Physical Basis of Quantum Mechanics

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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 will discuss the experimental evidence that led to the development of quantum mechanics, as well as some of the basic principles of the theory. In quantum mechanics, systems are described in terms of wave functions, which specify the probability of finding a system in a given state. This is in contrast to classical physics, which describes systems in terms of well-defined positions and velocities of particles. While many classical systems, such as fluids, are described in terms of statistical theories, said theories are only convenient approximations to the underlying deterministic theories that describe the individual particles that make up the system. Quantum mechaincs, on the other hand, is not a statistical approximation to some underlying deterministic theory. It proposes an inherent randomness to the behavior of physical systems, and all of its laws are written in terms of wave functions rather than definite properties of particles.

Due to this probabilistic nature, one can no longer think of particles as having definite positions and momenta. These become observables, which are linear operators that act on wave functions, in quantum mechanics. These operators allow one to extract from the wave function all of the possible outcomes of a given measurement and the probabilities of said outcomes. We will see, in fact, that quantities represented by non-commuting observables, such as position and momentum, cannot, even in principle, be simultaneously measured to infinite precision; this is known as the Heisenberg Uncertainty Principle.

The concept of a wave function leads to some of the basic ideas of quantum mechanics, such as wave-particle duality and the Principle of Complementarity. Wave-particle duality is the idea that particles possess a wave-like nature, and vice versa. The Principle of Complementarity states that it is impossible to observe both particle-like and wave-like behaviors in the same experiment.

The development of quantum mechanics started with attempts to understand the black-body radiation spectrum. If one attempts to use classical physics to describe this spectrum, one finds that the spectral intensity of the radiation diverges at short wavelengths, a phenomenon known as the ultraviolet catastrophe. Not only would this have disturbing implications if it actually happened, but it is in contradiction with experimental data, which shows that the spectral intensity decreases in the short wavelength limit. The German physicist, Max Planck, proposed that the radiation emitted by a black-body is quantized, that it is emitted in discrete packets called photons. This assumption is known as the Quantum Hypothesis. Under this assumption, he arrived at an expression for the spectral intensity that agreed with experiment.

Later, Albert Einstein proposed that this quantization of radiation could explain observations regarding the photoelectric effect. One would expect, from the classical picture of radiation as electromagnetic waves, that light of any frequency, if sufficiently intense, could liberate an electron from a metal. However, it was observed that only radiation of at least a certain frequency would result in a photocurrent. The way in which the idea of photons can explain these observations is that an individual photon would need to have sufficient energy to liberate a single electron from the sample. Since the energy of a photon is proportional to its frequency, this would mean that only a photon of sufficiently high frequency will result in a photocurrent, in agreement with experiment.

Additional experimental justification for the ideas of quantum mechanics came from the double slit experiment, which demonstrated that particles have a wave-like nature, and the Stern-Gerlach experiment, which demonstrated the quantization of angular momentum (and later led to the concept of spin, which is also quantized).

While quantum mechaincs is fundamentally different from classical mechanics, we know that classical mechanics describes sufficiently large objects very well; it is only at very small length scales (on the order of a nanometer) that classical mechanics begins to break down. This leads us to the Correspondence Principle, which states that quantum mechanics must reduce to classical mechanics at sufficiently large length and energy scales. It will, in fact, later become clear that the formalism that we develop for quantum mechanics does reduce to classical mechanics in the proper limit.

Chapter Contents

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