In our approach we don’t even need to consider the wave function of the measuring device. It is sufficient if we consider the effect of the electromagnetic field present in the measuring device on the wave function of the particle to be measured. From the considerations made in previous chapter we know that we have a susceptibility to initial conditions if there is absorption or emission of the radiation, which is always the case in any measurement process as we will show.

To understand what happens in the measurement we must look at how a measurement apparatus is designed:

If you are familiar with the concept of common set of eigenfunctions for two observables you can skip to *****skippers mark***** ...

Momentum and position are examples of physical observables. There are many other observables like angular momentum , energy etc.

The particular form of the wave function where an observable has an exact (infinitely sharp) value is called the eigenfunction of the observable. The sharp value corresponding to a given eigenfunction is called the eigenvalue of the observable. The plain waves are the eigenfunctions of the momentum. Each observable has in general infinite number of eigenfunctions each having a different form corresponding to a different eigenvalue. In the case of momentum these are plain waves with different wave lengths and different propagation directions. The energy eigenfunctions are the stationary solutions of the Schrodinger equation. Thus although the eigenfunctions of other observables don’t depend on the form of the external fields (namely their strength and variation in space), the energy eigenvalues and the form of the energy eigenfunctions depend on the form of the external fields for the particular problem.

If two different observables have common set of eigenfunctions they are said to be commuting observables. To give an example lets consider the momentum eigenfunctions. If there are no external fields, plane waves are stationary solutions of the Schrodinger equations thus they are also energy eigenfunctions.

Generally to different eigenfunctions of an observable correpond different eigen values. However there may be different eigenfunctions belonging to the same eigenvalue. Such eigenfunctions are called degenerated. For example two plane waves with the same wave length but propagating in different directions have the same energy.

*****skippers mark*****

Now we can understand what happens in a measurement process:

What one does in a measurement is trying to obtain a state that has a sharp value of the observable one wants to measure. How can we force the wave function to such a transition.

To achieve this goal the physicists subconsciously make use of the principle we arrived in the previous chapter. Lets remember it:

 Except for free particles non-stationary states are not equilibrium states because of the fluctuations in charge density distribution and the resulting interaction/energy exchange with electromagnetic field. Such a superposition can exist only during emission or absorption of radiation.

A measurement apparatus is prepared in such a way, that the corresponding energy eigenstates(stationary states)coincide with the eigenstates of the observable one wants measure. As a consequence, if the wave function is in a superposition of different eigenfunctions of the observable before the beginning of the measurement, namely if the observable to be measured has an unsharp value, its energy is also unsharp. The wave packet becomes instable when it enters the measuring device since it is not in a stationary state. Through emission or absorbtion process it proceeds towards an energy eigenstate which is because of the design also the eigenfunction of the measured observable.

You may think at this point that if there is a susceptibility to initial conditions as we claimed the energyeigenstate that has the greatest weight in the superposition initially should always be the final state.

The reality is different. The energy eigenstate with the greatest weight has the greatest probability but energy eigenstates with less weight have also a probability proportional to their weight in the superposition because of the following reason:

The situation is illustrated in an analogy in Figure figure 23 in a simplified form.

Figure 23 a)sphere on a plane(stable state before the wave packet enters the measuring device) , b)the plane begins to curve (the moment when the wave packet enters the measuring device) c)sphere on a convex surface (the wave packet is in the measuring device)

We said that the energy eigenstates of the measurement apparatus coincide with the eigenstates of the observable one wants measure. In reality however it is sufficient if the eigenstate of the "measurement apparatus" coincides not necessarily exactly but only approximately with the eigenstates of the observable to be measured, as for example in the "position measurement" with a scintillating crystal. The eigenstates of position are delta functions in space ¹ which are idealizations and never exist in nature. In reality the final state of a particle after the "position measurement" is not a delta function but one of the eigenstates of an atom in the scintillating material. It is this quasi-indeterministic collapse to a very localized state that led to the persistence of the unverified concept of a pointlike electron and to the persistence of the concept of duality. It is we, who give the meaning "position measurement" to such a transition.

