Schrödinger Equation and Its Application

Shu Hotta

Shu Hotta

Jing‐Bo Chen

Multisymplectic geometry for the Schrödinger equation in quantum mechanics is presented. This formalism of multisymplectic geometry provides a concise and complete introduction to the Schrödinger equation. The Schrödinger equation, its associated energy and momentum evolution equations, and the multisymplectic form are derived directly from the variational principle. Some applications are also explored.

Yisong Yang

Abstract The essence of quantum mechanics is the Schrödinger equation. Thus, the path to quantum mechanics lies in an understanding of the birth of the Schrödinger equation. This chapter derives the Schrodinger equation based on the photoelectric effect, which has found a wide range of applications in today’s electronics. It then discusses the Schrodinger equation’s statistical interpretations and applications. It goes on to present the quantum many-body problem, which is of contemporary research interest, as well as its many applications. Next it investigates the Hartree-Fock method and the Thomas-Fermi approach, as well as provides a discussion of the Rydberg formula. Finally, the chapter looks at and unpacks the density functional theory.

Paul Bracken

The first equation in a hierarchy of equations due to Vlasov is studied. The equation can be described as a Schrödinger equation for the probabilistic description of a system. It has applications in other areas as well. A physical meaning can be assigned to the phase of the wavefunction. The approach allows construction of solutions of the Schrödinger equation from known solutions and conversely much like a Bäcklund transformation. Finally, the process of introducing a potential into the Schrödinger equation is generalized to the case of Bohmian mechanics.