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 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 f0   Hz = s-1 [T]-1
Threshold Wavelength     m [L]
Work Function     J [M] [L]2 [T]-2
Stopping Potential V0   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




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 X-Ray source depends only

on scattering angle.

Moseley's Law

Frequency of most intense X-Ray Spectrum (K-α) line for an element,

Atomic Number Z.



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  

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


Projection of Angular


ml energy shift (orientation of

the subshell's shape)


Projection of Spin

Angular Momentum

ms spin of the electron:

-1/2 = counter-clockwise,

+1/2 = clockwise


Quantum Wave-Function and Probability edit

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  


m-n [L]-n
Wavefunction, Probability Density Function ρ  
Probability Amplitude A, N
Probability Current

Flow of Probability Density

J, I Non-Relativistic




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 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 useful, since they can be applied immediatley.

Operator (Common Name/s) Component Definitions General Definition SI Unit Dimension



  m [L]



  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


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:




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].


It can be obtained by inserting the quantum operators into the Momentum-Energy 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.

V0 = Constant Potential Energy

E0 = 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  

Multi-Electron Atoms, Perdiodic Table edit

References edit

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