Matter wave

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In quantum mechanics, the concept of matter waves or de Broglie waves 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

Propagation of de Broglie waves in 1d – real part of the complex amplitude is blue, imaginary part is green. The probability (shown as the colour opacity) of finding the particle at a given point x is spread out like a waveform, there is no definite position of the particle. As the amplitude increases above zero the curvature decreases, so the amplitude decreases again, and vice versa – the result is an alternating amplitude: a wave. Top: plane wave. Bottom: wave packet.

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

$E=h\nu$

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:

 “ When I conceived the first basic ideas of wave mechanics in 1923–24, I was guided by the aim to perform a real physical synthesis, valid for all particles, of the coexistence of the wave and of the corpuscular aspects that Einstein had introduced for photons in his theory of light quanta in 1905. ” —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

Quantum mechanics

The de Broglie equations relate the wavelength λ to the momentum p, and frequency f to the total energy E of a particle:[2]

 \begin{align} & \lambda = h/p\\ & f = E/h \end{align}

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

 \begin{align} & p = \hbar k\\ & E = \hbar \omega\\ \end{align}

using the definitions

• $\hbar=h/2\pi$ is the reduced Planck's constant (also known as Dirac's constant, pronounced "h-bar"),
• $k= 2\pi/\lambda$ is the angular wavenumber,
• $\omega=2\pi f$ is the angular frequency.

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

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}}} \end{align}

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

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

$v_g = \frac{\partial \omega}{\partial k} = \frac{\partial (E/\hbar)}{\partial (p/\hbar)} = \frac{\partial E}{\partial p}$

where E is the total energy of the particle, p is its momentum, ħ is the reduced Planck constant. For a free non-relativistic particle it follows that

\begin{align} v_g &= \frac{\partial E}{\partial p} = \frac{\partial}{\partial p} \left( \frac{1}{2}\frac{p^2}{m} \right),\\ &= \frac{p}{m},\\ &= v. \end{align}

where m is the mass of the particle and v its velocity.

Also in special relativity we find that

\begin{align} 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. \end{align}

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).

Both in relativistic and non-relativistic quantum physics, we can identify the group velocity of a particle's wave function with the particle velocity. Quantum mechanics has very accurately demonstrated this hypothesis, and the relation has been shown explicitly for particles as large as molecules.[citation needed]

Phase velocity

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

$v_\mathrm{p} = \frac{\omega}{k} = \frac{E/\hbar}{p/\hbar} = \frac{E}{p}.$

Using relativistic relations for energy and momentum, we have

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

where E is the total energy of the particle (i.e. rest energy plus kinetic energy in kinematic sense), p the momentum, $\gamma$ the Lorentz factor, c the speed of light, and β the speed as a fraction of c. The variable v can either be taken to be the speed of the particle or the group velocity of the corresponding matter wave. Since the particle speed $v < c$ for any particle that has mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e.

$v_\mathrm{p} > c, \,$

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.

Four-vectors

Using the four-momentum P = (E/c, p) and the four-wavevector K = (ω/c, k), the De Broglie relations form a single equation:

 $\mathbf{P}= \hbar\mathbf{K}$

which is frame-independent.

Experimental confirmation

Matter waves were first experimentally confirmed to occur in the Davisson-Germer experiment for electrons, and the de Broglie hypothesis has been confirmed for other elementary particles. Furthermore, neutral atoms and even molecules have been shown to be wave-like.

Electrons

In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow-moving electrons at a crystalline nickel target. The angular dependence of the reflected electron intensity was measured, and was determined to have the same diffraction pattern as those predicted by Bragg for x-rays. Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be only exhibited by waves. Therefore, the presence of any diffraction effects by matter demonstrated the wave-like nature of matter. When the de Broglie wavelength was inserted into the Bragg condition, the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons.[5]

This was a pivotal result in the development of quantum mechanics. Just as the photoelectric effect demonstrated the particle nature of light, the Davisson–Germer experiment showed the wave-nature of matter, and completed the theory of wave-particle duality. For physicists this idea was important because it means that not only can any particle exhibit wave characteristics, but that one can use wave equations to describe phenomena in matter if one uses the de Broglie wavelength.

