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.

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

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.

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.

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.

  Law of Probability Conservation for Quantum Mechanics 

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 useful, since they can be applied immediatley.

Wavefunction Equations
 Schrödinger's Equation 

General form proposed by Schrödinger:

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

 Dirac Equation 

The form proposed by Dirac is

where $$ \boldsymbol{\beta} $$ and $$ \boldsymbol{\alpha} $$ 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.

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

Quantum Numbers

Expressions for various quantum numbers are given below.