Measurements in Quantum Mechanics

Now we already know the measurement process "Which way a particle goes" in the double slit experiment:
At the same time it should be possible to observe it passes 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 say also about the wavefunction collapse or the state vector reduction since the total amplitude a1|1> + a2|2> is decreased either on a1|1> or a2|2>. That's to say when making "Which way a particle goes" measurements the sum is taken from the initial superposition (linear combination) using measurements. After making measurement the system is described only using function φ1(x) (and φ2(x) correspondingly). The a1 and a2 values are just constants and they can be neglected. And now this new function φ1(x) (and f2(x) correspondingly) determine the successive system development. One who doesn't understand it should be calmed: nobody understand it, but it works! Only lately scientists have begun to study this "wavefunction collapse" in more intensive way in order to understand better the measurement principle in quantum mechanics. Now we are satisfied with the statement that the "Which way" measuring operation thus influences on the wavefunction in a way that φ1 or φ2 only stays and is the system characteristics. So, the next step of 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 splitted by polarizer A into parallel (||) and perpendicular (^) components being || and ^ relative to the marked axis of A.

|Q> = a|||A||> + a^|A^>

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

<Q|Q> = 1 = (a||*<A||| + a^*<A^|)·(a|||A||> + a^|A^>) = |a|||2<A|||A||> + a^*a||<A^|A||> + a||*a^<A|||A^> + |a^|2<A^|A^>

= |a|||2 + |a^|2 = 1

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

The analyser A projects |Q> on the analyser states |A||> or |A^>. After being successively projected on, for instance, |A||> one will always 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 ||

<A|||Q>new = a||<A|||A||> = a||

The observation probability is then as follows

P|| = |<A|||Q>new|2 = |a|||2

For the ^ analysis:
<A^|Q>new = a||<A^|A||> = 0

P^ = |<A^|Q>new|2 = 0

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

|Q>new = a^|A^>

P^ = |<A^|Q>new|2 = |a^|2

P|| = |<A|||Q>new|2 = 0

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

|q> = b|||B||> + b^|B^>

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 |Qnew> = 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" |Qnew> for the ||-channel is a|||A||> = b|||B||> + b^|B^> = |q>. Therefore the B|| measurement gives finally: <B|||Qnew> = a||<B|||A||> = b||<B|||B||> + b^<B|||B^>, and from this it's clearly (<B|||B||> = 1 und <B|||B^> = 0):

b|| = a||<B|||A||>
and correspondingly: b^ = a||<B^|A||>

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:

P|| = |b|||2 = |a|||2 |<B|||A||>|2

or for ^ :

P^ = |b^|2 = |a|||2 |<B^|A||>|2

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.

P|| = 0

and
P^ = |a|||2

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 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 experiments. The probability amplitude projects A on state A|| (or A^) (analyser A). From now the amplitude is determined by state A|| (or A^). And now it's the initial point for successive experiments. I.e. when measuring polarization with the rotated polarizer C, now we 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 measurement of location then we project y on the "place location" x and from now y corresponds to x. Now a next measurement of the impulse will give another result than a measurement is determined with that the impulse without 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::

<after|instrument|before>

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:

<B^|M|A||>

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

<B^|M|A^>

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

<B|||M|A||> or <B|||M|A^>

Hence we have four possibilities:

after | before
| || | ^
|| | <B|||M|A||> | <B|||M|A^>
^ | <B^|M|A||> | <B^|M|A^>

And now let's try to write down 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:

y = Mφ

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 to do it and they are closely connected with the following persons:

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.