapezoidalbarrier\\ otherwise\end{array}$
Effective mass and parabolic band	$m^{\ast }=\frac{{\hslash }^2}{d^{2}E/d{k}^2}$	$E=E_c+\frac{{\hslash }^2{k}^2}{2m^{\ast }}$	$m^{\ast }:effectivemass,E(k):banddispersion$ $\begin{array}{r}{E}_c:conductionbandedge;negative\\ curvatureatthevalencebandtopgives\\ negativeelectronmass\end{array}$
Hydrogenic donor	$E_d=\frac{m_n^{\ast }{q}^4}{2K^2{\hslash }^2},K=4\pi {\epsilon }_0{\epsilon }_r$	$r_d=\frac{K{\hslash }^2}{m_n^{\ast }{q}^2}$	$\begin{array}{r}{E}_d:donorbindingenergy,m_n^{\ast }:\\ conductivityeffectivemass\end{array}$ ${\epsilon }_r:relativedielectricconstant$ $r_d:ground-stateorbitradius$

Important Questions

Ques: How does the Schrodinger equation follow from the energy equation?

By replacing momentum and energy with operators. Classically $p^2/2m+V=E$. Streetman substitutes $p_x\to (\hslash /j)\partial /\partial x$ and $E\to -(\hslash /j)\partial /\partial t$ and lets both sides act on $\Psi$, giving $-\frac{{\hslash }^2}{2m}{\partial }^2\Psi /\partial x^2+V\Psi =-\frac{\hslash }{j}\partial \Psi /\partial t$. Since $-\hslash /j=j\hslash$, this is the physics-book form with $i\hslash \partial \Psi /\partial t$ (background knowledge, Griffiths). Writing $\Psi =\psi (x)\varphi (t)$ separates it into $\varphi =e^{-jEt/\hslash }$ and the time-independent equation $[-\frac{{\hslash }^2}{2m}{d}^2/d{x}^2+V]\psi =E\psi$. Only V changes from problem to problem; the boundary conditions then select the allowed energies (Streetman). Streetman’s postulates add that $\Psi$ and its derivative must be continuous, finite and single valued, and that ${\Psi }^{\ast }\Psi$ is the probability density, normalized to one. The time factor $e^{-jEt/\hslash }=e^{-j\omega t}$ is the same for every eigenfunction, and a general state is a weighted sum of eigenfunctions, each measured energy $E_n$ occurring with probability $|c_n{|}^2$. Reference: ; B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed., Global Edition (Pearson, 2015), Secs. 2.2 to 2.5, 3.2.2, 3.2.4, 3.2.5, 6.4.7 and 9.5.2 (opened); D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed. (Cambridge University Press, 2018) (background knowledge).

Ques: Is the uncertainty limit hbar/2 or hbar?

Streetman writes $\Delta x\Delta p_x\ge \hslash /2$ and $\Delta E\Delta t\ge \hslash /2$, while Halliday, Resnick and Walker state the position-momentum relation as $\Delta x\Delta p_x\ge \hslash$ (background knowledge). The value $\hslash /2$ is more likely the correct bound, because it is the rigorous result for standard deviations and is reached exactly by a Gaussian wave packet (background knowledge, Griffiths); the $\hslash$ form is an order-of-magnitude statement. The factor matters in estimates: confining an electron to 0.1 nm gives a minimum kinetic energy of about 0.95 eV with $\hslash /2$ but 3.8 eV with $\hslash$ (computed), so the version used in any estimate should always be stated. Reference: ; B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed., Global Edition (Pearson, 2015), Secs. 2.2 to 2.5, 3.2.2, 3.2.4, 3.2.5, 6.4.7 and 9.5.2 (opened); D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, 10th ed. (Wiley, 2014) (background knowledge); D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed. (Cambridge University Press, 2018) (background knowledge).

Ques: How large are confinement energies in real quantum wells?

Large enough to matter at nanometre widths. From $E_n=n^2{\pi }^2{\hslash }^2/(2m{L}^2)$, a free electron in a 1 nm well has $E_1$ = 0.376 eV, and in the 0.59 nm well of Streetman’s Problem 2.11 the first five levels are 1.08, 4.32, 9.72, 17.3 and 27.0 eV (computed). In a semiconductor the mass becomes $m^{\ast }$: for the 50 angstroms GaAs well in Streetman’s Fig. 3-13, with an assumed 0.067 $m_0$, the infinite-well estimate is 0.22 eV (computed), close to the 0.28 eV conduction-band barrier shown. That estimate is therefore too high; Streetman notes that Eq. (2-33) must be modified for effective mass and finite barrier height, which lowers the level (my reading). Reference: ; B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed., Global Edition (Pearson, 2015), Secs. 2.2 to 2.5, 3.2.2, 3.2.4, 3.2.5, 6.4.7 and 9.5.2 (opened).

Ques: How quickly does tunnelling probability fall with barrier thickness?

