Atom and Wave Particle Duality

Bohr Atom:

  • classical physics fails to describe the properties of atoms, Planck’s constant served to bridge the gap between the classical world and the new physics
  • Bohr proposed a quantized shell model for the atom using the same basic structure as Rutherford, but restricting the behavior of electrons to quantized orbits
Perhaps the foremost scientists of the 20th century was Niels Bohr, the first to apply Planck’s quantum idea to problems in atomic physics. In the early 1900’s, Bohr proposed a quantum mechanical description of the atom to replace the early model of Rutherford.In 1913 Bohr proposed his quantized shell model of the atom to explain how electrons can have stable orbits around the nucleus. The motion of the electrons in the Rutherford model was unstable because, according to classical mechanics and electromagnetic theory, any charged particle moving on a curved path emits electromagnetic radiation; thus, the electrons would lose energy and spiral into the nucleus. To remedy the stability problem, Bohr modified the Rutherford model by requiring that the electrons move in orbits of fixed size and energy. The energy of an electron depends on the size of the orbit and is lower for smaller orbits. Radiation can occur only when the electron jumps from one orbit to another. The atom will be completely stable in the state with the smallest orbit, since there is no orbit of lower energy into which the electron can jump.

Bohr’s starting point was to realize that classical mechanics by itself could never explain the atom’s stability. A stable atom has a certain size so that any equation describing it must contain some fundamental constant or combination of constants with a dimension of length. The classical fundamental constants–namely, the charges and the masses of the electron and the nucleus–cannot be combined to make a length. Bohr noticed, however, that the quantum constant formulated by the German physicist Max Planck has dimensions which, when combined with the mass and charge of the electron, produce a measure of length. Numerically, the measure is close to the known size of atoms. This encouraged Bohr to use Planck’s constant in searching for a theory of the atom.

  • Bohr’s calculation produce an accurate map of the hydrogen atom energy levels
  • changes in electron orbits requires the release or gain of energy in the form of photons
  • Bohr’s atom perfectly explains the spectra in stars as gaps due to the absorption of photons of particular wavelengths that match the electron orbits of the various elements
  • larger formulations explain all the properties outlined by Kirchoff’s laws
Planck had introduced his constant in 1900 in a formula explaining the light radiation emitted from heated bodies. According to classical theory, comparable amounts of light energy should be produced at all frequencies. This is not only contrary to observation but also implies the absurd result that the total energy radiated by a heated body should be infinite. Planck postulated that energy can only be emitted or absorbed in discrete amounts, which he called quanta (the Latin word for “how much”). The energy quantum is related to the frequency of the light by a new fundamental constant, h. When a body is heated, its radiant energy in a particular frequency range is, according to classical theory, proportional to the temperature of the body. With Planck’s hypothesis, however, the radiation can occur only in quantum amounts of energy. If the radiant energy is less than the quantum of energy, the amount of light in that frequency range will be reduced. Planck’s formula correctly describes radiation from heated bodies. Planck’s constant has the dimensions of action, which may be expressed as units of energy multiplied by time, units of momentum multiplied by length, or units of angular momentum. For example, Planck’s constant can be written as h = 6.6×10-34 joule seconds.Using Planck’s constant, Bohr obtained an accurate formula for the energy levels of the hydrogen atom. He postulated that the angular momentum of the electron is quantized–i.e., it can have only discrete values. He assumed that otherwise electrons obey the laws of classical mechanics by traveling around the nucleus in circular orbits. Because of the quantization, the electron orbits have fixed sizes and energies. The orbits are labeled by an integer, the quantum number n.

With his model, Bohr explained how electrons could jump from one orbit to another only by emitting or absorbing energy in fixed quanta. For example, if an electron jumps one orbit closer to the nucleus, it must emit energy equal to the difference of the energies of the two orbits. Conversely, when the electron jumps to a larger orbit, it must absorb a quantum of light equal in energy to the difference in orbits. The Bohr model basically assigned discrete orbits for the electron, multiples of Planck’s constant, rather than allowing a continuum of energies as allowed by classical physics.

The power in the Bohr model was its ability to predict the spectra of light emitted by atoms. In particular, its ability to explain the spectral lines of atoms as the absorption and emission of photons by the electrons in quantized orbits.

