Abstract
A very concise overview of the basic notions and principles of quantum mechanics (physical states as rays in Hilbert space, observables as Hermitean operators, time development as unitary transformation) is given – leaving out all enduring problems with the interpretation of quantum mechanics. Next questions concerning symmetries are posed and answered anew in the context of quantum physics. This refers to Wigner’s theorem about the relation between symmetry transformations and either unitary and linear or anti-unitary and anti-linear symmetry operators. This relation ties intimately the quantum mechanical symmetries to the projective representations of the symmetry group. This allows for instance to derive the Schrödinger equation from representations of the Galilei group – or rather the Bargmann group. The chapter ends with an investigation of the Noether and Lie (point) symmetries of the Schrödinger equation.
Alle Quantenzahlen, mit Ausnahme der sog. Hauptquantenzahl, sind Kennzeichen von Gruppendarstellungen.
“All quantum numbers, with the exception of the so-called principal quantum number, are indices characterising representations of groups”. (H. Weyl in the introduction of “Gruppentheorie und Quantenmechanik” [550].) As a matter of fact because of the SO(4) invariance of the Coulomb potential also the principal quantum number is related to a group representation: W. Pauli, Z. Phys. 36 (1926) 336–363.
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Notes
- 1.
As prevailing in the (quantum) physics context, operators are denoted by a hat; in contexts in which the operator nature is obvious I refrain from the hat notation.
- 2.
For the purpose of this book it is not essential that strictly speaking a distinction of a self-adjoint and a Hermitean operator has to be made. However, if it comes to the mathematical foundation of quantum mechanics it is mostly the self-adjointness property which counts.
- 3.
This refers to “fermion-boson superselection”:
- 4.
Further quantum numbers completely labeling the state are supressed.
- 5.
Note that if you replace \(\psi \) by a real function \(u(x,t)\), \(u_t=u_{xx}\) is the heat equation. You can find this as a standard example in all books treating the Lie approach for solving differential equations; e.g. [284], [399], [485].
- 6.
These refer to the group representations which H. Weyl had in mind with his statement “All quantum numbers, with the exception of the so-called principal quantum number, are indices characterizing representations of groups”.
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Sundermeyer, K. (2014). Quantum Mechanics. In: Symmetries in Fundamental Physics. Fundamental Theories of Physics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-06581-6_4
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DOI: https://doi.org/10.1007/978-3-319-06581-6_4
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