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.
Contents
Applied Quantities, Definitions[edit]
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  f_{0}  Hz = s^{1}  [T]^{1}  
Threshold Wavelength  m  [L]  
Work Function  J  [M] [L]^{2} [T]^{2}  
Stopping Potential  V_{0}  J  [M] [L]^{2} [T]^{2} 
Wave Particle Duality[edit]
Massless Particles, Photons[edit]
Planck–Einstein Equation  
Photon Momentum 
Massive Particles[edit]
De Broglie Wavelength  
Heisenberg's Uncertainty
Principle 

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.


Compton Effect
Change in wavelength of photons from an XRay source depends only on scattering angle. 

Moseley's Law
Frequency of most intense XRay Spectrum (Kα) line for an element, Atomic Number Z. 
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}). 

The Assumptions of Quantum Mechanics[edit]
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[edit]
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  
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 

Magnetic
Projection of Angular Momentum 
m_{l}  energy shift (orientation of
the subshell's shape) 

Spin
Projection of Spin Angular Momentum 
m_{s}  spin of the electron:
1/2 = counterclockwise, +1/2 = clockwise 
Quantum WaveFunction and Probability[edit]
Born Interpretation of the Particle Wavefunction
Wavefunctions are probability distributions describing the spacetime behaviour of a particle, distributed through spacetime like a wave. It is the waveparticle 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 wavefunction 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) Wavefunction  ψ, Ψ 

m^{n}  [L]^{n} 
Wavefunction, Probability Density Function  ρ  
Probability Amplitude  A, N  
Probability Current
Flow of Probability Density 
J, I  NonRelativistic

Properties and Requirements[edit]
Normalization Integral
To be solved for probability amplitude.
R = Spatial Region Particle is definitley located in (including all space)
S = Boundary Surface of R.
Law of Probability Conservation for Quantum Mechanics
Quantum Operators[edit]
Observable quntities are calculated by operators acting on the wavefunction. 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 onedimensional considerations of problems the component forms are useful, since they can be applied immediatley.
Operator (Common Name/s)  Component Definitions  General Definition  SI Unit  Dimension 

Position 

m  [L]  
Momentum 

J s m^{1} = N s  [M] [L] [T]^{1}  
Potential Energy 

J  [M] [L]^{2} [T]^{2}  
Energy  J  [M] [L]^{2} [T]^{2}  
Hamiltonian 

J  [M] [L]^{2} [T]^{2}  
Angular Momentum 

J s = N s m^{1}  [M] [L]^{2} [T]^{1}  
Spin Angular Momentum 

J s = N s m^{1}  [M] [L]^{2} [T]^{1} 
Wavefunction Equations[edit]
Schrödinger's Equation
General form proposed by Schrödinger:
Commonly used corolaries are summarized below. A free particle corresponds to zero potential energy.
1D  3D  

Free Particle (V=0)  
Time Independant  
Time Dependant 
Dirac Equation
The form proposed by Dirac is
where and are Dirac Matrices satisfying:

KlienGorden 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 spin0 Bosons ^{[1]}.
It can be obtained by inserting the quantum operators into the MomentumEnergy invariant of relativity:
Common Energies and Potential Energies[edit]
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.
V_{0} = Constant Potential Energy
E_{0} = Constant Total Energy
Potential Energy Type  Potential Energy V 

Free Particle  0 
One dimensional box  
Harmonic Osscilator  
Electrostatic, Coulomb  
Electric Dipole  
Magnetic Dipole 
Infinite Potential well 

Wavefunction of a Trapped
Particle, One Dimensional Box 

Hydrogen atom, orbital energy 

Quantum Numbers
Expressions for various quantum numbers are given below.
spin projection quantum number  
Orbital Electron Magnetic Dipole Moment  
Orbital Electron Magnetic Dipole Components  
Orbital, Spin, Electron Magnetic Dipole Moment  
Orbital Electron magnetic dipole moment  
Orbital, Spin, Electron magnetic dipole moment Potential  
Orbital, Electron Magnetic Dipole Moment Potential  
Angular Momentum Components  
Spin Angular Momentum Magnitude  
Cutoff Wavelength  
Density of States  
Occupancy Probability 
Spherical Harmonics[edit]
The Hydrogen Atom[edit]
Hydrogen Atom Spectrum,
Rydberg Equation 

Hydrogen Atom, radial probability density 
MultiElectron Atoms, Perdiodic Table[edit]
References[edit]
 ↑ Particle Physics, B.R. Martin and G. Shaw, Manchester Physics Series 3rd Edition, 2009, ISBN 9780470032947