The Physics
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. 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.

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.


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) = − 2 ∇2Ψ(r,t) + V(r)Ψ(r,t)

can be separated into two halves

Ψ(rt) = ψ(r)φ(t)

spatial, time-independent half

Eψ(r) = − 2 ∇2ψ(r) + V(r)ψ(r)

time-dependent half

φ(t) = eiEt/ℏ


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

ΔpxΔx ≥ ℏ 

Momentum and position are measured in the same direction.

Δpx Δx ≥  ℏ    Δpy Δy ≥  ℏ    Δpz Δz ≥  ℏ 
2 2 2

Momentum–position becomes energy–time.

ΔEΔt ≥ ℏ 

Werner Heisenberg (1901–1976) Germany

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

electron microscopes


Electron Microscope

Related inventions

history and miscellany

Quantum mechanics is like a student who always gets the correct answer to the first question in an exam, but never the second.

Die Quantenmechanik ist sehr Achtung gebietend. Aber eine innere Stimme sagt mir, daß das noch nicht der wahre Jakob ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum näher. Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt.   Quantum mechanics is very respectable. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but hardly brings us any closer to the mystery of the old one. In any case, I am convinced that He is not rolling dice.
Brief an Max Born, 4 Dezember 1926   Letter to Max Born, 4 December 1926

Thus it seems Einstein was doubly wrong when he said, God does not play dice. Not only does God definitely play dice, but He sometimes confuses us by throwing them where they can't be seen.

Stephen Hawking, 1999

It is often stated that of all the theories proposed in this century, the silliest is quantum theory. In fact, some say that the only thing that quantum theory has going for it is that it is unquestionably correct.

Michio Kaku, 1995

Alice laughed. 'There's no use trying,' she said: 'one CAN'T believe impossible things.' 'I daresay you haven't had much practice,' said the Queen. 'When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast.

Lewis Carroll, 1871

What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school — and you think I'm going to explain it to you so you can understand it? No, you're not going to be able to understand it. Why, then, am I going to bother you with all this? Why are you going to sit here all this time, when you won't be able to understand what I am going to say? It is my task to convince you not to turn away because you don't understand it. You see my physics students don't understand it either. That is because I don't understand it. Nobody does.

Richard Feynman, 1983

A good simple application. Electrons have very little mass and therefore occupy a lot of volume. Protons are much more massive and occupy very little volume.

Absolute zero does not imply absolute rest. Zero point energy.

virtual particles

Casimir Effect — Hendrik Brugt Gerhard Casimir

F =  π2c  A


I was saying off the air: I hate quantum mechanics. More scalawags have hidden in the mires of quantum mechanics and legitimate science.

Perry DeAngelis, 2006