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Opus in profectus

Matter Waves

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de Broglie

Combine the two equations for momentum.

p = mv  &  p =  h    ⇒  λ =  h
λ mv

This was basically Louis de Broglie's doctoral thesis of 1924.

Davisson-Germer experiment. Electrons diffracted after being reflected off a single crystal of nickel.

A person entering a room with more than one entrance will always enter through one of them, not all of them at the same time. An electron, on the other hand, can and does enter enter a room through all doors simultaneously.

Thomson-Reid experiment. Electrons diffracted after passing through gold foil. Powder pattern.

Estermann-Stern experiment. Helium atoms diffracted after being reflected off a single crystal of lithium fluoride (LiF).

Neutron diffraction. Ernest O. Wollan, Lyle B. Borst and Walter H. Zinn were all able to observe neutron diffraction in 1944 using the X-10 graphite reactor and the CP-3 heavy water reactor (built as part of the Manhattan Project) in Oak Ridge, Tennessee. Subsequent work by Wollan and Clifford G. Shull laid the foundations for neutron diffraction as an important research tool

Kapitza-Dirac effect. Diffraction of a particle beam by a standing wave of light. Proposed in 1933. Observed by Herman Batelaan at the University of Nebraska in 2001.

schrödinger

Erwin Schrödinger (1887–1961) Austria, Abhandlungen zur Wellenmechanik. Wave equation for matter reminiscent of Maxwell's equations for electromagnetic waves. The story I heard is that Schrödinger went to Switzerland with two goals: to keep his mistress happy and to derive a wave equation for matter. How successful he was with the former is open to speculation.

full, time-dependent form

iℏ ∂ Ψ(r,t) = − ℏ2 ∇2Ψ(r,t) + U(r)Ψ(r,t)
∂t2m

can be separated into two halves

Ψ(r,t) = ψ(r)φ(t)

spatial, time-independent half

Eψ(r) = − ℏ2 ∇2ψ(r) + U(r)ψ(r)
2m

time-dependent half

φ(t) = e−iEt/ℏ

heisenberg

Quote to get things started.

Two seemingly incompatible conceptions can each represent an aspect of the truth… They may serve in turn to represent the facts without ever entering into direct conflict.

Louis de Broglie, 1948

The defining feature of the microscopic world is the wave-particle duality. Whenever we observe elementary entities (like electrons or photons) they appear as a localized events. A single photon can be observed as a tiny dot on a photographic plate. A single electron can be observed as a tiny flash on a television screen. This locality (existing at a particular place) and temporality (occurring at a specific time) is what it means for a thing to exist as a particle. It interacts with its environment in a specific place at a specific time.

In contrast, when we are not observing these entities interacting with their environment, they behave in a wavelike manner — extended in space, diffracting around obstacles and through openings, interfering with other elementary entities of the same type (that is, electrons interfere with electrons, and photons with photons). The nature of the waves associated with elementary entities are probability waves — unitless numbers, numerical ratios. They tell you the probability of finding a particular particle at a particular place and time and nothing else. They do not measure the value of any physical quantity. The waves themselves carry no mass, no charge, no energy, no momentum, no angular momentum, no information of any sort other than the likelihood of existence. In essence, they carry information only. Nothing else.

The conflict between these two aspects of microscopic reality results in the Uncertainty Principle

∆px∆x ≥ ℏ 
2

Momentum and position are measured in the same direction.

∆px ∆x ≥  ℏ    ∆py ∆y ≥  ℏ    ∆pz ∆z ≥  ℏ 
2 2 2

Momentum-position becomes energy-time.

∆E∆t ≥ ℏ 
2

Werner Heisenberg (1901–1976) Germany

Formally, a function and its Fourier transform cannot both be compactly supported.

electron microscopes

thomson-reid

Electron Microscope

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