Quantum wave functions come alive! May the Bohr Model rest in peace

-July 26, 2013

Physicists from the Canadian Institute for Measurement Standards are the first to measure a quantum mechanical wave function. And it only took 88 years from the formulation of Schroedinger’s equation!

The difficulty of the measurement originates with the fundamental uncertainty between position and momentum, energy and time, and a host of other so-called conjugates: complementary pairs of quantities that cannot be measured to arbitrary accuracy at the same time. Yes, it’s the Heisenberg Uncertainty Principle.

The Heisenberg Uncertainty Principle came straight from Fourier and the mathematics of waves. To form a localized wave packet, we have to add up many waves of different wavelengths; the tighter, more localized that wave packet, the larger its bandwidth. In the limit of perfect position accuracy, the packet is described by a Dirac delta function which has infinite bandwidth. The tradeoff between localization and bandwidth is the heart of the uncertainty principle.



Figure 1: Contrasting the spatial nature of waves, particles, and wave packets (Graphic by Ransom).

Let’s make a quantum leap (pardon the expression, couldn’t help it): Think of the wave packet as a particle whose position isn’t well known. The spread of the wave packet is the uncertainty in the particle’s position. If we know nothing about its position, then the particle is smeared out across space like a wave approaching a beach, the top graphic in Figure 1. If we know the particle’s position perfectly, then it’s not smeared out at all, the center graphic, but we don’t know anything about its wavelength.

Either we can know the position or wavelength, but not both. For waves, it seems silly. Waves are spread out, not localized like particles. For classical waves, the trade-off is localization vs bandwidth.

Fourier’s uncertainty principle:       

Now, if we replace wavelength with the reciprocal of momentum and a constant to get the units right (Planck’s constant, h), we get Heisenberg’s version: the more accurately you localize something, the greater its momentum “spectrum”—that is, the more accurate the position, the less accurate the momentum and vice versa,

Heisenberg’s uncertainty Principle        

In quantum mechanics particles are described as wave packets with probability “amplitudes.” They exhibit all the properties of waves, like interference and diffraction. The absolute square of the probability amplitude gives the probability distribution of where you will find the particle, should you measure its position. This is where the whole “particle existing in two places at once” comes from. It makes much more sense if you think of waves smeared across space according to a wave function. Waves are always in more than one place, except in the special case of a tight wave packet, they’re not local phenomena.

All of which brings me to the Bohr model of the atom and one of my pet peeves. You recall the Bohr Model, it describes an atom as though it were a planetary system with electrons playing the role of planets and the nucleus acting like the sun.

Since Bohr was a genius, it’s a shame that his name is associated with the most misleading conceptual aid in all of science. The Bohr model successfully predicts hydrogen energy levels with good accuracy, but it’s a wrong in every other way.

The first problem with the Bohr model is that, like the solar system, it describes atoms as two-dimensional discs but they are obviously three dimensional. Second, if an electron really rotated about a proton, it would radiate off its energy and collapse. Remember, accelerated charges radiate and going in a circle requires acceleration. Third, the Bohr model prediction of the electron’s angular momentum is wrong.

Fortunately, the solution to Schroedinger’s equation for hydrogen, an electron in the static electric field of a proton, yields beautiful solutions. Solutions explicitly observed in experiments performed at the Canadian Institute for National Measurement Standards. The measurements elegantly reproduce the orbitals (not orbits!) for the discrete states of hydrogen—just as shown in chemistry and physics textbooks for at least 70 years.


Figure 2: Measurement of hydrogen atom wave functions (Physical Review Letters via New Scientist)


Figure 3: 3D simulation of electron orbitals of the hydrogen atom (from askamathematician.com).

The orbital geometry, which is to say the wave equations, were mapped by measuring the positions of electrons in hydrogen atoms over and over again.

Think of electrons bound to atomic nuclei as being smeared out like waves. If you pluck a guitar string and set up a standing wave, you don’t point at one spot on the string and say, “there it is!” In two dimensions, you’d think of a drum head. Similarly, atomic orbitals are three-dimensional electron standing waves. Just as a guitar string has a discrete set of harmonic tones, so does an atom have a set of discrete energy states, each with a corresponding orbital.

In practice, the Heisenberg Uncertainty Principle requires that we think about measurements as though we are part of the system. By measuring position, we destroy information about momentum; in jargon, we “collapse the wave function” onto a single position. Similarly, by measuring momentum, we collapse the wave function onto a sinusoidal wave but surrender all position information.

When the guys from the Canadian Institute for National Measurement Standards resolve the electron bound to a specific hydrogen atom, they don’t measure that standing wave, they collapse the wave function down to a tiny wave-packet whose spread is limited only by their experimental spatial resolution. Since the wave function describes the probability that they’ll find the electron at a specific position, as they repeat the measurement they reproduce the wave function.

So...
  • Why do you think it took 88 years for this experimental verification?
  • This result has been assumed for at least 70 years, should verification have merited greater attention?
  • Do you see how the popularity of a phrase like “can be in two places at once,” while true, is misleading?

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