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What Happened Before the Big Bang?
The Bad Astronomer writes to tell us that a recent advance in Loop Quantum Gravity theory appears to allow the mathematics of cosmology to be extended to the time before the Universe underwent the Big Bang. Bad Astronomer also attempts to simplify things a bit with his own explanation of the new discovery.
There are are lots of different ways to understand the Heisenberg Uncertainty Principle physically -- most of them not very satisfying without acclimating to the lingo and concepts of quantum theory. Nevertheless, I think one can gain an intellectual foothold into the idea, before even digging into the quantum theory, by realizing that ALL wave behavior (sound, water, radio, light, etc.) obeys something akin to a HUP. If you can get the basic idea down for sound or water waves, then you can start to build a conceptual bridge to matter waves. Since you are an EE, the conceptual underpinnings will probably look quite familiar.
Lots of mathematical qualifications aside, basically ALL waveforms can be represented by a sum of harmonic waves (pure sine and cosine functions). A single pure sine or cosine has a well defined frequency, wavelength, and wave velocity. However, in contrast, an arbitrary waveform does NOT have a single wavelength or frequency -- it has many, given by the distribution of sines and cosines that were used to construct it. A handy variable to use is called the wavenumber, which is basically the number of cycles per meter (proportional to 1/wavelength) of a harmonic wave. An interesting thing to do is plot a particular waveform, say a snapshot of a water wave shaped like a lump at a moment in time, and then also plot the distribution of wavenumbers from all the sines and cosines making up that lump. They are two representations of the same object. One looks like a water lump in space, and the other will look like another lump telling you the distribution of sines and cosines in "wavenumber space." What you find is that if your water lump in space is narrow, then it takes many sines and cosines of many wavenumbers to make that happen. If the water lump is very spread out, you only need a narrow range of wavenumbers of harmonic waves to make this happen. Many engineers are very familiar with this bandwidth effect in the context of transmission theory, but the same will be true for ANY waveform. It is a byproduct of wave theory: the width of the spatial distribution of an arbitrary wave is inversely related to the width of its wavenumber distribution. If you allow the wave to change in time, you get a similar inverse relation for the distribution of the wave in time and the distribution of frequencies in the wave. You are probably familiar with all that in the context of Fourier analysis etc. One says that wavenumber and position are "complimentary" (so are frequency and time).
The big leap in quantum mechanics is that the momentum of a particle is inversely related to the wavelength of some harmonic wave "associated with" the particle. The larger the momentum, the shorter the wavelength of the matter wave and vice versa. That is, momentum and position are complimentary variables. Keep in mind, the wave isn't the particle itself but rather tells you where the particle is likely to be. Once you accept the rather odd idea that momentum and wavelength are inversely related, *wave theory alone* tells you that the more likely a particle is to be at a particular location in space, the wider its distribution of wavenumbers is -- and thus the wider range of momenta it can have. Similarly, if you have a very narrow range of wavenumbers, the wider the spatial extent of the matter wave -- thus for a well defined momentum the particle has a wider range of spatial positions available to it. This is basically the heart of the Heisenberg Uncertainty Principle.
Since this matter wave tells you about probabilities, you need to prepare an ensemble of identical objects and do a statistical analysis of their positions and momenta to see the effect of the HUP. For example, lets say you prepare 100 particles each with a well defined position. Now you perform a position measurement followed by a momentum measurement for each particle. Taking your raw data, you made a plot of the number
i\hbar\dot{\psi}=\hat{H}\psi