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===3.1. A few words about the quantum formalism=== | ===3.1. A few words about the quantum formalism=== | ||
Denote by <math display="inline">\mathcal{H}</math> a complex Hilbert space. For simplicity, we assume that it is finite dimensional. Pure states of a system <math>S</math> are given by normalized vectors of <math display="inline">\mathcal{H}</math> and mixed states by density operators (positive semi-definite operators with unit trace). The space of density operators is denoted by <math>S</math> (<math display="inline">\mathcal{H}</math>). The space of all linear operators in <math display="inline">\mathcal{H}</math> is denoted by the symbol <math display="inline">\mathcal{L}(\mathcal{H})</math> . In turn, this is a linear space. Moreover, <math display="inline">\mathcal{L}(\mathcal{H})</math> is the complex Hilbert space with the scalar product, <math display="inline"><A|B>=TrA^*B</math>. We consider linear operators acting in <math display="inline">\mathcal{L}(\mathcal{H})</math>. They are called ''superoperators.'' | Denote by <math display="inline">\mathcal{H}</math> a complex Hilbert space. For simplicity, we assume that it is finite dimensional. Pure states of a system <math>S</math> are given by normalized vectors of <math display="inline">\mathcal{H}</math> and mixed states by density operators (positive semi-definite operators with unit trace). The space of density operators is denoted by <math>S</math> (<math display="inline">\mathcal{H}</math>). The space of all linear operators in <math display="inline">\mathcal{H}</math> is denoted by the symbol <math display="inline">\mathcal{L}(\mathcal{H})</math> . In turn, this is a linear space. Moreover, <math display="inline">\mathcal{L}(\mathcal{H})</math> is the complex Hilbert space with the scalar product, <math display="inline"><A|B>=TrA^*B</math>. We consider linear operators acting in <math display="inline">\mathcal{L}(\mathcal{H})</math>. They are called ''superoperators.'' | ||
The dynamics of the pure state of an isolated quantum system is described by ''the Schrödinger equation:'' | The dynamics of the pure state of an isolated quantum system is described by ''the Schrödinger equation:'' | ||
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| width="33%" align="right" |<math>(4)</math> | | width="33%" align="right" |<math>(4)</math> | ||
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