Chapter 14: Heisenberg's Equations of Motion for Tunneling

Volodymyr Krasnoholovets

This chapter discusses important peculiarities of quantum systems by using the submicroscopic concept. It focuses on the derivation of the Schrodinger equation, the physical interpretation of the wave ψ-function and looks at the interior of other well-established notions of quantum mechanics. The chapter considers the basic notions of quantum mechanics, such as tunneling and Heisenberg's uncertainty principle. It discusses inertons emitted by the particle at its motion. The inerton cloud spreads out to the distance Λ from the particle in the transverse directions. The inerton cloud is able to interact with obstacles and transfers the appropriate information to the particle. Since the inerton cloud carries the particle's mass, the fluid element in question can be specified with both mass and volume. The chapter considers the motion of inertons emitted by the moving particle. The mother-particle, which creates inertons, moves in ajerky manner caused by its continuous periodic stopping, and its inerton cloud behaves similarly.

D. Dubbers, H.‐J. Stöckmann

Guoping Zhang, Mingsu Si, Thomas F. George

This textbook provides ample opportunities for practice and real experimental demonstrations. Conceptual understanding and mastering key techniques are enhanced by rigorous derivations, numerous worked examples, more than 300 exercises, about 150 problems and 16 computer codes. The preface summarizes all of the key concepts and formulas, along with a detailed schedule for teaching. The first three chapters introduce the quantum idea, wave-particle duality, operators and measurement. The Noether theorem is invoked to introduce the Schrödinger equation, followed by applications to infinite and finite quantum wells, quantum tunneling, harmonic oscillators, Heisenberg equation of motion, uncertainty principle, blackbody radiation and photoelectric effect. Chapters 4 and 5 are on angular momentum, the hydrogen atom and time-independent approximate methods. Chapters 6 and 7 are on spin and time-dependent perturbation theory. Chapters 8, 9 and 10 are on molecular orbitals, energy bands, quantum transport, scanning tunneling microscopy, lattice vibrations, Berry phase and quantum computing. The book is intended for a one-semester or one-year course and is also appropriate for researchers in related fields.

Gordon Baym

The position operator in the Heisenberg representation obeys the usual classical equation of motion. The similarity between the Heisenberg equations of motion and the classical Hamiltonian equations of motion arises from the fact that the commutator of the operators for the physical quantities plays a similar role in quantum mechanics, to that of the Poisson bracket of the classical quantities A and B, in classical mechanics. Generally, the Heisenberg equations of motion are more difficult to solve than the corresponding classical equations because of the lack of commutation of quantum mechanical operators. The chapter explains the notion of a time-ordered product of operators, and then can be written much more compactly.