Schrödinger Equation

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Quantum Mechanics A
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian \mathcal{H}, it describes how a state |\Psi\rangle evolves in time.
Basic Concepts and Theory of Motion
UV Catastrophe (Black-Body Radiation)
Photoelectric Effect
Stability of Matter
Double Slit Experiment
Stern-Gerlach Experiment
The Principle of Complementarity
The Correspondence Principle
The Philosophy of Quantum Theory
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
Linear Vector Spaces and Operators
Commutation Relations and Simultaneous Eigenvalues
The Schrödinger Equation in Dirac Notation
Transformations of Operators and Symmetry
Time Evolution of Expectation Values and Ehrenfest's Theorem
One-Dimensional Bound States
Oscillation Theorem
The Dirac Delta Function Potential
Scattering States, Transmission and Reflection
Motion in a Periodic Potential
Summary of One-Dimensional Systems
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

In this chapter, we introduce the Schrödinger equation, the most fundamental equation in quantum mechanics. In 1926, it had been well established by Rutherford that atoms are comprised of a dense, positively-charged nucleus which is surrounded by a region of negative charge. It had also been established by Thompson that the negatively-charged region was comprised of particles called electrons. In the period between the discovery of these basic atomic attributes and quantum theory, little was known as to why the electrons in atoms are not instantly pulled into the nucleus. After all, there is an electrostatic potential between the nucleus and the electrons, the latter of which are constantly accelerating and thus emitting radiation, losing energy in the process. What kept the electrons from falling inward, thereby causing the atom to collapse?

Austrian physicist Erwin Schrödinger confronted the problem, utilizing a new approach that was emerging at the time. He used the idea that matter in motion possesses wave-like attributes. This idea, which may seem unusual to those first exposed to it, was not based on pure speculation. In fact, much work was being produced, largely by Planck and Einstein, at this point which showed that light exists as both a particle and a wave, a concept that originated from experiments concerning the diffraction of light around barriers and black body radiation (the ultraviolet catastrophe). French physicist Louis Victor de Broglie had applied this particle-wave concept to matter and had devised an expression which relates the momentum of an object with the wavelength of its "matter wave".

Not only did Schrödinger utilize the idea of waves in his new theory, but also the idea of quantization. Danish physicist Niels Bohr had attempted to work with this idea after he realized that the emission spectrum of atomic hydrogen consisted of specific frequencies, rather than continuous bands thereof. This indicated to him that the electron in a hydrogen atom only makes certain energy transitions within the atom, meaning that the energy that the electron could possess was restricted to a discrete set of values - in other words, the energy of the electron was quantized. As it turns out, standing waves do in fact display a sort of quantization in that there can only exist integer amounts of antinodes and nodes along the wave.

Schrödinger's model for electron behavior is usually referred to as the wave-mechanical model. This model states that all of the possible positions which an electron can occupy can be represented by a wave. It does so by ascribing a specific function to the electron. Since the function is based on wave properties, it is referred to as a wave function. From this wave function, all information which one wishes to ascertain about the electron can be extracted. Wave functions will vary depending on the situation. The only conditions placed on them is that they must be continuous, single valued and square-integrable. The Schrödinger equation is the mathematical relation between an electron's wavefunction and the Hamiltonian describing the system. The Schrödinger equation is a very general equation, making it versatile in terms of its application to various types of electron behavior.

In the following sections, we will give a basic introduction to the mathematical details of the Schrödinger equation. We will also introduce the Heisenberg Uncertainty Principle and discuss some of its consequences.

Chapter Contents

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