Schrödinger's original struggles with a complex wave function

2020 ◽  
Vol 88 (6) ◽  
pp. 433-438
Author(s):  
Ricardo Karam
Keyword(s):  
Wave Function ◽  
Complex Wave

scholarly journals Quaternionic wave function

2019 ◽  
Vol 34 (02) ◽  
pp. 1950001 ◽  
Author(s):  
Pavel A. Bolokhov
Keyword(s):  
Wave Function ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Variational Calculus ◽  
Nonrelativistic Limit ◽  
Natural Choice ◽  
Complex Wave ◽  
Spinor Formalism ◽  
Function Language

We argue that quaternions form a natural language for the description of quantum-mechanical wave functions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper, we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the nonrelativistic limit to find the Schrödinger’s equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the noninteracting nonrelativistic limit, the quaternionic description effectively reduces to the regular complex wave function language. We provide an easy-to-use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell’s equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.

scholarly journals Synthesis and Characterization of a Cobalt(III) Corrole with an Sbound DMSO Ligand

2021 ◽  
Author(s):  
Nicolás I. Neuman ◽  
Arijit Singha Hazari ◽  
Julia Beerhues ◽  
Fabio Doctorovich ◽  
Santiago E. Vaillard ◽  
...  
Keyword(s):  
Wave Function ◽  
Electrochemical Properties ◽  
Spectroscopic Data ◽  
Synthesis And Characterization ◽  
Frontier Orbitals ◽  
Complex Wave

A cobalt(III) corrole complex with an apical dmso ligand is presented. Crystallographic and spectroscopic data are used to unequivocally establish the dmso(O) vs. dmso(S) coordination in this complex. Wave function based methods were used to calculate the frontier orbitals for the complexes with O-bound and S-bound dmso ligands. Electrochemical properties of the complex is presented as well.<br>

scholarly journals A novel view on successive quantizations, leading to increasingly more “miraculous” states

2019 ◽  
Vol 34 (23) ◽  
pp. 1950186 ◽  
Author(s):  
Matej Pavšič
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Wave Function ◽  
Wave Packet ◽  
Time Derivative ◽  
Point Particle ◽  
Order Equation ◽  
Quantum Field ◽  
First Order ◽  
Complex Wave

A series of successive quantizations is considered, starting with the quantization of a non-relativistic or relativistic point particle: (1) quantization of a particle’s position, (2) quantization of wave function, (3) quantization of wave functional. The latter step implies that the wave packet profiles forming the states of quantum field theory are themselves quantized, which gives new physical states that are configurations of configurations. In the procedure of quantization, instead of the Schrödinger first-order equation in time derivative for complex wave function (or functional), the equivalent second-order equation for its real part was used. In such a way, at each level of quantization, the equation a quantum state satisfies is just like that of a harmonic oscillator, and wave function(al) is composed in terms of the pair of its canonically conjugated variables.

scholarly journals The Domain of the Quantum Matter Wave

2001 ◽  
Vol 56 (1-2) ◽  
pp. 216-219
Author(s):  
B. Mills
Keyword(s):  
Wave Function ◽  
Particle System ◽  
Physical Space ◽  
Quantum Systems ◽  
Matter Wave ◽  
Quantum Matter ◽  
Multiple Particles ◽  
Complex Wave ◽  
The Individual ◽  
Individual Particles

AbstractIn quantum mechanics a particle is represented by a complex wave function. This is similar to the classical description of a distributed particle by a real valued density function. However, while classically the combined density of multiple particles can be decomposed into individual densities; the wave function of a multi-particle system can not always be decomposed into wave functions for the individual particles. Hence the classical assumption that density is a property of physical space, which is only possible because of the special topology of classical configuration space, does not work for quantum systems.

scholarly journals Uncertainty Relations in the Madelung Picture

Entropy ◽  
2021 ◽  
Vol 24 (1) ◽  
pp. 20
Author(s):  
Moise Bonilla-Licea ◽  
Dieter Schuch
Keyword(s):  
Wave Function ◽  
Quantum System ◽  
Momentum Space ◽  
Coherent States ◽  
Quantum Mechanical ◽  
Uncertainty Relations ◽  
Hydrodynamic Equations ◽  
Generalized Coherent States ◽  
Complex Wave ◽  
Classical Hydrodynamic

Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position–momentum correlations. This will be illustrated explicitly using generalized coherent states as examples.

scholarly journals Synthesis and Characterization of a Cobalt(III) Corrole with an Sbound DMSO Ligand

2021 ◽  
Author(s):  
Nicolás I. Neuman ◽  
Arijit Singha Hazari ◽  
Julia Beerhues ◽  
Fabio Doctorovich ◽  
Santiago E. Vaillard ◽  
...  
Keyword(s):  
Wave Function ◽  
Electrochemical Properties ◽  
Spectroscopic Data ◽  
Synthesis And Characterization ◽  
Frontier Orbitals ◽  
Complex Wave

A cobalt(III) corrole complex with an apical dmso ligand is presented. Crystallographic and spectroscopic data are used to unequivocally establish the dmso(O) vs. dmso(S) coordination in this complex. Wave function based methods were used to calculate the frontier orbitals for the complexes with O-bound and S-bound dmso ligands. Electrochemical properties of the complex is presented as well.<br>

Analytic integration of contrast transfer functions

1988 ◽  
Vol 46 ◽  
pp. 822-823
Author(s):  
Peter Rez
Keyword(s):  
Wave Function ◽  
Transfer Function ◽  
Transfer Functions ◽  
Spherical Aberration ◽  
Continuous Distribution ◽  
Objective Lens ◽  
Direction Parallel ◽  
Scattering Angle ◽  
Analytic Integration ◽  
Image Position

In high resolution microscopy the image amplitude is given by the convolution of the specimen exit surface wave function and the microscope objective lens transfer function. This is usually done by multiplying the wave function and the transfer function in reciprocal space and integrating over the effective aperture. For very thin specimens the scattering can be represented by a weak phase object and the amplitude observed in the image plane is1where fe (Θ) is the electron scattering factor, r is a postition variable, Θ a scattering angle and x(Θ) the lens transfer function. x(Θ) is given by2where Cs is the objective lens spherical aberration coefficient, the wavelength, and f the defocus.We shall consider one dimensional scattering that might arise from a cross sectional specimen containing disordered planes of a heavy element stacked in a regular sequence among planes of lighter elements. In a direction parallel to the disordered planes there will be a continuous distribution of scattering angle.

Atomic Imaging of Crystals using Large-Angle Electron Scattering in STEM

1990 ◽  
Vol 48 (1) ◽  
pp. 74-75
Author(s):  
D.E. Jesson ◽  
S. J. Pennycook
Keyword(s):  
Wave Function ◽  
Electron Scattering ◽  
Numerical Solutions ◽  
Large Angle ◽  
Real Space ◽  
High Angle ◽  
Analytical Treatment ◽  
Order Zero ◽  
Experimental Conditions ◽  
Ewald Sphere

It is well known that conventional atomic resolution electron microscopy is a coherent imaging process best interpreted in reciprocal space using contrast transfer function theory. This is because the equivalent real space interpretation involving a convolution between the exit face wave function and the instrumental response is difficult to visualize. Furthermore, the crystal wave function is not simply related to the projected crystal potential, except under a very restrictive set of experimental conditions, making image simulation an essential part of image interpretation. In this paper we present a different conceptual approach to the atomic imaging of crystals based on incoherent imaging theory. Using a real-space analysis of electron scattering to a high-angle annular detector, it is shown how the STEM imaging process can be partitioned into components parallel and perpendicular to the relevant low index zone-axis.It has become customary to describe STEM imaging using the analytical treatment developed by Cowley. However, the convenient assumption of a phase object (which neglects the curvature of the Ewald sphere) fails rapidly for large scattering angles, even in very thin crystals. Thus, to avoid unpredictive numerical solutions, it would seem more appropriate to apply pseudo-kinematic theory to the treatment of the weak high angle signal. Diffraction to medium order zero-layer reflections is most important compared with thermal diffuse scattering in very thin crystals (<5nm). The electron wave function ψ(R,z) at a depth z and transverse coordinate R due to a phase aberrated surface probe function P(R-RO) located at RO is then well described by the channeling approximation;

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