Experimenting with a series of polaroids

Let us find out what happens when we send a random photon through various combinations and arrangements of polaroids.

Example 1 — three polaroids

0o 3 polaroids with successive 45o rotation of planes of polarisation

Using Malus's law,

Probability that the photon gets through all three polaroids

One photon in four gets through. Example 2 — five polaroids

5 polaroids with successive 22.5o rotation of planes of polarisation

Probability

More than half the photons will get through, and yet the three polaroids from example 1 are included in the row of five!

Example 3 — 91 polaroids

Let us give the photon a real test. What is the probability that a random photon gets through a series of 91 polaroids as depicted in the diagram below? The axis of each successive polaroid is rotated by 1° relative to the previous one. Note again that the axes of the first and last polaroids are perpendicular to one another.

91 polaroids in strict order of axis orientation

91 polaroids in strict order of axis orientation

Probability = cos180 (1°) = 0.99985180 = 0.97296 Over 97 photons out of 100 get through this system!*

Don't shuffle the polaroids!

We must be careful that all polaroids are in their correct order. Even one out of place will change the result completely! Order does matter.

Example 4 — the importance of order

In this example we take the 2° polaroid in the previous example out of its rightful place and insert it near the end of the line — say, between the 88° and the 89° polaroids. The fact that there is now a gap between the second and fourth polaroids makes very little difference. (If we want to be pedantic we can replace cos4 (1°) (= 0.99939) by cos2 (2°) (= 0.9976).) On the other hand, inserting it near the other end has a dramatic effect. We are

* We can write a general formula for the probability of a photon passing through n polaroids relatively rotated through equal angles 6 as P = [cos2(6)]n-1.

effectively creating the equivalent of two crossed polaroids, making the probability of the photon getting through the system negligible.

P = cos174 (1°) cos2 (2°) cos2 (86°) cos2 (87°) = 0.99985174 x 0.999392 x 0.069762 x 0.052332 = 0.0000062

About six photons in a million get through! 12.3.4 The uncertainty principle

The intrinsic lack of knowledge which we can have of the 'quantum world' can be stated in a slightly different form which does not involve matrix operators or commutation relations. This is known as the Heisenberg uncertainty principle. The principle forms the basis of quantum mechanics and states that:

There is a fundamental limit with which we can simultaneously know the position coordinate q and the corresponding momentum coordinate p of a particle.

If the uncertainty in position is Aq, then there is an uncertainty in momentum Ap, such that the product

Dp ¥ Dq ~ — (The Heisenberg uncertainty principle) 2p

The more exact is one's knowledge of the position of a particle such as an electron, the less is it possible to know its momentum, and vice versa. Somehow, the laws of Nature do not allow us to know these parameters precisely at the same time.

The uncertainty principle defines the limits beyond which the concepts of classical physics cannot be employed.

The commutation relation between matrices which represent position and momentum (12.1) is another way of expressing the uncertainty principle.

In Newtonian mechanics, things have an independent existence, with precise values for everything observed and not observed. That is why the theory of quantum mechanics is so strange. We are used to an unambiguous kind of physical reality. If there is uncertainty in our knowledge of any physical attribute it is due to the imperfection of our methods of measurement. Instinctively we feel that, the better our measuring instruments are, the closer we can get to the value of a physical quantity. That is why Einstein found these and other concepts in quantum mechanics hard to swallow.

Classical theory assumes that, in principle, it is possible to know everything about a physical system at a given instant. It follows that, by using such knowledge and applying the laws of physics, it should, again in principle, be possible to determine the future of such a system. For that reason Newtonian or classical mechanics is termed deterministic. It applies to the 'household world'.

In the world of the fundamental entities such as photons, electrons, atoms and atomic nuclei, the Heisenberg uncertainty principle takes on a dominant role. Since you cannot have exact knowledge of the present it is hardly surprising that you cannot determine the future! As a consequence, probability and not determinism is inherent in the laws which govern them.

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