classical mechanics

Examples of classical mechanics in the following topics:

• The Bohr Model of the Atom

  • Bohr suggested that electrons in hydrogen could have certain classical motions only when restricted by a quantum rule.
  • The laws of classical mechanics predict that the electron should release electromagnetic radiation while orbiting a nucleus (according to Maxwell's equations, accelerating charge should emit electromagnetic radiation).
  • He suggested that electrons could only have certain classical motions:
  • In these orbits, the electron's acceleration does not result in radiation and energy loss as required by classical electrodynamics.
  • The significance of the Bohr model is that the laws of classical mechanics apply to the motion of the electron about the nucleus only when restricted by a quantum rule.

• Overview of Temperature and Kinetic Theory

  • Classical mechanics defines the translational kinetic energy of a gas molecule as follows:
  • The distribution of the speeds (which determine the translational kinetic energies) of the particles in a classical ideal gas is called the Maxwell-Boltzmann distribution.
  • In kinetic theory, the temperature of a classical ideal gas is related to its average kinetic energy per degree of freedom Ek via the equation:

• The Wave Function

  • A wave function is a probability amplitude in quantum mechanics that describes the quantum state of a particle and how it behaves.
  • In quantum mechanics, a wave function is a probability amplitude describing the quantum state of a particle and how it behaves.
  • The laws of quantum mechanics (the Schrödinger equation) describe how the wave function evolves over time.
  • This figure shows some trajectories of a harmonic oscillator (a ball attached to a spring) in classical mechanics (A-B) and quantum mechanics (C-H).
  • This "energy quantization" does not occur in classical physics, where the oscillator can have any energy.

• Relativistic Kinetic Energy

  • In classical mechanics, the kinetic energy of an object depends on the mass of a body as well as its speed.
  • The classical kinetic energy of an object is related to its momentum by the equation:
  • At a low speed ($v << c$), the relativistic kinetic energy may be approximated well by the classical kinetic energy.
  • Thus, the total energy can be partitioned into the energy of the rest mass plus the traditional classical kinetic energy at low speeds.
  • Compare classical and relativistic kinetic energies for objects at speeds much less and approaching the speed of light

• Lasers

  • Principles of laser operation are largely based on quantum mechanics.
  • (One exception would be free-electron lasers, whose operation can be explained solely by classical electrodynamics. ) When an electron is excited from a lower-energy to a higher-energy level, it will not stay that way forever.
  • This is the mechanism of fluorescence and thermal emission.
  • However, an external photon at a frequency associated with the atomic transition can affect the quantum mechanical state of the atom .

• Simple Machines

  • They can be described as the simplest mechanisms that use mechanical advantage (or leverage) to multiply force.
  • Usually, the term "simple machine" is referring to one of the six classical simple machines, defined by Renaissance scientists.
  • For example, a bicycle is a mechanism made up of wheels, levers, and pulleys.
  • The ratio of the output force to the input force is the mechanical advantage of the machine.
  • For instance, the mechanical advantage of a lever is equal to the ratio of its lever arms.

• Implications of Quantum Mechanics

  • Quantum mechanics has also strongly influenced string theory.
  • The application of quantum mechanics to chemistry is known as quantum chemistry.
  • Relativistic quantum mechanics can, in principle, mathematically describe most of chemistry.
  • A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than classical computers.
  • Explain importance of quantum mechanics for technology and other branches of science

• Atomic Structure

  • So far we have used classical and semi-classical approaches to understand how radiation interacts with matter.
  • We have generally treat the electrons (the lightest charged particle so the biggest emitter) classically and the radiation either classically or as coming in quanta (i.e. semi-classically).
  • We also derived some important relationships between how atoms emit and absorb radiation, but to understand atomic processes in detail we will have to treat the electrons quantum mechanically.
  • In quantum mechanics we characterize the state of a particles (or group of particles) by the wavefunction ($\Psi$).
  • We can imagine the operator $H$ as a matrix that multiplies the state vector $\psi$, so this equation is an eigenvalue equation with $E$ as the eigenvalue and $\psi$ as an eigenvector (or eigenfunction) of the matrix (or operator) $H$.The Hamiltonian classically is the sum of the kinetic energy and the potential energy of the particles.

• Philosophical Implications

  • Since its inception, many counter-intuitive aspects of quantum mechanics have provoked strong philosophical debates.
  • According to this interpretation, the probabilistic nature of quantum mechanics is not a temporary feature which will eventually be replaced by a deterministic theory, but instead must be considered a final renunciation of the classical idea of causality.
  • This is due to the quantum mechanical principle of wave function collapse.
  • One of the most bizarre aspect of the quantum mechanics is known as quantum entanglement.
  • Formulate the Copenhagen interpretation of the probabilistic nature of quantum mechanics

• Diffraction

  • In classical physics, the diffraction phenomenon is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings.
  • As physical objects have wave-like properties (at the atomic level), diffraction also occurs with matter and can be studied according to the principles of quantum mechanics.