A Quantum Algebra Supplement - Syntax and Semantics

Syntax and Semantics

In the following, the notations, their associated FullForm constructs and the semantics are shown.

1.  Operator
   
   1. An Operator has the following Full Form: Operator[Name, "H", { Lower Indices }, { Upper Indices }]
      The lists of upper and lower indices may also be empty, the indicator for an adjoint operator "H" may be missing.
   2. Sample notation
      See Notations, Operator, Sample notation
   3. Semantics
      Name indicates the name of the operator, "H" indicates the associated adjoint operator, the two lists contain the lower and upper indices of the notation. The addition to Quantum Algebra often makes use of the convention that the lower indices include wavenumber or momentum variables in subscript representation. The 3 - momentum would consist of the indices p₁, p₂, p₃ and would be supplemented by p₀ as energy for the 4 - momentum.
2. Hermitian
   1. Full Form Syntax
      The Hermitian function determines the adjoint operator for an operator, operator expression or an operator matrix:
      op2 = Hermitian[op1];
   2. Sample Notation
      The super-dagger symbol on an expression can either indicate an adjoint operator or cause the generation of the adjoint of a complete expression, see e.g. Klein - Gordon - Equation a.
   3. Semantics
      An adjoint operator is a mirror operator for a dual space (see literature on quantum mechanics).
3. NonCommutativeMultiply
   1. Full Form Syntax
      Operators in quantum mechanics are generally noncommutative. Their multiplication therefore requires an extra symbol:
      NonCommutativeMultiply[op1, op2] or op1 ** op2 or use the shortcut icon CenterDot.
   2. Sample Notations
      See e.g. Notations, CenterDot.
   3. Semantics
      The noncommutative multiplicative link.
4. QAIntegrate
   1. Full Form Syntax
      QAIntegrate[Integrand, Area, Dim, Var(s)]
   2. Sample Notation
      See e.g. Notations, QAIntegrate.
   3. Semantics
      This is an integral over a generally multidimensional area Area. Dim is the dimension and Var is an integration variable. You can also specify more than one, with the associated notation supporting a maximum of 2. The range may either be finite - a symbol (e.g., V) is given - or infinity - here the infinity symbol is given.
5. QASum
   1. Full Form Syntax
      The syntax is similar to integration, with no scope or dimension. In the end, it is always summed from -Infinity to + Infinity, and the dimensionality is assumed to be 3.
      QASum[Expression, Var(s)]
   2. Sample Notation
      See e.g. Notations, QASum.
   3. Semantics
      This is a summation over an expression, the 3-dimensional variable Var runs over all integer values. You can also specify 2 of them or the expression will be simplified accordingly.
6. QAD
   1. Full Form Syntax
      QAD[Quexp, Var]
   2. Sample Notation
      See e.g. Notations, QAD
   3. Semantics
      This is a derivative of quantum expressions Quexp to variables Var.