Wednesday, June 13, 2007

Schrödinger’s legacy

HΨ = iℏ ∂Ψ/∂t

It is safe to say that nobody understands quantum mechanics. Richard Feynman

It is difficult, maybe impossible to wrap ones head around the strange events occurring at the quantum level; but occur they do. It is a frustrating exercise to imagine any intuitive familiarity regarding the schizophrenic behavior of matter that occurs at these truly tiny dimensions.Trying to relate them to “real world” experience becomes futile. Our sensory “common sense” sensory and cognitive make up simply can not quite put this dualistic reality together, even though, in the same breath, quantum mechanics, a human endeavor, predicts its effects so well.

The quantum wave nature of a human is tiny and undetectable. So much so that it exist within realms where space itself loses meaning and as Al-Khalili (Quantum) states: “we need not be concerned with wavy cows.” An electron traveling at several m/s has a comparatively huge wavelength comparable to the width of a human hair. This is large enough for its quantum wave nature to readily reveal itself even during routine behavior. This is a critical point, a mental "milestone", where many people tend to stumble and confuse what an electron quantum wave can do and think that a human quantum wave can do the same.

The wave function has been described as an abstract mathematical “quantity” that allows for the prediction and calculation of atomic properties. It does not necessarily actually describe the atom itself, or directly imply the atom is the wave function, though whatever is going on, it describes with exquisite accuracy.

Erwin Schrödinger was able to use de Broglies idea of matter waves to explain Bohr’s general model of the atom (electrons move around in atoms in quantized [“fixed”] orbits). He developed an equation that described not the way a particle moves, but how a wave evolves (there are other mathematical methodologies describing quantum system behavior- Schrödinger’s seems the one most taught in basic physics per Al-Khalili in his book Quantum). This formula shed light on the concept of wave functions, their behavior, and revealed the probabilistic nature of quantum mechanics.

The Schrödinger equation is described as an “eigenvalue” equation. That is, it produced what physicists called sensible solutions for specific values of the energy variable of the equation- other solutions produced absurd (i.e. infinity) energy values and were considered nonphysical. In this way, Schrödinger was able to calculate the permitted quantum energy states in hydrogen atoms, and this had broad implications for quantum physics.

Later, Max Born realized the significance and potential of this equation. He found that the Schrödinger wave function itself was an unobservable quantity but that the square of this wave function was observable and, most importantly, probabilistic in nature. These findings had a huge impact on understanding how to predict the quantum behavior of matter. Jim Al-Khalili adds: “In general, a wave function does not simply oscillate like a water wave. It is far more complicated than that…at each point in space the wave function is defined by two numbers known as its real and imaginary parts. Joining all the “real” numbers together produces one wave and the “imaginary” numbers another, and the full wave function is a combination of the two. In addition, a typical wave function will, if plotted on a graph, have a quite complicated shape depending on the system it is describing.”

Today, the wave function is considered by most physicists as being an abstract mathematical “thing” or entity used to extract information about nature, though others consider it some type of actual independent reality. Richard Feynman was able to get correct quantum results using infinite particle paths (sum over histories) instead of wave interactions. Even so, quantum phenomena so closely resembles wave behavior it is most commonly used to describe the “reality” of nature. This however, is not the critical issue as Al-Khalili explains: “Strangely enough, what is important here is that it doesn’t matter whether the wave function is real or not, its mathematical properties are the same; and what it can tell us about the way nature behaves on the subatomic scale is not in any doubt.”

Whether waves or some other concept can describe these strange events Kenneth Ford (The Quantum World) sums up the real issue: “…a quantum entity… is a particle when it is created and annihilated (emitted and absorbed). It is a wave in the interval between those events. Still, adopting this point of view is not sufficient for overcoming the impression of quantum weirdness. You may still want to ask: If the wave starts to propagate outward from the point where the particle is created, how does the wave know where and when to “collapse” to signal the annihilation (the detection) of the particle? The only answer to this question is that the wave is a wave of probability, of potentiality. It gives the likelihood that the particle will end its life at some future time and some other place.”

The link between wave function and probability opened the doors to even stranger findings regarding the nature of quantum phenomena including concepts such as the uncertainty principle, superposition, nonlocality, and entanglement (which will be topics for future posts). As bizarre as these concepts can be, it is important to have some understanding of quantum mechanics and refrain from jumping into non sequitors and false assumptions as physicists expand their knowledge and share their thoughts with us.


Al-Khalili, J. Quantum.Weidenfeild & Nicolson. UK. 2004

Ford, K. The Quantum World.Harvard Univ Press.Mass.2005

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