ISSYP Virtual

A “derivation” of the Schrodinger wave equation based on simple calculus.
Learning Outcomes:
• How to express the de Broglie wave of a free particle, i.e. a complex traveling wave, in terms of the particle’s energy and momentum, and how to differentiate this wave with respect to its space and time variables (x and t).
• How to combine the above mathematical results with the Newtonian expression for the total energy of a particle to get Schrodinger’s wave equation.
• Dirac’s extension of these ideas to Einstein’s expression for the total energy of a particle: introduction to spin, antimatter, and the Standard Model of particle physics.

The de Broglie waves we have been using thus far were assumed to be real functions; we discuss why this is wrong and how to fix the problem.
Learning Outcomes:
• Understanding why there is a serious flaw with using real de Broglie waves, and how using a complex wave (one with both a real and an imaginary part) solves the problem.
• Understanding how the de Broglie wave corresponding to a free particle is like a moving corkscrew, with a magnitude that is uniform across space and constant in time.
• When right- and left-travelling de Broglie waves (“corkscrews”) are added, as happens for a particle in a box, we get a complex standing wave whose magnitude is constant in time.

Learning Outcomes:
• How the complex standing wave states of an electron in a one-dimensional box are “stationary states” in that the electron probability pattern is static (not changing with time).
• However, if the electron is put in a superposition of two such stationary states (with different energies), its probability pattern is not static, but rather oscillates back and forth; understanding how this oscillation is connected with photon emission and absorption.
• Understanding how these ideas apply to the atom, and how a LASER works (Light Amplification by Stimulated Emission of Radiation).

Understanding the zero point energy of the quantum harmonic oscillator as a consequence of the Heisenberg Uncertainty Principle.
Learning Outcomes:
• Understanding why the minimum energy of a ball in a bowl must be greater than zero based on the Heisenberg Uncertainty Principle.
• How the Heisenberg Uncertainty Principle adds a purely quantum mechanical kinetic energy to the ball, in addition to its classical potential energy.
• Understanding graphically how the total energy – the sum of the classical potential energy and the new quantum kinetic energy – has a minimum that is greater than zero: the zero point energy.

A discussion of how the zero point energy of atoms is what makes possible their existence in our universe – atoms are purely quantum mechanical objects.
Learning Outcomes:
• Continuation of QM-9: A calculus-based derivation of the zero point energy of the quantum harmonic oscillator.
• How our previous understanding of energy quantization and zero point energy can be applied also to the hydrogen atom.
• Why classical atoms cannot exist in our universe, and how the Heisenberg Uncertainty Principle and associated zero point energy stabilizes atoms, making their existence possible.

A discussion of the space and time axes of a moving observer and an introduction to length contraction.
Learning Outcomes:
• Understanding why and by how much a moving observer’s position axis is “tilted in time.”
• Understanding how a moving platform appears to a stationary observer.
• Beginning to understand the cause of length contraction.

A continuation of the SR-10 discussion on length contraction. Resolving Principle 2*.
Learning Outcomes:
• Relativity of simultaneity revisited – gaining a deeper understanding of what it means.
• A full understanding of the nature of length contraction based on relativity of simultaneity.
• Resolving a key paradox in special relativity: Principle 2*, introduced in SR-4. How it is possible to measure the same speed for the light whether you are running toward or away from a flashlight.

By applying our understanding of the quantum harmonic oscillator to the electromagnetic field we learn what a photon is, and are introduced to “quantum field theory” and the amazing “Casimir effect.”
Learning Outcomes:
• Understanding that classical electromagnetic waves bouncing around inside a mirrored box will exist as standing waves with only certain allowed frequencies.
• How each of these standing waves oscillates harmonically, and thus why – at the quantum level – their energies must be discrete, which is interpreted as the presence of a discrete number of photons.
• What the zero point energy of the electromagnetic field represents, and its relationship to a remarkable property of the quantum vacuum called the “Casimir effect.”

Taking our intuitive understanding of the quantum world gained by studying a particle in a one-dimensional box, we generalize to understand a quantum harmonic oscillator.
Learning Outcomes:
• Introduction to the classical physics of a ball rolling back and forth in a bowl, a simple example of a very important type of bounded motion called a “harmonic oscillator.”
• The quantization of allowed energies of a harmonic oscillator: even spacing between energy levels, and zero point energy.
• Being able to sketch the allowed wavefunctions and particle probability patterns of a quantum harmonic oscillator, including a new phenomenon called “tunnelling.”

A discussion of how to synchronize clocks that are separated in space, and how this leads to the relativity of simultaneity.
Learning Outcomes:
• Understanding that clock synchronization is a physical process, and exploring various methods of synchronization using spacetime diagrams.
• How to measure distance with a clock: the concept of radar ranging distance.
• A profound realization about the nature of spacetime: Events that are simultaneous for one observer might not be simultaneous for another.

Learning to use Minkowskian geometry to understand, very simply, a variety of aspects of Einstein’s spacetime.
Learning Outcomes:
• How a straight line is the longest path between two points in spacetime.
• How a light particle experiences space and time: its journey from one location in the universe to another involves zero spacetime distance, and is thus instantaneous!
• How Einstein’s special relativity has no difficulty handling accelerated observers.

Space obeys the rules of Euclidean geometry. Spacetime obeys the rules of a new kind of geometry called Minkowskian geometry.
Learning Outcomes:
• Triangles in spacetime obey a Pythagoras-like theorem, but with an unusual minus sign.
• The true nature of time as geometrical distance in spacetime.
• How to analyse and resolve the Twins’ Paradox using spacetime diagrams in combination with Minkowskian geometry.