An Improved Semi-Classical Approximation Based on Heisenberg's Matrix Mechanics

Виктор Павлович Маслов, M. V. Fedoriuk

Ulf O Karlsson

Mihai V. Putz

Abstract ................................................................................................... 66 2.1 Introduction .................................................................................... 66 2.2 Wave Function Picture ................................................................... 67 2.2.1 Green and Dirac Functions ................................................ 67 2.2.2 Momentum and Energy Operators ..................................... 69 2.2.3 Klein-Gordon and Schrödinger Equations ......................... 72 2.2.4 Electronic and Photonic Spins ........................................... 74 2.2.5 Eigen-Functions and Eigen-Values .................................... 75 2.2.6 Hermitic Operators ............................................................ 76 2.2.7 Heisenberg Uncertainty Theorem ...................................... 81 2.2.8 Ehrenfest Theorem ............................................................. 85 2.2.9 Current Density Probability Conservation Theorem ......... 89 2.3 Classical to Quantum Mechanics’ Correspondence ....................... 92 2.3.1 Classical Euler-Lagrange, Hamilton, and Hamilton-Jacobi Equations ................................................ 92 2.3.2 Wave-Function Quantum Field .......................................... 97 2.3.3 Semi-classical Expansion and the WKB Approximation ................................................................. 102 2.3.4 From Field Internal Symmetry to Current Conservation .................................................................... 106 2.3.5 From Poisson Parentheses to Quantum Commutators and Heisenberg Picture ............................. 108 2.4 Bra-ket (Dirac) Formalism ............................................................115 2.4.1 Vectors in Hilbert Space....................................................115 2.4.2 Linear Operators in Hilbert Space ....................................118 2.4.3 Spectral Representations of Vectors and Operators ......... 126 2.4.4 Coordinate and Momentum Representations ................... 136 2.4.5 Energy Representation ..................................................... 148 2.4.6 Heisenberg Matrix Quantum Mechanics: The Harmonic Oscillator ................................................. 151 2.5 Conclusion ................................................................................... 159 Keywords .............................................................................................. 160 References ............................................................................................. 161 Author’s Main Reference ............................................................. 161 Further Readings .......................................................................... 161 ABSTRACT Although equivalent to certain points, various approaches of quantum phenomena may add new insights and perspective on studied systems either in isolate or interaction states. Here, the main formalisms are reviewed and illustrated, especially those making the ground-nous of the quantum mechanics itself, i.e., the undulatory quantum mechanics, while introducing the quantum propagator (Green function) concept, the semi-classical approach based on Euler-Lagrange and Hamiltonian, while the bra-ket Dirac picture opens the new perspectives of further developing quantum theory by abstract yet logical and formalized reasoning.

Abraham Klein, Ching‐Teh Li

Abraham Klein

The WKB approximation for bound states is derived from Heisenberg matrix mechanics. It is shown that the result can be obtained from the Bohr–Sommerfeld quantization of a suitably chosen solution of the classical equations of motion. It is also derived from the commutation relation and from a quantum form of Hamilton’s variational principle.