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单词 DiracEquation
释义

Dirac equation


The Dirac equationMathworldPlanetmath is an equation derived by Paul Dirac in 1927 that describes relativistic spin 1/2 particles (fermions). It is given by:

(γμμ-im)ψ=0

The Einstein summation convention is used.

0.1 Derivation

Mathematically, it is interesting as one of the first uses of the spinor calculus in mathematical physics. Dirac began with the relativistic equation of total energy:

E=p2c2+m2c4

As Schrödinger had done before him, Dirac then replaced p with its quantum mechanical operator, p^i. Since he was looking for a Lorentz-invariant equation, he replaced with the D’Alembertian or wave operatorMathworldPlanetmath

=2-1c22t2

Note that some authors use 2 for the D’Alembertian. Dirac was now faced with the problem of how to take the square root of an expression containing a differential operatorMathworldPlanetmath. He proceeded to factorise the d’Alembertian as follows:

2-1c22t2=(a0x+a1y+a2z+a3ict)2

Multiplying this out, we find that:

(a0)2=(a1)2=(a2)2=(a3)2=1

And

a0a1+a1a0=a0a2+a2a0=a0a3+a3a0=a1a2+a2a1=a1a3+a3a1=a2a3+a3a2=0

Clearly these relations cannot be satisfied by scalars, so Dirac sought a set of four matrices which satisfy these relations. These are now known as the Dirac matrices, and are given as follows:

γ0=-ia0=(1000010000-10000-1),γ1=-ia1=(000100100-100-1000)
γ2=-ia2=(000-i00i00i00-i000),γ3=a3=(0010000-1-10000100)

These matrices are also known as the generators of the special unitary group of order 4, i.e. the group of 4×4 matrices with unit determinantDlmfMathworldPlanetmath.Using these matrices, and switching to natural units (=c=1) we can now obtain the Dirac equation:

(γμμ-im)ψ=0

0.2 Feynman slash notation

Richard Feynman developed the following convenient notation for terms involving Dirac matrices:

γμqμ:=q

Using this notation, the Dirac equation is simply

(-im)ψ=0

0.3 Relationship with Pauli matrices

The Dirac matrices can be written more concisely as matrices of Pauli matricesMathworldPlanetmath, as follows:

γ0=(σ000-σ0)
γ1=(0σ1-σ10)
γ2=(0σ2-σ20)
γ3=(0σ3-σ30)
TitleDirac equation
Canonical nameDiracEquation
Date of creation2013-03-22 17:54:46
Last modified on2013-03-22 17:54:46
OwnerRaphanus (20453)
Last modified byRaphanus (20453)
Numerical id21
AuthorRaphanus (20453)
Entry typeDefinition
Classificationmsc 35Q40
Classificationmsc 81Q05
Related topicSpinor
Related topicKleinGordonEquation
Related topicSchrodingersWaveEquation
Related topicPauliMatrices
DefinesFeynman slash notation
DefinesDirac matrices
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