Schrödinger's Cormorant

5.8 Not Sure About That .......

The uncertainty principle. Exploring the consequences of the trade off between reducing positional uncertainty (by restricting a wave packet to a smaller space) and increasing momentum uncertainty.

We are now at a very important moment – one which will introduce to one of the strangest and most important consequences of a universe based on quantum mechanics. Nefertiti and Cormorant are excited. Another ghostly physicist appears: it is Werner Heisenberg and he has something to say: “You cannot be certain that I was a Nazi.” Nefertiti raises an eyebrow. Cormorant has no eyebrows. “Or perhaps you are more concerned with the other issue of uncertainty. Very well. We will proceed.” The next question will lead you to a profound realisation .....

Question 5-8

And there we have it: localisation of a wave (to form a wave packet) can be understood as a superposition of simple sinusoidal wave forms, with different wavelengths. When we are concerned with matter waves, each individual sinusoidal wave has an associated momentum (defined by the de Broglie equation). Therefore a wave packet, corresponding to a particle localised in a specific region of space, is associated not with a single, specific momentum but with a distribution of momenta. Like the spatial distribution this is effectively a probability distribution for the particle momentum, where the probability reflects the weighting of an individual sinusoidal component in constructing the wave packet.

"Now hold on there, Bald Eagle" interjects Cormorant, "I thought you were all telling me that I should think of a particle as a standing wave." (He is, of course, right: see section 5.5, for example). "So, if it's standing still, how does it have momentum?" That's a fair point, but Nefertiti is quick to remind him that we are thinking of our quantum mechanical waves as "probability waves", rather than conventional physical waves. A wave packet is a superposition of such waves and, therefore, carries with it a range of possible momenta, which might be measured, if a measurement were to be made. Moreover, a standing wave is (as we established in section 4.5) a superposition of waves travelling in opposite directions and, therefore, with momenta of opposite sign. Again this can be interpreted as meaning that if a measurement of a particle's momentum is made, we cannot be certain even of the direction of the momentum arising from its waviness.

"We must not lose time getting upset about these physical interpretations" insists Werner. "As my friend Niels Bohr has said, "if anybody says he can think about quantum theory without getting giddy it merely shows that he hasn't understood the first thing about it!." Cormorant does his best to look consoled. The key result we have found is this: the more we try to narrow a wave packet – in effect, to pin down a particle to a specific region of space – the greater the required range of wavelengths of the component sinusoids and therefore the greater the uncertainty there will be in the particle's momentum. What we have here is an enormously important principle, which is at the heart of quantum physics:

We cannot simultaneously know exactly both the position and the momentum of a particle.

"I was the first to make this connection", explains Werner: “You will refer to it as the uncertainty principle, and it will be obeyed, without question, at all times.”

All of this is quite shocking. As Cormorant flies majestically across the sky, he knows where he is and he knows where he’s going and he knows how fast. Sir Isaac backs him up, so is he going to let Werner tell him that he can’t really be sure? He is not. But, as we have seen before, it could be just a matter of scale. So we need to try to quantify things a bit.

In our formulation, we have been generating wave packets by superposing y = cos (kx) over a range of k, the angular wavenumber (k = 2π/λ). Prince Louis’ equation (p = h/λ) then establishes the momentum connection: p = (hk/2π). So the uncertainty in any measurement of momentum would be related to the spread in the values of k. The uncertainty in position is related to the width of the wave packet – and, more specifically, to the probability density distribution obtained by squaring the wavefunction.

Recall that we worked out, in section 5.7, an expression for the function we get by superposing all functions y = cos (kx), over a range of k from k1 to k2. The equation we got, scaled so that it is in effect the mean of the superposition, is:

In the case of matter waves, we know that the probability density distribution for the particle’s position is given by ψ2 (multiplied by a constant, but we won’t worry about that for now, as it doesn’t affect the width of the peak). So we get:

Let's consider two cases, where the range of k over which we integrate is different. In the picture below, the left hand panel shows this probability density distribution we get when k varies from 0 to 1, together with the individual waveform with the biggest k value in our range, y = cos(x).

The right hand panel does the same for the case where we increase the range of k, so that it now extends from 0 to 2. So here, again, the two curves are the probability density function and the waveform with the biggest value of k, y = cos(2x).

This comparison shows clearly what we already figured out in section 5.7, that a bigger range of k makes the central, dominant peak in the probability density distribution narrower.

