Sample Applications - Continuation

Dirac - Equation

The free fields that solve the field equations are again specified:

Ψ := p r = 1 2 ( Sqrt [ m / ( V p 0 ) ] c ^ p 1 , p 2 , p 3 r u r Exp [ exp ] ) + p r = 1 2 ( Sqrt [ m / ( V p 0 ) ] d ^ p 1 , p 2 , p 3 r , v r Exp [ + exp ] ) Ψ_ ¯ := ( Hermitian [ Ψ ] γ 0 ) // QASimplify [ ]

  1. The Hamilton operator is defined and evaluated:

    H := ( Ψ ¯ // QASimplify[] ) i = 1 3 ( γ i ( D x i ( Ψ // Evaluate ) /. p p 1 ) ) + m ( Ψ ¯ // QASimplify[] ) ( Ψ /. p p 1 ) H := V H 3 x H // QAEvaluate2 // Timing { 154.141 , 2 1 ^ p p 0 + p c ^ p 1 , p 2 , p 3 1 , c ^ p 1 , p 2 , p 3 1 p 0 + p c ^ p 1 , p 2 , p 3 2 , c ^ p 1 , p 2 , p 3 2 p 0 + p d ^ p 1 , p 2 , p 3 1 , d ^ p 1 , p 2 , p 3 1 p 0 + p d ^ p 1 , p 2 , p 3 2 , d ^ p 1 , p 2 , p 3 2 p 0 }

  2. The 3 - momentum is defined and evaluated:

    P i_ := ( Ψ ¯ // QASimplify [ ] ) ( γ 0 ( D x i ( Ψ // Evaluate ) /. p p1 ) ) P i_ := V P i 3 x Map [ ( P i // QAEvaluate2 ) & , { 1 , 2 , 3 } ] // Timing { 116.688 , { p c ^ p 1 , p 2 , p 3 1 , c ^ p 1 , p 2 , p 3 1 p 1 p c ^ p 1 , p 2 , p 3 2 , c ^ p 1 , p 2 , p 3 2 p 1 p d ^ p 1 , p 2 , p 3 1 , d ^ p 1 , p 2 , p 3 1 p 1 p d ^ p 1 , p 2 , p 3 2 , d ^ p 1 , p 2 , p 3 2 p 1 , p c ^ p 1 , p 2 , p 3 1 , c ^ p 1 , p 2 , p 3 1 p 2 p c ^ p 1 , p 2 , p 3 2 , c ^ p 1 , p 2 , p 3 2 p 2 p d ^ p 1 , p 2 , p 3 1 , d ^ p 1 , p 2 , p 3 1 p 2 p d ^ p 1 , p 2 , p 3 2 , d ^ p 1 , p 2 , p 3 2 p 2 , p c ^ p 1 , p 2 , p 3 1 , c ^ p 1 , p 2 , p 3 1 p 3 p c ^ p 1 , p 2 , p 3 2 , c ^ p 1 , p 2 , p 3 2 p 3 p d ^ p 1 , p 2 , p 3 1 , d ^ p 1 , p 2 , p 3 1 p 3 p d ^ p 1 , p 2 , p 3 2 , d ^ p 1 , p 2 , p 3 2 p 3 } }

  3. The 4 - current is defined and the charge is evaluated (without multiplicative charge constant):

    J i : = ( Ψ QASimplify [ ] ) ( γ i ( Ψ /. p p 1 ) ) J i : = v J i 3 x QAEvaluate2 [ J 0 ] // Timing { 41.5313 , 2 1 ^ p 1 + p c ^ p 1 , p 2 , p 3 1 , c ^ p 1 , p 2 , p 3 1 + p c ^ p 1 , p 2 , p 3 2 , c ^ p 1 , p 2 , p 3 2 p d ^ p 1 , p 2 , p 3 1 , d ^ p 1 , p 2 , p 3 1 p d ^ p 1 , p 2 , p 3 2 , d ^ p 1 , p 2 , p 3 2 }

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