Basic Concepts of Classical and Quantum Mechanics

Classical or Newtonian mechanics appeared on the 17^th century and was very successful in describing the physics of the macroscopic world. However, in the beginning of the 20^th century, experiments revealed a series of atomic and subatomic phenomena that could not be explained by the old theory. This triggered the development of Quantum mechanics.

In classical mechanics a system can be described by a number of dynamic variables. The goal is to be able to calculate the exact values of these variables at any given time and determine how these values evolve as a function of time t. Once the position x(t) of a particle within an one-dimensional system is known, a series of other variables such as the velocity ν(t), acceleration α(t), momentum p(t), potential V and kinetic energy K can be calculated. Newton's Law is the basis of classical mechanics:

F = mα or F = dp/dt

where F is the force acting on the particle and m the mass of the particle

Classical mechanics failed to describe an atomic system using the above principles, and further attempts and experimental observations led to the development of quantum theory. Given the new theory, a system can now be described by a state function Ψ and not a set of dynamic variables. This complicated wave function is the solution of Schrödinger's equation, the equivalent to Newton's Law in classical mechanics.

The statistical nature of the state function indicates the probability of finding a particle within a spatial range, in contrast to the Newtonian mechanics where a particle's position can be accurately defined.

Quantum mechanics also accepts the quantization of energy, momentum, etc, when the classical view can only accept a continuous range of values for each variable. In addition, a particle in Quantum mechanics contains an amount of energy even at ground state. This is due to the Heisenberg Uncertainty Principle, where the exact values of two complimentary variables can never be measured simultaneously. The energy at ground level for the classical mechanics is always zero.

Quantum Mechanics became fundamental in the modern world after having explained a series of phenomena:

• Blackbody radiation: The classical approach could not predict and explain the spectrum curve of the blackbody radiation in the UV frequencies. This is also known as the “UV catastrophe.” Max Planck assumed that the oscillators' energy takes discrete instead of continuous values. The quantization of energy gave a new prediction that matched the experimental values.
• Photoelectric effect: From a classical point of view, light behaves as an electromagnetic wave that induces the ejection of electrons from a metal's surface due to the oscillations caused by the incident wave. However, the experiment showed that the kinetic energy of the ejected electrons could not be calculated accurately by the wave theory. Einstein introduced the idea of quantized packets of energy; the photons and discovered that the electrons' kinetic energy is a function of the photon energy and a material's constant.
• Duality of wave/particle: Apart from photons, electrons can also behave as particles or waves, depending on the situation. This duality, accepted only by quantum mechanics, explained the interferences phenomena occurring in Young's Double Slit experiment.

Other phenomena such as the Tunneling Effect are only possible if we accept that the subatomic world is ruled by quantum laws.

The classical view is still successful when referring to large and low velocity systems. Atomic and subatomic systems can only be described by quantum mechanics. However the two theories are not irrelevant. Quantum mechanics converges with classical mechanics as the order of the system's magnitude in scope increases.

• “Classical mechanics and quantum mechanics: an elementary approach to the comparison of the two viewpoints”, L.Paolonia
• “Classical Mechanics vs Quantum Mechanics”, C.L. Tang, Cambrigde University Press
• “Blackbody Radiation, Photoelectric Effect, Wave-Particle Duality” U. Wisconsin, Physics 104, Fall 2005 (PDF of PowerPoint slides)