Physics Formulae/Quantum Mechanics Formulae

Lead Article: Tables of Physics Formulae

This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Quantum Mechanics.

The nature of Quantum Mechanics is formulations in terms of probabilities, operators, matrices, in terms of energy, momentum, and wave related quantites. There is little or no treatment of properties encountered on macroscopic scales such as force.

Applied Quantities, Definitions

Many of the quantities below are simply energies and electric potential differances.

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Unit	Dimension
Threshold Frequency	f₀	$f_0 \,\!$	Hz = s^-1	[T]^-1
Threshold Wavelength	$\lambda_0 \,\!$	$\lambda_0 = \frac{c}{f_0} \,\!$	m	[L]
Work Function	$\phi, \Phi \,\!$	$\Phi = h f_0\,\!$	J	[M] [L]² [T]^-2
Stopping Potential	V₀	$V_0 = e \Phi_0 \,\!$	J	[M] [L]² [T]^-2

Wave Particle Duality

Massless Particles, Photons

Planck–Einstein Equation	$E = hf = \frac{hc}{\lambda}\,\!$
Photon Momentum	$p = \frac{hf}{c} = \frac{h}{\lambda}\,\!$

Massive Particles

De Broglie Wavelength	$p = \frac{h}{\lambda} = \hbar k \,\!$
Heisenberg's Uncertainty Principle	$\Delta x \Delta p_x \geqslant \frac{\hbar}{2} \,\!$ $\Delta E \Delta t \geqslant \frac{\hbar}{2} \,\!$

Typical effects which can only explained by Quantum Theory, and in part brought rise to Quantum Mechanics itself, are the following.

Photoelectric Effect: Photons greater than threshold frequency incident on a metal surface causes (photo)electrons to be emmited from surface.	$E_\mathrm{k\;\;\!\!max} = hf - \Phi\,\!$
Compton Effect Change in wavelength of photons from an X-Ray source depends only on scattering angle.	$\Delta \lambda = \frac{h}{2m} \left( 1 - \cos \theta \right ) \,\!$
Moseley's Law Frequency of most intense X-Ray Spectrum (K-α) line for an element, Atomic Number Z.	$f = \frac{c}{\lambda} = M_{K_\alpha} (Z - 1)^2$ $M_{K_\alpha} = 2.47 \times 10^{15}$ Hz
Planck's Radiation Law I is spectral radiance (W m^-2 sr^-1 Hz^-1 for frequency or W m^-3 sr^-1 for wavelength), not simply intensity (W m^-2).	$I(\nu,T) =\frac{ 2 h\nu^{3}}{c^2}\frac{1}{ e^{\frac{h\nu}{kT}}-1} \,\!$ $I (\lambda,T) =\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{\lambda kT}}-1}$

The Assumptions of Quantum Mechanics

1: State of a system	A system is completely specified at any one time by a Hilbert space vector.
2: Observables of a system	A measurable quantity corresponds to an operator with eigenvectors spanning the space.
3: Observation of a system	Measuring a system applies the observable's operator to the system and the system collapses into the observed eigenvector.
4: Probabilistic result of measurement	The probability of observing an eigenvector is derived from the square of its wavefunction.
5: Time evolution of a system	The way the wavefunction evolves over time is determined by Shrodinger's equation.

Quantum Numbers

Quantum numbers are occur in the description of quantum states. There is one related to quantized atomic energy levels, and three related to quantized angular momentum.

Name	Symbol	Orbital Nomenclature	Values
Principal	n	shell	$1 \le n , n \in \mathbf{N} \,\!$
Azimuth Angular Momentum	l	subshell; s orbital is listed as 0, p orbital as 1, d orbital as 2, f orbital as 3, etc for higher orbitals	$0 \le \ell \le n-1, \ell \in \mathbf{N} \$
Magnetic Projection of Angular Momentum	m_l	energy shift (orientation of the subshell's shape)	$-\ell \le m_\ell \le \ell, m_\ell \in \mathbf{N}\$
Spin Projection of Spin Angular Momentum	m_s	spin of the electron: -1/2 = counter-clockwise, +1/2 = clockwise	$- \begin{matrix} \frac{1}{2} \end{matrix} , \begin{matrix} \frac{1}{2} \end{matrix} \$

Quantum Wave-Function and Probability

Born Interpretation of the Particle Wavefunction

Wavefunctions are probability distributions describing the space-time behaviour of a particle, distributed through space-time like a wave. It is the wave-particle duality characteristics incorperated into a mathematical function. This interpretation was due to Max Born.

Quantum Probability

Probability current (or flux) is a concept; the flow of probability density.

The probability density is analogous to a fluid; the probability current is analogous to the fluid flow rate. In each case current is the product of density times velocity.

Usually the wave-function is dimensionless, but due to normalization integrals it may in general have dimensions of length to negative integer powers, since the integrals are with respect to space.

