Matter wave

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In quantum mechanics, the concept of matter waves or de Broglie waves /dəˈbrɔɪ/ reflects the wave–particle duality of matter. The theory was proposed by Louis de Broglie in 1924 in his PhD thesis.[1] The de Broglie relations show that the wavelength is inversely proportional to the momentum of a particle and is also called de Broglie wavelength. Also the frequency of matter waves, as deduced by de Broglie, is directly proportional to the total energy E (sum of its rest energy and the kinetic energy) of a particle.[2]

Historical context[edit]


At the end of the 19th century, light was thought to consist of waves of electromagnetic fields which propagated according to Maxwell’s equations, while matter was thought to consist of localized particles (See history of wave and particle viewpoints). This division was challenged when, in his 1905 paper on the photoelectric effect, Albert Einstein postulated that light was emitted and absorbed as localized packets, or "quanta" (now called photons). These quanta would have an energy


where \scriptstyle \nu is the frequency of the light and h is Planck’s constant. In the modern convention, frequency is symbolized by f as is done in the rest of this article. Einstein’s postulate was confirmed experimentally by Robert Millikan and Arthur Compton over the next two decades. Thus it became apparent that light has both wave-like and particle-like properties. De Broglie, in his 1924 PhD thesis, sought to expand this wave-particle duality to all particles:


—De Broglie[3]

In 1926, Erwin Schrödinger published an equation describing how this matter wave should evolve—the matter wave equivalent of Maxwell’s equations—and used it to derive the energy spectrum of hydrogen. That same year Max Born published his now-standard interpretation that the square of the amplitude of the matter wave gives the probability of finding the particle at a given place. This interpretation was in contrast to De Broglie’s own interpretation, in which the wave corresponds to the physical motion of a localized particle.

de Broglie relations[edit]

Quantum mechanics[edit]


& \lambda = h/p\\
& f = E/h

where h is Planck's constant. The equation can be equivalently written as

& p = \hbar k\\
& E = \hbar \omega\\

using the definitions

In each pair, the second is also referred to as the Planck-Einstein relation, since it was also proposed by Planck and Einstein.

Special relativity[edit]

Using the relativistic momentum formula from special relativity

p = \gamma m_0v

allows the equations to be written as[4]

\begin{align}&\lambda = \frac {h}{\gamma m_0v} = \frac {h}{m_0v} \sqrt{1 - \frac{v^2}{c^2}}\\
& f = \frac{\gamma\,m_0c^2}{h} = \frac {m_0c^2}{h\sqrt{1 - \frac{v^2}{c^2}}}

where m0 is the particle's rest mass, v is the particle's velocity, γ is the Lorentz factor, and c is the speed of light in a vacuum. See below for details of the derivation of the de Broglie relations. Group velocity (equal to the particle's speed) should not be confused with phase velocity (equal to the product of the particle's frequency and its wavelength). In the case of a non-dispersive medium, they happen to be equal, but otherwise they are not.

Group velocity[edit]

Albert Einstein first explained the wave–particle duality of light in 1905. Louis de Broglie hypothesized that any particle should also exhibit such a duality. The velocity of a particle, he concluded then (but may be questioned today, see above), should always equal the group velocity of the corresponding wave. De Broglie deduced that if the duality equations already known for light were the same for any particle, then his hypothesis would hold. This means that



    &= \frac{p}{m},\\
    &= v.


Also in special relativity we find that

  v_g &= \frac{\partial E}{\partial p} = \frac{\partial}{\partial p} \left( \sqrt{p^2c^2+m^2c^4} \right),\\
    &= \frac{pc^2}{\sqrt{p^2c^2 + m^2c^4}},\\
    &= \frac{p}{m\sqrt{\left(\frac{p}{mc}\right)^2+1}},\\
    &= \frac{p}{m\gamma},\\
    &= \frac{mv\gamma}{m\gamma},\\
    &= v.

where m is the mass of the particle, c is the speed of light in a vacuum, \gamma is the Lorentz factor, and v is the velocity of the particle regardless of wave behavior.

Group velocity (equal to an electron's speed) should not be confused with phase velocity (equal to the product of the electron's frequency multiplied by its wavelength).


Phase velocity[edit]

In quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypothesis, we see that


Using relativistic relations for energy and momentum, we have

v_\mathrm{p} = \frac{E}{p} = \frac{\gamma m c^2}{\gamma m v} = \frac{c^2}{v} = \frac{c}{\beta}


v_\mathrm{p} > c, \,

and as we can see, it approaches c when the particle speed is in the relativistic range. The superluminal phase velocity does not violate special relativity, as it carries no information. See the article on signal velocity for details.




which is frame-independent.

Experimental confirmation[edit]





Neutral atoms[edit]





Spatial Zeno effect[edit]

The matter wave leads to the spatial version of the quantum Zeno effect. If an object (particle) is observed with frequency Ω >> ω in a half-space (say, y < 0), then this observation prevents the particle, which stays in the half-space y > 0 from entry into this half-space y < 0. Such an "observation" can be realized with a set of rapidly moving absorbing ridges, filling one half-space. In the system of coordinates related to the ridges, this phenomenon appears as a specular reflection of a particle from a ridged mirror, assuming the grazing incidence (small values of the grazing angle). Such a ridged mirror is universal; while we consider the idealised "absorption" of the de Broglie wave at the ridges, the reflectivity is determined by wavenumber k and does not depend on other properties of a particle.[8]

See also[edit]


  6. ^ R.B.Doak; R.E.Grisenti, S.Rehbein, G.Schmahl, J.P.Toennies2, and Ch. Wöll (1999). "Towards Realization of an Atomic de Broglie Microscope: Helium Atom Focusing Using Fresnel Zone Plates". Physical Review Letters 83 (21): 4229–4232. Bibcode:1999PhRvL..83.4229D. doi:10.1103/PhysRevLett.83.4229. 
  10. ^ Pierre Cladé; Changhyun Ryu, Anand Ramanathan, Kristian Helmerson, William D. Phillips (2008). "Observation of a 2D Bose Gas: From thermal to quasi-condensate to superfluid". arXiv:0805.3519.
  14. ^ Gerlich, S.; S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüxen, M. Mayor & M. Arndt (5 April 2011). "Quantum interference of large organic molecules". Nature Communications 2 (263): 263–. Bibcode:2011NatCo...2E.263G. doi:10.1038/ncomms1263. PMC 3104521. PMID 21468015. 

Further reading[edit]

External links[edit]