     ## 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
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.
`QAD[Quexp, Var]`