Operator (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Unit	Dimension
(Quantum) Wave-function	ψ, Ψ	$\psi = \psi\left ( \mathbf{r}, t \right ) \,\!$ $\Psi = \Psi\left ( \mathbf{r}, t \right ) \,\!$	m^-n	[L]^-n
Wavefunction, Probability Density Function	ρ	$\rho(\mathbf{r},t) = \left \| \Psi (\mathbf{r},t) \right \|^2 \mathrm{d}V \,\!$
Probability Amplitude	A, N
Probability Current Flow of Probability Density	J, I	Non-Relativistic $\mathbf{j} = \frac{\hbar}{2mi}\left(\Psi^* \nabla \Psi - \Psi \nabla \Psi^\right)$ $= \frac\hbar m \mathrm{Im}(\Psi^\nabla\Psi)$ $=\mathrm{Re} \left ( \Psi^* \frac{\hbar}{im} \nabla \Psi \right )$

Properties and Requirements

Normalization Integral

To be solved for probability amplitude.

R = Spatial Region Particle is definitley located in (including all space)

S = Boundary Surface of R.

\int_{\mathbf{r} \in R} \left | \Psi \right |^2 \mathrm{d} V = 1 \,\!

Law of Probability Conservation for Quantum Mechanics

\frac{\partial}{\partial t} \int_V |\Psi|^2 \mathrm{d}V + \int_S \mathbf{j} \cdot \mathrm{d}\mathbf{A} = 0

Quantum Operators

Observable quntities are calculated by operators acting on the wave-function. The term potential alone often refers to the potential operator and the potential term in Schrödinger's Equation, but this is a misconception; rather the implied quantity is potential energy .

It is not immediatley obvious what the opeators mean in their general form, so component definitions are included in the table. Often for one-dimensional considerations of problems the component forms are usefull, since they can be applied immediatley.

Operator (Common Name/s)	Component Definitions	General Definition	SI Unit	Dimension
Position	$\hat{x} = x \,\!$ $\hat{y} = y \,\!$ $\hat{z} = z \,\!$	$\mathbf{\hat{r}} = \mathbf{r} \,\!$	m	[L]
Momentum	$\hat{p}_x = -i \hbar \frac{\partial }{\partial x} \,\!$ $\hat{p}_y = -i \hbar \frac{\partial }{\partial y} \,\!$ $\hat{p}_z = -i \hbar \frac{\partial }{\partial z} \,\!$	$\mathbf{\hat{p}} = -i \hbar \nabla \,\!$	J s m^-1 = N s	[M] [L] [T]^-1
Potential Energy	$\hat{V}_x = V(x) \,\!$ $\hat{V}_y = V(y) \,\!$ $\hat{V}_z = V(z) \,\!$	$\hat{V} = V\left ( \mathbf{r}, t \right ) = V \,\!$	J	[M] [L]² [T]^-2
Energy		$\hat{E} = i \hbar \frac{\partial }{\partial t} \,\!$	J	[M] [L]² [T]^-2
Hamiltonian		$\hat{H} = \hat{T} + \hat{V} \,\!$ $\hat{H} = -\frac{\hbar^2}{2m}\nabla^2+ V \,\!$	J	[M] [L]² [T]^-2
Angular Momentum	$\hat{L}_x = -i\hbar \left(y {\partial\over \partial z} - z {\partial\over \partial y}\right)$ $\hat{L}_y = -i\hbar \left(z {\partial\over \partial x} - x {\partial\over \partial z}\right)$ $\hat{L}_z = -i\hbar \left(x {\partial\over \partial y} - y {\partial\over \partial x}\right)$	$\mathbf{\hat{L}} = -i\hbar \mathbf{r} \times \nabla$	J s = N s m^-1	[M] [L]² [T]^-1
Spin Angular Momentum	$\hat{S}_x = {\hbar \over 2} \sigma_x$ $\hat{S}_y = {\hbar \over 2} \sigma_y$ $\hat{S}_z = {\hbar \over 2} \sigma_z$		J s = N s m^-1	[M] [L]² [T]^-1

Wavefunction Equations

Schrödinger's Equation

General form proposed by Schrödinger:

\hat{H} \Psi = \hat{E} \Psi

Commonly used corolaries are summarized below. A free particle corresponds to zero potential energy.

	1D	3D
Free Particle (V=0)	$- \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d} x^2} \Psi = \hat{E} \Psi \,\!$	$- \frac{\hbar^2}{2m} \nabla^2 \Psi = \hat{E} \Psi \,\!$
Time Independant	$\left ( - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x^2} + V \right ) \Psi = \hat{E} \Psi \,\!$	$\left ( - \frac{\hbar^2}{2m} \nabla^2 + V \right ) \Psi = \hat{E} \Psi \,\!$
Time Dependant	$\left ( - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V \right ) \Psi = i \hbar \frac{\partial}{\partial t} \Psi \,\!$	$\left ( - \frac{\hbar^2}{2m} \nabla^2 + V \right ) \Psi = i \hbar \frac{\partial}{\partial t} \Psi \,\!$