Neutral atoms

Experiments with Fresnel diffraction[6] and specular reflection[7][8] of neutral atoms confirm the application of the de Broglie hypothesis to atoms, i.e. the existence of atomic waves which undergo diffraction, interference and allow quantum reflection by the tails of the attractive potential.[9] Advances in laser cooling have allowed cooling of neutral atoms down to nanokelvin temperatures. At these temperatures, the thermal de Broglie wavelengths come into the micrometre range. Using Bragg diffraction of atoms and a Ramsey interferometry technique, the de Broglie wavelength of cold sodium atoms was explicitly measured and found to be consistent with the temperature measured by a different method.[10]

This effect has been used to demonstrate atomic holography, and it may allow the construction of an atom probe imaging system with nanometer resolution.[11][12] The description of these phenomena is based on the wave properties of neutral atoms, confirming the de Broglie hypothesis.

Molecules

Recent experiments even confirm the relations for molecules and even macromolecules, which are normally considered too large to undergo quantum mechanical effects. In 1999, a research team in Vienna demonstrated diffraction for molecules as large as fullerenes.[13] The researchers calculated a De Broglie wavelength of the most probable C60 velocity as 2.5 pm. More recent experiments prove the quantum nature of molecules with a mass up to 6910 amu.[14] In general, the De Broglie hypothesis is expected to apply to any well isolated object.

Spatial Zeno effect

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]

References

1. ^ L. de Broglie, Recherches sur la théorie des quanta (Researches on the quantum theory), Thesis (Paris), 1924; L. de Broglie, Ann. Phys. (Paris) 3, 22 (1925).
2. ^ a b Resnick, R.; Eisberg, R. (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-87373-X.
3. ^
4. ^ Holden, Alan (1971). Stationary states. New York: Oxford University Press. ISBN 0-19-501497-9.
5. ^ Mauro Dardo, Nobel Laureates and Twentieth-Century Physics, Cambridge University Press 2004, pp. 156–157
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.
7. ^ F. Shimizu (2000). "Specular Reflection of Very Slow Metastable Neon Atoms from a Solid Surface". Physical Review Letters 86 (6): 987–990. Bibcode:2001PhRvL..86..987S. doi:10.1103/PhysRevLett.86.987. PMID 11177991.
8. ^ a b D. Kouznetsov; H. Oberst (2005). "Reflection of Waves from a Ridged Surface and the Zeno Effect". Optical Review 12 (5): 1605–1623. Bibcode:2005OptRv..12..363K. doi:10.1007/s10043-005-0363-9.
9. ^ H.Friedrich; G.Jacoby, C.G.Meister (2002). "quantum reflection by Casimir–van der Waals potential tails". Physical Review A 65 (3): 032902. Bibcode:2002PhRvA..65c2902F. doi:10.1103/PhysRevA.65.032902.
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.
11. ^ Shimizu; J.Fujita (2002). "Reflection-Type Hologram for Atoms". Physical Review Letters 88 (12): 123201. Bibcode:2002PhRvL..88l3201S. doi:10.1103/PhysRevLett.88.123201. PMID 11909457.
12. ^ D. Kouznetsov; H. Oberst, K. Shimizu, A. Neumann, Y. Kuznetsova, J.-F. Bisson, K. Ueda, S. R. J. Brueck (2006). "Ridged atomic mirrors and atomic nanoscope". Journal of Physics B 39 (7): 1605–1623. Bibcode:2006JPhB...39.1605K. doi:10.1088/0953-4075/39/7/005.
13. ^ Arndt, M.; O. Nairz, J. Voss-Andreae, C. Keller, G. van der Zouw, A. Zeilinger (14 October 1999). "Wave-particle duality of C60". Nature 401 (6754): 680–682. Bibcode:1999Natur.401..680A. doi:10.1038/44348. PMID 18494170.
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.