Remembering that k=2π/λ, we can see why this is so: all the other sinusoids in the superposition have smaller k values than the one shown, corresponding to longer wavelengths. They can, therefore, add together to cause destructive interference outside the central peak of the y=cos(kmaxx) waveform but not inside it. The width of this central peak, around x=0, in the y=cos(kmaxx) function is therefore the key to this irreducuble limit to the width of the probability density distribution.

Now let’s have a go at analysing this a bit more quantitatively:

Question 5-9

This analysis gives a remarkable) result:

If we denote the uncertainties in position and momentum of the particle as Δx and Δp respectively, they are inversely proportional, related by:

ΔxΔp = 2h

Remembering that h is Planck’s constant (h = 6.63 x 10-34 kgm2s-1): a universal constant. This means that there's no escaping it. Whatever we do to narrow down the position uncertainty, the momentum uncertainty will always increase in accord with this equation and we can't do anything about it. Planck's constant is a very small number, so you can see that this represents an uncertainty which is tiny relative to our everyday scale but which may be very important on the atomic scale.

Now we have to remember that this has been a far-from-rigorous definition for at least a couple of reasons:

Our superposition model, in which each of the contributing wavefunctions is weighted equally, is artificial. In reality there could be a superposition with varying weights across the range of k.
We have not properly defined what we mean by "uncertainty". In particular, we haven't taken account of the fact that the particle is much more likely to be found close to x=0 than at the edges of that central peak in the ψ2 distribution. This might lead us to guess that we have probably overestimated the likelihood of significant deviation from the mean position.

The bad news is that a rigorous derivation is not straightforward. It's not going to happen; not on this day. It was originally done by Earle Kennard (using a totally different approach), in the year after Werner established the principle, and it can be done in various ways - for example with the help of Joseph Fourier’s theory - but this is a whole world of maths we can’t really get into at this point. What then to do?

Cormorant has an idea: “let’s not worry about it.”

Nefertiti isn't going to give up quite so easily, though. If you're with her, you can pursue the question further by clicking below. This will take you to a deeper analysis, which takes a more statistical view of what uncertainty means. So, especially if you know a bit about normal distributions, standard deviations and the like, you might find this interesting. Be warned, though, it will be less than elegant and will still require a bit of cheating. It will take us closer to Earle's accepted result but we still won't be quite there. If that seems like too much work for a modest gain, you can skip this little extension activity.

Extension: Deeper Into Uncertainty

Earle's full treatment takes a statistical view of uncertainty, expressing it as the standard deviation (σ) of a probability density distribution, rather than a simple range. His result confirms the reciprocal relationship that we have derived which means that there is a fundamental limit on how precisely it is possible to know both the position and the momentum of a particle. However, as we suspected, the size of this uncertainty is a bit smaller than we estimated in our simplified treatment. Earle's analysis leads to:

σxσp = h/4π.

This value is the fundamental limit on the precision that is obtainable, because of the waveform superposition inherent in localising a particle. In a real experiment, the may be all kind of practical reasons why the actual precision achievable will be lower than this. For this reason, the relationship is generally written as an inequality. So Heisenberg's uncertainty principle is generally expressed in quantitative form as:

σxσp ≥ h/4π (V-12)

Now, let’s see how strong a challenge the uncertainty principle offers to Cormorant’s confidence in his simultaneous awareness of his position and momentum.

Question 5-15

Planck's constant, h, is a very small quantity when we're thinking on the macroscopic scale. So the position-momentum uncertainty that results from the waviness of a particle is completely negligible when that particle is, for example, a cormorant. As we found with special relativity, the weirdness inherent in quantum theory is not apparent in everyday experience. But on the atomic scale, it’s a different story. The fuzziness embodied in the uncertainty principle is the key to quantum mechanics. We'll look at an example in the next section.

Before we move on, though, let's think about the energy implications of the uncertainty principle.

We're used to the idea that the more we localise a particle, the bigger the uncertainty in the momentum, σp, will become. Now we know that the mean momentum is always zero - because it is equally likely to be forward or backward momentum - but the mean of the magnitude of the momentum, <lpl>, (ie the mean if we ignore the + or - sign) will necessarily increase as σp increases.

In turn, of course, we know that momentum is related to kinetic energy. So, by virtue of its waviness, a particle has some intrinsic kinetic energy, given - in classical physics - by

Ek = <lpl>2/2m

In a localised particle, there will therefore be a probability distribution for its kinetic energy too. The increased value of <lpl> required to decrease the uncertainty in position therefore means that the mean value of the kinetic energy in this distribution must also be increased. The implication of this is very profound: it means that pinning down the position more precisely is achievable only at a cost of increased energy. This is sometimes called the “localisation energy” of the particle.

Next: why this matters.

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