Full knowledge of polarisation angle.

[yuiop]

Reading around the subject, I get the impression that it is impossible to have full knowledge of the polarisation angle of a photon. For example is a linearly polarised photon passes through a vertical polariser it is said to be "vertically polarised" on exiting the polariser, but in fact all we know is that before entering the the polariser the vertical component was great than the horizontal component of the polarisation and that after exiting the polariser, the horizontal component is undetermined and almost certainly not the same as its horizontal polarisation before it entered the polariser.

It seems if set up various experiments, even including entangled photons, we can not full determine what the vertical and horizontal components were at any given time. Can anyone offer a counter-example?

Is this correct? What is the correct terminology for this effect? Is vertical and horizontal polarisation complementary? I get the impression it is not but I am not clear why.

 

[Chopin]

Vertical and horizontal polarization can be known simultaneously--if you put light through a vertical polarizer, none of it will then go through a horizontal polarizer, so knowing one always allows you to know the other. Vertical and 45 degrees don't commute, though--vertically polarized light has a 50/50 chance of going through a 45 degree polarizer.

 

[yuiop]

Vertical and horizontal polarization can be known simultaneously--if you put light through a vertical polarizer, none of it will then go through a horizontal polarizer, so knowing one always allows you to know the other. Vertical and 45 degrees don't commute, though--vertically polarized light has a 50/50 chance of going through a 45 degree polarizer.

If we a light source that emits photons with random polarisation angles and pass them through a vertical polariser, 50% of the photons got through. This suggest that photons with a polarisation angle of anywhere between -45 and +45 degrees can pass through the vertical polariser. Now if they came out the polariser with an exact angle of 0 degrees, they would be unable to pass through a second polariser orientated at say 60 degrees to the first, because only photons with an orientation of of between 15 and 105 degrees can enter the second polariser. However we know this is not the case because photons have a 25% chance of passing through the 60 degree polariser according to Mallus law.

You also say "Vertical and 45 degrees don't commute". Now Wikipedia http://en.wikipedia.org/wiki/Commutativity#Non-commuting_operators_in_quantum_mechanics defines commutativity like this:

According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary which means that they cannot be simultaneously measured or known precisely

which seems to have the exact opposite of your intention. I would of thought that H polarisation and V polarisation do not commute and while V polarisation and 45 degree polarisation do commute because we can measure both for a single photon. Then again, I am not that familiar with the terminology and I am probably missing the point here.

 

[Chopin]

No, that's the central point I'm getting at--horizontal and vertical DO commute. You can measure both of them at the same time. What doesn't commute is vertical and 45 degrees (or, by symmetry, horizontal and 45 degrees.)

The way to think about this process is the following. We have light coming in with random polarization. This means you can model each photon as being in a superposition of both vertical and horizontal at the same time. The vertical polarizer acts as a measuring device, so it collapses the wavefunction of each photon into either 100% vertical or 100% horizontal. The ones which end up being horizontal get blocked, and the ones that end up being vertical get passed. So each photon coming out of this polarizer is 100% vertical. You can check this by putting it through another polarizer--if it's vertical, 100% of the light will go through (if the polarizer is ideal, of course), and if it's horizontal, 0% will go through. So the light is now 100%, definitively vertical, and 0% horizontal.

Now, if you put it through a 45 degree polarizer, you have to change bases. Instead of thinking in a "X percent vertical and Y percent horizontal" basis, you have to change to a "X percent positive 45 degrees and Y percent negative 45 degrees" basis. Note that these two axes are still perpendicular, just like our original two, we've just rotated. Now, due to the way you transform between these two bases, it turns out that light which is 100% vertical and 0% horizontal can also be viewed as light which is 50% positive 45 degrees and 50% negative 45 degrees. So our beam of light, which we thought was purely one state, can also be thought of as an equal superposition of two other states, +45 and -45. So now, when it goes through the second filter, it again acts like a measuring device, and collapses the wavefunction to one of those two. Just like the first time, if it's positive 45 degrees, it's blocked, and if it's negative 45 degrees, it's passed (or vice versa, depending on how you oriented the plate.) So you will see half of the light transmitted, and half absorbed, just like in the classical description.

This concept is very central to how quantum mechanics works--a system that is purely, definitively, 100% in one state can also be viewed as a superposition of multiple other states, if they don't commute with the first state. It's exactly analogous to how you can think of a vector that goes along the X axis as having only an X coordinate, and a Y coordinate of 0. But there's nothing really special about that vector--if you looked at it in a rotated coordinate system, it would have nonzero values for both components. So nothing is special about one basis or another, you just transform it into whatever basis you need for that problem. The way the state transforms under a change of basis determines its behavior. In this case, the equations for how you transform from one polarization basis into another are what give rise to the classical equation of Malus's Law.

 

[DrChinese]

You also say "Vertical and 45 degrees don't commute". Now Wikipedia http://en.wikipedia.org/wiki/Commutativity#Non-commuting_operators_in_quantum_mechanics defines commutativity like this: which seems to have the exact opposite of your intention. I would of thought that H polarisation and V polarisation do not commute and while V polarisation and 45 degree polarisation do commute because we can measure both for a single photon. Then again, I am not that familiar with the terminology and I am probably missing the point here.

Chopin is correct: 0 and 45 degrees do not commute. 0 and 90 do commute. It is different for photons vs. electrons. For electrons, 0 and 90 do not commute.

 

[yuiop]

To Chopin and DrChinese, thanks for taking the time to help me come to terms with the terminology. The "mental picture" of what what is going on, that I had formed in my mind might be correct mathematically, but it is not the formal interpretation, so thanks for clearing that up. I might be better able to understand the literature now, so thanks again.