# Re submission Modelling non ideal apparatus

1. Oct 22, 2011

### deneve

I’m playing around with possibilities and my own ideas about “the measurement problem” and have a query about the validity of my approach to modelling non ideal types of apparatus. If anyone can offer critical/helpful suggestions or comments about my approach below, then I would be very grateful. I tried to submit this before but the equations did not show.

My reasoning goes as follows:

Let S be the state space of a quantum system and A that of the experimental apparatus ( also considered as a quantum system)for measuring the system S

Consider the development of the combined system SA (tensor product)and let |s1>,|s2> belonging to S be two eigenstates of the object corresponding to two different results of the experiment. These results must leave the apparatus in different states |a1> and |a2> (describing say, different positions of a pointer.). Suppose the apparatus is initially in another eigenstate |a0>. The experiment therefore consists of allowing the object and the apparatus to interact in such a way that if the object state is |s1>, then after the experiment, the object will still be in the state |s1> and the apparatus will record the appropriate result, i.e. will be in the state |a1>. A similar argument holding for |s2> and |a2>. This is what I believe to represent an ideal measurement where the measuring devices works properly.

Thus during the experiment the Hamiltonian H in exp(-iHt/hbar) must be such that:

exp(-iHt/hbar) (|s1>|a0>)=|s1>|a1>

exp(-iHt/hbar) (|s2>|a0>)=|s2>|a1> (1)

Where t is the time taken for the experiment to yield a definite result.

Now, if before the experiment the system was in the state

|s0> = c1|s1> + c2|s2> (2)

Then after it, the system and the apparatus together will be in the state

exp(-iHt/hbar) (|s0>|a0>)

= exp(-iHt/hbar) (c1|s1>|a0> + c2|s2>|a0>) (3)

I’ve wondered about how things would be in real non ideal measurements where say the apparatus was less reliable. Being a macroscopic device with many degrees of freedom then there are many ways one could imagine from the classical point of view that false readings might occur – pointer sticks for some reason etc. From the quantum microscopic point of view errors could occur simply because it may not be possible to make systems with Hamiltonians that are exactly appropriate. However I am not as sure about things at this level so if anyone can give me valid reasons then I would be grateful. Anyway to account for non ideal functioning of an apparatus I am guessing that during the experiment we might instead have a Hamiltonian which is such that

exp(-iHt/hbar)(|s1>|a0>)=|s1>(a|a0> + b|a1> + c|a2>)

exp(-iHt/hbar)(|s2>|a0>)=|s2>(a|a0> + c|a1> + b|a2>)

where |a|^2 + |b|^2 + |c|^2 =1 and |c1|^2 + |c2|^2 = 1. Also

|a|^2, |c|^2 are very small whilst |b|^2 is nearly = 1 (4)

Now, following the original argument, if before the experiment the system was in the state
|s0> = c1|s1> + c2|s2> (5)

Then after it, the system and the apparatus together will be in the state

exp(-iHt/hbar)(|s0>|a0>) = exp(-iHt/hbar) (c1|s1>|a0>) + c2|s2>|a0>)

= c1|s1>(a|a0> + b|a1> + c|a2>)+ c2|s2>(a|a0> + c|a1> + b|a2>)

= b c1|s1>|a1> + b c2|s2>|a2> + c c1|s1>|a2> + a c2|s2>|a0> +

c c2|s2>|a1>+ a c1|s1>|a0> (6)

The first two terms are “almost those” of equation (3), but the probability amplitudes of the others are far less dominant because of the relative sizes of the
|a|^2, |c|^2 and |b|^2 terms.

I am guessing that I can interpret this then as an experiment where the measuring apparatus usually works correctly but on rare occasions

(i) Does not detect anything when in fact it should have done - e.g. the presence of the |s1>|a0> or |s2>|a0> terms.
(ii) Very occasionally records the eigenvalue of one state |s1> when it should have recorded the eigenvalue of the other state |s2> and vice versa.

I really don’t know whether I can use this approach and would like comments and criticisms as to whether my analysis can be considered valid. For example if the effect is due to macroscopic failure (like the pointer sticking or a fuse blows and therefore the reading is in the |a0> state irrespective of the system state) is a valid interpretation of (6) or microscopically say for example, the direction of motion of a particle gets redirected because it recoils from a rogue air molecule and goes the wrong way in the detector chamber.

Many thanks for anyone who takes the trouble to answer my query.