Why are complex numbers needed in quantum mechanics? …

Why are complex numbers needed in quantum mechanics ?Some answers for the introductory levelRicardo Karam Department of Science Education, University of Copenhagen, DenmarkAbstractComplex numbers are broadly used in physics, normally as a calculation tool that makes thingseasier due to Euler s formula. In the end, it is only the real component that has physical meaning orthe two parts (real and imaginary) are treated separately as real quantities. However, the situationseems to be different in quantum mechanics , since the imaginary unitiappears explicitly in itsfundamental equations. From a learning perspective, this can create some challenges to this article, four conceptually different justifications for the use/need of complex numbers inquantum mechanics are presented and some pedagogical implications are INTRODUCTIONC omplex numbers were invented (or discovered?) in 16th-century Italy as a calculationtool to solve cubic equations.

Quantum mechanics seems to use complex numbers in a more fundamental way. It su ces to look at some of the most basic equations, both in the matrix ([^p;x^] = i~) and wave (i~@ @t = H ^ ) formulations, to wonder about the presence of the imaginary unit. What is essentially di erent in quantum mechanics is that it deals with complex quantities ...

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  Number, Mechanics, Complex, Quantum, Needed, Quantum mechanics, Complex number, In quantum mechanics, Complex numbers needed in quantum mechanics

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