Friday, 27 January 2017

Measurement probabilities

In the last post we mentioned that waves on a guitar string could be 'pure' waves- each with a fixed number of equal vibrating segments and a specific rate of vibration (or frequency)- or, more generally, the string could vibrate as a more-complicated jumble of pure states.

The overall shape of a jumbled wave depends on exactly what combination of pure states make it up. If you think of the pure states as being ingredients in a recipe for a complicated wave, then we can adjust the relative quantities of each of the ingredients to make different hybrid waves.

You can actually hear this effect if you pluck a guitar string in different places along its length. If you pluck the string near one end you will get a more' twangy' sound than if you pluck it in the middle. The reason  is that plucking it near the end sets off more of the higher frequency pure waves, thus changing the recipe of the sound somewhat.

The same holds true in quantum theory for the particle waves. Depending on the surroundings the particle finds itself in, its wave can be any mix of pure energy states (or eigenstates). In such a jumbled state the particle doesn't have a well-defined energy. As we mentioned in the last post, when you measure the particle's energy you do always find that the particle's jumbled wave does suddenly change into one of the pure energy waves, and the energy you measure is the energy associated with that pure wave.

The weird thing is that you can never tell which of the pure energy waves the particle's jumbled wave will switch into, and so you can never be sure what result you will get if you try to measure the energy of a particle in a jumbled state. Quantum theory contains an essential degree of uncertainty.

However, quantum theory does tell you the probability of getting a particular result when you measure the energy of a particle with a jumbled wave. It turns out that the probability of measuring a particular energy value is proportional to how much of that value's associated pure energy wave was in the recipe for the jumbled wave.

That's a difficult idea to express without maths, so let's go over it again in a different way. Let's suppose that 10% of particular pure energy wave was one of the ingredients for a jumbled wave of a particle. If you measure that particle's energy, there's a 10% chance that the answer you get will be the energy of that pure wave. If you have a particle with a different jumbled energy wave, which includes 50% of a certain pure energy wave, then there will be a 50% chance that the result of measuring the particle's energy will be the energy associated with that pure energy wave.

So, to recap, particles can have 'pure' energy waves, each of which has a specific energy associated with it. Conversely a particle can have a jumbled wave, which is made up of a bit of one pure wave, and a bit of some other pure wave, and so on. When you measure the energy of a particle, the particle's jumbled wave switches at random to become one of the pure waves in its recipe, and the probability of it switching to a particular pure wave depend upon how much of that pure wave is in the recipe.

As an analogy, imagine you made a jar of mixed spices with some pepper, some mace, some ginger, some coriander, some cumin, etc. Then you asked someone to taste it and say which single spice it tastes like. You won't know for sure which answer you will get, but probably the chance of getting one particular answer- ginger say,- will depend on how much ginger is in the mixture compared with anything else.

The weird thing about quantum theory, however, is that once you've measured the energy of a particle with a jumbled wave, its wave switches to a single pure energy wave. It's as if your bottle of mixed spice turns into pure ginger if someone tastes it and thinks it is most like ginger!

he answer you get is one of the eigenvalues ButA particle wavemiddle you'll get a more mellow sound than if you

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