  • Heisenberg and Schroedinger formalize Bohr’s model and produce quantum mechanics
  • quantum mechanics is an all encompassing science that crosses over into many fields
Our current understanding of atomic structure was formalized by Heisenberg and Schroedinger in the mid-1920’s where the discreteness of the allowed energy states emerges from more general aspects, rather than imposed as in Bohr’s model. The Heisenberg/Schroedinger quantum mechanics have consistent fundamental principles, such as the wave character of matter and the incorporation of the uncertainty principle.In principle, all of atomic and molecular physics, including the structure of atoms and their dynamics, the periodic table of elements and their chemical behavior, as well as the spectroscopic, electrical, and other physical properties of atoms and molecules, can be accounted for by quantum mechanics => fundamental science.

de Broglie Matter Waves:

 

  • early quantum physics did not ask the question of `why’ quantum effects are found in the microscopic world
Perhaps one of the key questions when Bohr offered his quantized orbits as an explanation to the UV catastrophe and spectral lines is, why does an electron follow quantized orbits? The response to this question arrived from the Ph.D. thesis of Louis de Broglie in 1923. de Broglie argued that since light can display wave and particle properties, then perhaps matter can also be a particle and a wave too.
 

  • One way of thinking of a matter wave (or a photon) is to think of a wave packet. Normal waves look with this:
 

  • having no beginning and no end. A composition of several waves of different wavelength can produce a wave packet that looks like this:
  • the wave packet interpretation requires the particle to have no set position
  • momentum of a particle is proportional to the wavelength of the particle
So a photon, or a free moving electron, can be thought of as a wave packet, having both wave-like properties and also the single position and size we associate with a particle. There are some slight problems, such as the wave packet doesn’t really stop at a finite distance from its peak, it also goes on for every and every. Does this mean an electron exists at all places in its trajectory?de Broglie also produced a simple formula that the wavelength of a matter particle is related to the momentum of the particle. So energy is also connected to the wave property of matter.
  • Lastly, the wave nature of the electron makes for an elegant explanation to quantized orbits around the atom. Consider what a wave looks like around an orbit, as shown below
  • only certain wavelengths will fit into orbit, so quantiziation is due to wavelike nature of particles
The electron matter wave is both finite and unbounded (remember the 1st lecture on math). But only certain wavelengths will `fit’ into an orbit. If the wavelength is longer or shorter, then the ends do not connect. Thus, de Broglie explains the Bohr atom in that on certain orbits can exist to match the natural wavelength of the electron. If an electron is in some sense a wave, then in order to fit into an orbit around a nucleus, the size of the orbit must correspond to a whole number of wavelengths.
 

  • wavelike nature also means that a particles existence is spread out, a probability field
Notice also that this means the electron does not exist at one single spot in its orbit, it has a wave nature and exists at all places in the allowed orbit. Thus, a physicist speaks of allowed orbits and allowed transitions to produce particular photons (that make up the fingerprint pattern of spectral lines). And the Bohr atom really looks like the following diagram:

  • the idea of atoms being solid billiard ball type objects fails with quantum physics
  • quantum effects fade on larger scales since macroscopic objects have high momentum values and therefore small wavelengths
While de Broglie waves were difficult to accept after centuries of thinking of particles are solid things with definite size and positions, electron waves were confirmed in the laboratory by running electron beams through slits and demonstrating that interference patterns formed.How does the de Broglie idea fit into the macroscopic world? The length of the wave diminishes in proportion to the momentum of the object. So the greater the mass of the object involved, the shorter the waves. The wavelength of a person, for example, is only one millionth of a centimeter, much to short to be measured. This is why people don’t `tunnel’ through chairs when they sit down.

Probability Fields:

  • wave interpretation requires a statistical or probability mathematical description of the position of a particle
  • where wave represents the probability of finding the particle at a particular point
The idea that an electron is a wave around the atom, instead of a particle in orbit begs the question of `where’ the electron is at any particular moment. The answer, by experimentation, is that the electron can be anywhere around the atom. But ‘where’ is not evenly distributed. The electron as a wave has a maximum chance of being observed where the wave has the highest amplitude. Thus, the electron has the highest probability to exist at a certain orbit.Where probability is often used in physics to describe the behavior of many objects, this is the first instance of an individual object, an electron, being assigned a probability for a Newtonian characteristic such as position. Thus, an accurate description of an electron orbit is one where we have a probability field that surrounds the nucleus, as shown below:

 

  • for higher orbits the probability field becomes distorted
For more complicated orbits, and higher electron shells, the probability field becomes distorted by other electrons and their fields, like the following example:

 

  • meaning of existence has an elusive nature in the quantum world
Thus, for the first time, the concept of existence begins to take on an elusive character at the subatomic level.
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