Dirac Equation

The form proposed by Dirac is

\left(\boldsymbol{\beta} mc^2 + c \sum_{k = 1}^3 \boldsymbol{\alpha}_k p_k \right) \Psi = i \hbar \frac{\partial}{\partial t} \Psi

where $\boldsymbol{\beta}$ and $\boldsymbol{\alpha}$ are Dirac Matrices satisfying:

\boldsymbol{\alpha}_i^2=\boldsymbol{\beta}^2=\mathbf{I}_4

$\boldsymbol{\alpha}_i\boldsymbol{\alpha}_j + \boldsymbol{\alpha}_j\boldsymbol{\alpha}_i = 0 \,$

$\boldsymbol{\alpha}_i\boldsymbol{\beta} + \boldsymbol{\beta}\boldsymbol{\alpha}_i = 0 \,$

Klien-Gorden Equation

Schrödinger and De Broglie independantly proposed the relativistic form before Gorden and Klein, but Gorden and Klein included electromagnetic interactions into the equation, useful for charged spin-0 Bosons ^[1].

\left(-\frac{1}{c^2}\frac{\partial^2}{\partial t^2} + \nabla^2\right)\Psi = \left ( \frac{m_0c}{\hbar} \right )^2 \Psi

It can be obtained by inserting the quantum operators into the Momentum-Energy invariant of relativity:

$\frac{E^2}{c^2} - p^2 = \left ( m_0c \right )^2$

Common Energies and Potential Energies

The following energies are used in conjunction with Schrödinger's equation (and other variants). In fact the equation cannot be used for calculations unless the energies defined for it.

The concept of potential energy is important in analyzing probability amplitudes, since this energy confines particles to localized regions of space; the only exception to this is the free particle subject to zero potential energy.

V₀ = Constant Potential Energy

E₀ = Constant Total Energy

Potential Energy Type	Potential Energy V
Free Particle	0
One dimensional box	$V = \begin{cases} V_0 & x \in [a,b] \\ 0 & x \notin [a,b] \end{cases} \,\!$
Harmonic Osscilator	$V = \frac{1}{2}kx^2$
Electrostatic, Coulomb	$V = \frac{q_1 q_2}{4\pi\epsilon_0 r}$
Electric Dipole	$V = - \mathbf{p} \cdot \mathbf{E}$
Magnetic Dipole	$V = - \mathbf{m} \cdot \mathbf{B}$

Infinite Potential well	$E_n = \left ( \frac{hn}{2L} \right )^2 \frac{1}{2m}\,\!$ $n \in \mathbf{N} \,\!$
Wavefunction of a Trapped Particle, One Dimensional Box	$\Psi_n(x) = A \sin \left ( \frac{n\pi x}{L} \right ) \,\!$ $n \in \mathbf{N} \,\!$
Hydrogen atom, orbital energy	$E_n = -\frac{me^4}{8\epsilon_0^2h^2n^2} = \frac{13.61eV}{n^2}\,\!$ $n \in \mathbf{N} \,\!$

Quantum Numbers

Expressions for various quantum numbers are given below.

spin projection quantum number	$m_s \in \{-1/2,+1/2\}\,\!$
Orbital Electron Magnetic Dipole Moment	$\mathbf{\mu}_{orb} = -e\mathbf{L}/2m\,\!$
Orbital Electron Magnetic Dipole Components	$\mathbf{\mu}_{orb,z} = -m_\mathcal{L}\mu_B\,\!$
Orbital, Spin, Electron Magnetic Dipole Moment	$\mathbf{\mu_s} = -e\mathbf{S}/m = gq\mathbf{S}/2m\,\!$
Orbital Electron magnetic dipole moment	$\mathbf{\mu}_{orb}=-e\mathbf{L}_{orb}/2m\,\!$
Orbital, Spin, Electron magnetic dipole moment Potential	$U = -\mathbf{\mu}_s\cdot\mathbf{B}_{ext} = -\mu_{s,z}B_{ext}\,\!$
Orbital, Electron Magnetic Dipole Moment Potential	$U = -\mathbf{\mu}_{orb}\cdot\mathbf{B}_{ext} = -\mu_{orb,z}B_{ext}\,\!$
Angular Momentum Components	$L_z = m\mathcal{L}\hbar\,\!$
Spin Angular Momentum Magnitude	$S = \hbar\sqrt{s(s+1)}\,\!$
Cutoff Wavelength	$\lambda_{min} = hc/K_0\,\!$
Density of States	$N(E) = 8\sqrt{2}\pi m^{3/2}E^{1/2}/h^3\,\!$
Occupancy Probability	$P(E) = 1/(e^{(E-E_F)/kT}+1)\,\!$

Spherical Harmonics

The Hydrogen Atom

Hydrogen Atom Spectrum, Rydberg Equation	$\frac{1}{\lambda} = R_H \left ( \frac{1}{n^2_2} - \frac{1}{n^2_1} \right )\,\!$
Hydrogen Atom, radial probability density	$P(r) = \frac{4r^2}{a^3e^{2r/a}}\,\!$

Referances

↑ Particle Physics, B.R. Martin and G. Shaw, Manchester Physics Series 3rd Edition, 2009, ISBN 978-0-470-03294-7

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