Fundamentals of Physics, II - Audio

The various postulates of quantum mechanics treated in previous lectures are reviewed and summarized. The uncertainty principle is again discussed and a new one between energy and time is introduced. The quantum mechanical behavior of an electron in a hydrogen atom is described. The principles of quantum mechanics are then generalized to describe two or more quantum particles. It is shown that identical particles have to be bosons or fermions, the latter obeying the Pauli exclusion principle, which in turn is key to explaining the periodic table.

The time-dependent Schrödinger Equation is introduced as a powerful analog of Newton's second law of motion that describes quantum dynamics. It is shown how given an initial wave function, one can predict the future behavior using Schrödinger's Equation. The special role of stationary states (states of definite energy) is discussed.

The allowed energy states of a free particle on a ring and a particle in a box are revisited. A scattering problem is studied to expose more quantum wonders: a particle can tunnel into the classically forbidden regions where kinetic energy is negative and a particle incident on a barrier with enough kinetic energy to go over it has a nonzero probability to bounce back.

It is shown how to extract the odds for getting different values of momentum from a generic wave function by writing it as a sum over functions of definite momentum. A recipe is given for finding states of definite energy, which requires solving a differential equation that depends on what potential the particle is experiencing. The particle in a box is considered and the allowed energies derived.

The fact that the wave function provides the complete description of a particle's location and momentum is emphasized. Measurement collapses the wave function into a spike located at the measured value. The quantization of momentum for a particle on a ring is deduced.

Lecture begins with a detailed review of the double slit experiment with electrons. The fate of an electron traversing the double slit is determined by a wave putting an end to Newtonian mechanics. The momentum and position of an electron cannot both be totally known simultaneously. The wave function is used to describe a probability density function for an electron. Heuristic arguments are given for the wave function describing a particle of definite momentum.

The double slit experiment, which implies the end of Newtonian Mechanics is described. The de Broglie relation between wavelength and momentum is deduced from experiment for photons and electrons. The photoelectric effect and Compton scattering, which provided experimental support for Einstein's photon theory of light are reviewed. The wave function is introduced along with the probability interpretation. The uncertainty principle is shown arise from the fact that the particle's location is determined by a wave and that waves diffract when passing a narrow opening.

Young's double slit experiment shows clearly that light is a wave. (In order to observe the wave behavior of light, the slit size and separation should be comparable or smaller than the wavelength of light.) Interference is described using real and complex numbers (in anticipation of quantum mechanics). Grating and crystal diffraction are analyzed.

Ray diagrams are used to investigate the behavior of light incident on mirrors and lenses. The principle of least time is used to show that all rays from an object in front of a concave mirror focus on the image point if they are not too far from the axis. The experiments describing the breakdown of geometric optics are discussed.

Geometric optics is discussed as an approximation to wave theory when the wavelength is very small compared to other lengths in the problem (such as the size of openings). Many results of geometric optics involving reflection, refraction (mirrors and lenses) are derived in a unified way using Fermat's Principle of Least Time.

The physical meaning of the components of the wave equation and their applications are discussed. The power carried by the wave is derived. The fact that unlike Newton's laws, Maxwell's equations are already consistent with relativity is discussed. The existence of magnetism is deduced from a thought experiment using relativity.

Waves on a string are reviewed and the general solution to the wave equation is described. Maxwell's equations in their final form are written down and then considered in free space, away from charges and currents. It is shown how to verify that a given set of fields obeys Maxwell's equations by considering them on infinitesimal cubes and loops. A simple form of the solutions is assumed and the parameters therein fitted using MEs. The wave equation follows, along with the wave speed equal to that of light (3 x 108), suggesting (correctly) that light is an electromagnetic wave. The vector relationship between the electric field, the magnetic field and the direction of wave propagation is described.

The mathematics underlying LCR circuit theory for AC currents is discussed. Complex numbers are used to convert differential equations to algebraic equations. The notion of impedance is introduced. The radio is used to illustrate the concepts of resonance and variable capacitance. The body of classical electromagnetism treated so far is reviewed and summarized. The displacement current is introduced, leading to the complete Maxwell equations.

The electric effect of a changing magnetic field is described using Faraday's Law. The direction of the current so generated is given by Lenz's Law. The operation and energy accounting of the generator are described. The concept of inductance is introduced. The Betatron is described as an example of Faraday's law. Self and mutual inductance are introduced. The energy density in a magnetic field is derived.

Like capacitors, inductors act as energy storage devices in circuits. The relationship between voltage, inductance and current in a variety of circuits with DC voltages is described.

After a description of more complicated electric circuits, the basic ideas underlying magnetism are discussed and the relationship between electrical charges and magnetic fields is explored. Magnetism is caused and experienced only by moving charges. The Lorentz force on a charge is described and used to deduce the force on a current carrying wire. The cyclotron and velocity selector are described.

Ampere's law is used to find the magnetic field generated by currents in highly symmetric geometries like the infinitely long wire and the solenoid. It is shown how magnetism can be used to convert macroscopic mechanical energy to do microscopic electrical work. Lenz's and Faraday's Laws are introduced. The latter says that a changing magnetic field generates a non-conservative electric field.

The mechanism by which electric currents produce a magnetic field (Law of Biot-Savart) is discussed in greater detail. The field due to a single loop and an infinite wire are computed. Ampere's Law is derived. The operation of the DC electric motor is used to illustrate the torque generated on moving charges in a magnetic field.

Lecture begins with a discussion of electric potential distribution in conductors. Image charges are introduced and exploited. Capacitance is explained in greater detail and illustrated using the parallel plate capacitor. The energy stored in the electric field is derived. The forces acting on an electric current flowing through a conducting wire are examined. The RC circuits and its energetics are discussed. The EMF due to a battery is explained.

The electric potential is defined for the electric field. It is introduced as an integral of the electric field making the field the derivative of the potential. After discussing the ideas of electric potential and field as presented in the previous lecture, the concept of capacitance is introduced as a means of storing charge and energy.

The law of conservation of energy is reviewed using examples drawn from Newtonian mechanics. The work-energy theorem is derived from first principles and used to initiate a discussion of the vector calculus underlying the law of conservation of energy.

Lecture begins with a recap of Gauss's Law, its derivation, its limitation and its applications in deriving the electric field of several symmetric geometries—like the infinitely long wire. The electrical properties of conductors and insulators are discussed. Multiple integrals are briefly reviewed.

The electric field is discussed in greater detail and field due an infinite line charge is computed. The concepts of charge density and electric flux are introduced and Gauss's Law, which relates the two, is derived. It is applied to the study of the electric field generated by a spherical charge distribution.

The electric field is introduced as the mediator of electrostatic interactions: objects generate the field which permeates all of space, and charged objects in the field experience a force with magnitude proportional to their charge. Several instructive examples are given, including the field of an electric dipole and the notion of the electric dipole and dipole moment. The notion of field lines is introduced.

The second half of the course begins with a discussion of electricity. The concept of charge is introduced, and the properties of electrical forces are compared with those of other familiar forces, such as gravitation. Coulomb's Law, along with the principle of superposition, allows for the calculation of electrostatic forces from a given charge distribution.