Exponentially, through $e^{-2\kappa W}$ with $\kappa =\sqrt{2m(V_0-E)}/\hslash$ (background knowledge, Griffiths). For a free electron 1 eV below the barrier top, $\kappa$ = 5.12 per nm, so $e^{-2\kappa W}$ is 3.5e-5 for 1 nm and 1.3e-9 for 2 nm (computed). With E = 1 and $V_0$ = 2 eV, the prefactor $16(E/V_0)(1-E/V_0)$ is 4, giving T = 1.4e-4 at 1 nm (computed). For the 3.1 eV Si-SiO2 barrier with an assumed $m^{\ast }$ = 0.5 $m_0$, a 2 nm oxide gives 8.3e-12 (computed). Streetman explains the qualitative picture: the wavefunction decays inside the barrier but is nonzero beyond it, so thicker barriers make tunnelling negligible. Reference: ; D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed. (Cambridge University Press, 2018) (background knowledge); B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed., Global Edition (Pearson, 2015), Secs. 2.2 to 2.5, 3.2.2, 3.2.4, 3.2.5, 6.4.7 and 9.5.2 (opened).

Ques: What sets the Fowler-Nordheim current through a gate oxide?

The oxide field and the barrier height. Streetman gives $I_{FN}\propto {\mathcal{E}}_{ox}^2\exp (-B/{\mathcal{E}}_{ox})$, with B depending on $m^{\ast }$ and barrier height, for electrons tunnelling through a triangular barrier into the SiO2 conduction band. The full form is $J=A{\mathcal{E}}^2{e}^{-B/\mathcal{E}}$ with $A=q^3/(8\pi h{\varphi }_B)$ and $B=8\pi \sqrt{2m^{\ast }}{\varphi }_B^{3/2}/(3qh)$ (background knowledge, Sze). For 3.1 eV and an assumed $m^{\ast }$ = 0.42 $m_0$, B is 242 MV per cm and J is 2.4e-6 A per square centimetre at 8 MV per cm, rising to 1.6e-3 at 10 MV per cm (computed). Plotting $\ln (J/{\mathcal{E}}^2)$ against $1/\mathcal{E}$ gives a line of slope $-B$, as in Streetman’s Fig. 6-24, and the slope gives the barrier height. Reference: ; B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed., Global Edition (Pearson, 2015), Secs. 2.2 to 2.5, 3.2.2, 3.2.4, 3.2.5, 6.4.7 and 9.5.2 (opened); S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2007) (background knowledge).

Ques: Is the Rydberg constant 109,678 or 109,737 per cm?

Both are right for different cases. Streetman gives R = 109,678 per cm for hydrogen, while his Example 2-1 uses 1.097e7 per m, which rounds either value. Evaluating Bohr’s $m{q}^4/(2K^2{\hslash }^{2}hc)$ with the free-electron mass gives 1.0974e7 per m, the infinite-nucleus value of 109,737 per cm (computed). Replacing m by the electron-proton reduced mass lowers it by a factor of 1.000544 to 1.0968e7 per m, exactly Streetman’s 109,678 per cm (computed). For hydrogen spectra 109,678 is therefore more likely the correct value, and 109,737 is the reference constant (background knowledge, Griffiths). The same Bohr energy gives -13.6 eV and a 0.529 angstroms radius for n = 1 (computed). Reference: ; B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed., Global Edition (Pearson, 2015), Secs. 2.2 to 2.5, 3.2.2, 3.2.4, 3.2.5, 6.4.7 and 9.5.2 (opened); D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed. (Cambridge University Press, 2018) (background knowledge).

Ques: Does Streetman’s donor-orbit example use h or hbar?

It must use $\hslash$, although the printed substitution shows 6.63e-34. Streetman’s Example 3-3 computes the donor radius in silicon from $r=4\pi {\epsilon }_r{\epsilon }_0{\hslash }^2/(m_n^{\ast }{q}^2)$ with ${\epsilon }_r$ = 11.8 and $m_n^{\ast }$ = 0.26 $m_0$ and reports 24.1 angstroms. Using $\hslash$ = 1.055e-34 J s gives 24.0 angstroms, while using h would give 948 angstroms (computed), so the printed constant is a typesetting slip and 24 angstroms is correct (my reading). The orbit spans more than four lattice constants of 5.43 angstroms, as Streetman notes. The hydrogenic binding energy is 25 meV (computed), close to his statement that donors in Si lie 0.03 to 0.06 eV below the band edge. Reference: ; B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed., Global Edition (Pearson, 2015), Secs. 2.2 to 2.5, 3.2.2, 3.2.4, 3.2.5, 6.4.7 and 9.5.2 (opened).

References

1. B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed., Global Edition. Harlow: Pearson, 2015. Sections 2.2 to 2.5 (photoelectric effect, Bohr model, uncertainty principle, Schrodinger equation, potential well, tunnelling, hydrogen atom), 3.2.2 (effective mass), 3.2.4 (donor levels), 3.2.5 (quantum wells), 6.4.7 (Fowler-Nordheim and direct tunnelling) and 9.5.2 (flash memory erase).
2. D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed. Cambridge: Cambridge University Press, 2018 (cited from background knowledge).
3. D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, 10th ed. Hoboken, NJ: Wiley, 2014 (cited from background knowledge).
4. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. Hoboken, NJ: Wiley, 2007 (cited from background knowledge).