Measurements in Quantum Mechanics

We are already familiar with the measurement process required to determine "Which way a particle goes" in the double slit experiment:
It should be possible to simultaneously observe it pass through either slit 1 or 2. The probability amplitude y(x) = a₁φ₁(x) + a₂φ₂(x) will change during measurements. It takes value y(x) = a₁φ₁(x) if a particle pass through slit 1 and it takes y(x) = a₂φ₂(x) if a particle crosses slit 2. Therefore, the wavefunction will also change during measurements (either a₁φ₁(x) or a₂φ₂(x)). One can also describe the wavefunction as collapsing or the state vector being reduced since the total amplitude a₁|1> + a₂|2> is decreased either on a₁|1> or a₂|2>. This means that when one makes "Which way a particle goes" measurements, the sum is taken from the initial superposition (linear combination) using measurements. After making measurements, the system is described only with function φ₁(x) (and φ₂(x) correspondingly). The a₁ and a₂ values are just constants and they can be neglected. This new function φ₁(x) (and f₂(x) correspondingly) determines how the system develops successively. If you don't understand this, don't worry. Nobody understands it, but it works! Only recently have scientists began to study this "wavefunction collapse" intensely in order to better understand the measurement principle in quantum mechanics. Now we shoulb be satisfied with saying that the "which way" measuring operation influences the wavefunction in a way that φ₁ or φ₂ only stays and is the system characteristics. So, the next step of the system is described by y = φ₁ (and y = φ₂(x) correspondingly). Let's have a look at a simple experiment with polarizers.

The unpolarized light beam Q is split by polarizer A into parallel (||) and perpendicular (^) components being || and ^ relative to the marked axis of A. 

There are two states of A: |A_||> and |A_^> that are detected by the analyzer. The state vector |Q> is then |Q> = a_|||A_||> + a_^|A_^>. For the total probability amplitude of parallel and perpendicular polarization (also called the sum of all possibilities) we have:

|a_^|² is the ^ polarization probability.
|a_|||² is the || polarization probability.

The analyzer A projects |Q>  on the analyser states |A_||> or |A_^>. After being successively projected on, for instance, |A_||> we will observe |A_||> on the mounted A_|| analyser and never |A_^>. The system transmits only |A_||> state and the probability amplitude should be equal only to a_|||A_||> now.

Measurement || :  |<A_|||Q>|²  =  |a_|||²
Measurement ^:  |<A_^|Q>|²  =  |a_^|²

After passing of analyser A it is |Q>_new = a_|||A_||>. Each successive ideal analyser with the same qualities measures only ||

The observation probability is then as follows

And also we can have a look to the projection on state A_^> passing through one analyser:

Let's now rotate second polarizer relatively to the first one. Since it is another analysing device we will call it B having the states  |B_||> and |B_^>. The total probability amplitude B is equal to

We must always pay attention to the fact that || and ^ should be determined relative to polarizers! Let's observe the exit of A analyser, i.e. we initially have |Q_new> = a_|||A_||>.

b_|| is the component of |B_||> state and b_^ is the component of  |B_^> state. These components are defined by the projection of |q> vector on |B_||> and |B_^>-axis correspondingly. The "initial point" |Q_new> for the ||-channel is a_|||A_||> = b_|||B_||> + b_^|B_^> = |q>. Therefore the B_|| measurement gives finally: <B_|||Q_new> = a_||<B_|||A_||> = b_||<B_|||B_||> + b_^<B_|||B_^>, and from this it's clearly (<B_|||B_||> = 1 und <B_|||B_^> = 0):

One follows <B_|||A_||>¹ 1 and <B_^|A_||>¹ 0, since B_|| isn't parallel to A_|| and B_^ isn't perpendicular to A_||.

Then we will have for the probability we will observe || polarized beam:

If the analyser B is rotated in a way that A_||^ B_||, then one will have <B_|||A_||> = 0 and <B_^|A_||> = 1, i.e.
 

Now let us put the third analyser C between A and B in such a way that this one angle J takes 0° < J < 90° but it is still B ^ A.

In this arrangement, light comes from B_^. If we put away C there is no light passing through B_^! The particles (photons) after moving through analyser C possess a new state c_||f_|| + c_^f_^, having c_|| or c_^ which depends on the probability amplitude (as a function of  rotation angle J). Now it's the initial state for all others successive measurements and hence there is also a component B_^. However, the component B_^ is zero without analyser C, because a vertical arrangement was chosen between A and B.

The measuring state is always defined relative to polarizer with which one carries out the experiments. The probability amplitude projects A on state A_|| (or A_^) (analyser A). Now the amplitude is determined by state A_|| (or A_^). Now it's the initial point for successive experiments. I.e. when measuring polarization with the rotated polarizer C, we now study c_|| and c_^ relative to this new polarizer C.
Certainly, we can measure other values as for instance the wavelength of the light without influence from the measurement of the polarization.  

If we make a measurement of the location, then we project y on the "place location" x and from now y corresponds to x. Now the next measurement of the momentum will give a different result than without the previous local measurement. 

Now let's C will not be an analyser, but it will be a new instrumentation or a machine M which rotates polarization in any way. The light of particular state  (|A_||> or |A_^>) then goes through this instrument M and finally we will have B state (|B_||> or |B_^> . Hence we write in a known way from right to left::

If we have the initial state |A_||> and then try to make measurement after instrument M, we will have the following noting for the probability amplitude:

And if we have the initial state |A_^> then we will have the following noting for the probability amplitude to detect photons:

which has another value than <B_^|M|A_||>. And finally the probability amplitude to detect a photon in state <B_||| is as follows

Hence we have four possibilities:
 

And now let's try to write a more general equation for a photon state (which initially has had arbitrary state φ) that has just passed through instrument M. So, the instrument M somehow changes the polarization state of incident photon and therefore we will have another state y after the instrument:

The symbol M is neither the probability amplitude nor state (nor vector). It's something new that is called the Operator. The operator acts upon the initial state φ and produces new state y. We will mark operators using bold letters. Sometimes they are marked by sign  ^{^} above the letter of operator, for instance: Ô. Here you can find a bit more information about operators.

Everything is wonderful and good, however how will we write down amplitudes and wavefunctions now? 
There are three possibilities and they are closely connected with the following people:
 

1) Matrix algebra	(Werner Heisenberg)	}	They are all equivalent in mathematical way
2) Operators and ODE	(Erwin Schrödinger)
3) Curve integral	(Richard Feynman)

One can find out more details about Operators and the Schrödinger Differential equation in these two links.

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