Lets look at another example of measurement namely the spin measurement with stern gerlach apparatus in figure 24

 Figure 24 - Stern-Gerlach apparatus

A beam with arbitrary aligned spin 1/2 particles enters the magnet field. Lets name the direction of the magnet field z direction. When the beam comes out of the magnet field it is split into two beams. One beam consisting of particles with a parallel alignment of the spin relative to magnet field (spin up beam) and the other beam consisting of particles with anti-parallel alignment of the spin relative to magnet field (spin down beam). Thus whatever the initial alignment is the particle leaves the magnet either totally parallel or totally anti-parallel to magnet field . Classically one would expect that according to the projection of the spin on z axis angular momentum values in between should be observed. Quantum mechanically a particle with for example a spin that is aligned on the xy plane initially (no contribution from z direction) has equal probabilities to be found in spin up beam and spin down beam state. Any arbitrary initial alignment of the spin can be written as the superposition of spin up and spin down states of any fixed direction of choice(z direction in our case). The weight in superposition determines the probability for the particle leaving the magnet in the up beam or the down beam. Thus if we have an initial beam of particles each having the same spin alignment (for example prepared as an up beam of another Stern-Gerlach apparatus that is placed prior to the second one) with a Stern-Gerlach apparatus by measuring the intensity of the outcoming up and down beams one can get information about the initial spin state.

And lets take a look what happens actually in the magnet field:

In the absence of the magnetic field the energy of a particle is independent of its spin. In magnetic field spins aligned parallel to magnet field have lower energy then the spins aligned antiparallel to the magnetic field Thus the spin degeneracy of energy is removed with magnetic field. Only spins aligned parallel or antiparallel to the magnet field represent energy eigenstates. An arbitrarily aligned spin is a superposition of these energy eigenstates. This leads to instability for an arbitrarily aligned spin because of the interaction with electromagnetic field. It in this case the reason for the instability is not the fluctuations in charge density since spin is a separate degree of freedom independent of the spatial wave function. The reason is that an arbitrarily aligned magnetic moment in external magnetic field is not in equilibrium. There is a torque acting upon it. The situation is in some aspects similar to that of the compass needle in the magnetic field. There are two equilibrium states for the compass needle. The one we usually observe when pointing to the north direction and the other one pointing in opposite direction. The second state of the needle is however instable so that we can never observe the needle in this state. Although the torque acting in this state is 0 (namely although it is in equilibrium) slightest deviation from this position would proceed towards stabile position so that we can never observe the needle in this instable position. In these aspects the situation our spin ½ particle in the magnet field are very similar. There is some aspect however that is not similar. In the case of compass needle the energy difference between stable and unstable equilibrium is very high (at macroscopical scales) . In the case of the particles with spin in the magnet field the difference between the energies is so small that at laboratory temperatures the thermodynamic occupation probability is very close to each other. This means the photon density is high enough so that probability for the higher energy state to “fall down” by spontaneous emission is comparable to the probability that a particle in the lower energy state jumps up to the higher state by absorption of a photon that is available at these temperatures. Therefore the system can not remain in this superposition and proceeds until either the higher energy state is achieved by absorbtion of electromagnetic radiation or until the lower energy state is achieved by emission of radiation . Thus it is not a measurement in known sense but it is simply enforcing the particle to parallel or antiparallel alignment.

Thermodynamically however the transition towards the lower energy state is slightly (the degree depending on temperature and the energy difference) more probable then a transition to the higher energy state because the thermodynamical occupation probabilities not exactly the same. Thus if the tempretaure is lowered the difference in occupation probabilities would increase distorting the conclusion one can make about the initial spin state. The measurement can only be regarded as accurate as long as the energy difference between the split states are so small that at the temperatures at wich the experiment is conducted the termodynamical occupation probabilities can be considered as almost equal.

 Back to the position measurement. Our explanation is not sufficient as it may seem at first look.

Thus the quasiindeterminacy can be understood by the instability of nonstationary states and the sensitivity of the oscillation modes to fluctuations. A concept of fundamental irreducibel indeterminacy is not needed.

1 delta function is an infinitesimally narrow and infinitely high peak. One obtains it as a limit when one makes a peak narrower and at the same time higher so that the area under the peak remains the same.