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## 1. Atomic Structure and Quantum Mechanics

In: Dykman M ed Fluctuating nonlinear oscillators: from nanomechanics to quantum superconducting circuits. Breuer H, Petruccione F The theory of open quantum systems. Doherty A, Jacobs K Feedback-control of quantum systems using continuous state-estimation. Dong D, Lam J, Tarn T Rapid incoherent control of quantum systems based on continuous measurements and reference model. Gough J, James MR The series product and its application to quantum feedforward and feedback networks. Grigoriu A, Rabitz H, Turinici G Controllability analysis of quantum systems immersed within an engineered environment.

We assume that the conditions. If at t 0 when the interaction between S and M starts it is possible to write. This relation must necessarily be fulfilled in the ideal measurement scheme. As a consequence, it can be said that in those cases where the initial state of S is an eigenstate of the operator A S representing the physical quantity A S to be measured, the corresponding conservation law of A is valid.

But in view of the Projection Postulate such a superposition is broken down. In this last case,. It is worth noticing that inequalities 35 are obtained under the assumptions that the individual interpretation of the state vector and the Projection Postulate are valid. We are thus forced to conclude that if the initial state of S is not an eigenvector of AS, the physical quantity A is not conserved in processes of measurement of A S. In [ 52 - 55 ] we give examples of processes of measurement of the type analyzed in this section; and in [ 45 ] we deal with the continuous case.

The same result is obtained. A similar conclusion resulting from a different analysis has been obtained by P. If the process is repeated N times, the resulting average is. To sum up, in individual processes of measurement of AS the conservation law of A is in general not valid; but this law still has a statistical sense. We shall start this section with some remarks concerning the concept of probabilities.

Following tradition, we are going to adopt the expression subjective probabilities for probabilities related to the lack of knowledge in processes governed by deterministic laws; and objective probabilities for probabilities where the process is not ruled by deterministic laws. Accordingly, there is no room for objective probabilities in classical mechanics, electromagnetism and relativity.

Then, we explicitly state that a system cannot be in two different states at the same time. There is no doubt that quantum mechanics has been extremely successful in explaining radioactivity, electron-phonon scattering, interactions between light and matter and many other phenomena which involve, supposedly, only spontaneous processes. So, in principle one could expect that the analysis of these processes does not involve projections for they play a role just in cases measurements are performed.

Nevertheless, reading quantum mechanics textbooks one is forced to conclude the opposite; see for instance any book of the following list: [ 6 - 10 , 18 , 57 ]. To deal with spontaneous processes involved in phenomena such as those previously mentioned, in most cases time-dependent perturbation theory is necessary.

## Control of Quantum Systems

Consider a system with Hamiltonian H 0 which does not depend explicitly on time. It is assumed that the eigenvalues equations of H 0 have previously been solved.

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And which one should be the physical quantity to be measured? But let us come back to the declared aim of time-dependent perturbation theory. But perturbations do not induce transitions. In this sense Messiah is very clear. On the contrary, other authors seem to have forgotten that in the framework of OQM measurements are absolutely necessary in order to obtain the transition probability P i f t f.

Projections are disliked for many reasons; one of them is that they imply discontinuities. But is there a way of give an account for processes of emission and absorption of light without invoking discontinuities? We think there is not, as shown in the following. To start with, let us face this question in an intuitive and nearly classic way.

Consider the absorption of one photon by one atom. This implies that the photon must be absorbed by the atom at once and that the energy of the atom must be increased in an instantaneous way. A few authors have considered the possibility that projections may happen at the microscopic level. One of them is H. Discretion and good taste, born from experience, allow us to use quantum theory with marvelous success, despite the ambiguity of the concepts named above in quotation marks.

A postulate ensuring the statistical sense of conservation laws is adopted. Taking into account this postulate the concept of preferential states is introduced. By contrast, if the system has preferential states projections may happen. Spontaneous and measurement processes are treated on the same footing.

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SPA is compatible with epistemological realism: we assume that the world exists independently of the knowing subject and that it is possible to know it, at least in a partial way. So our discourse will be about what happens, not about what is measured or observed.

This does not mean, obviously, that it has to be right; it could happen that it be completely wrong. The primitive undefined notions of SPA are: system, physical quantity, state system and probability ; the term probability will be used as a synonym of objective probability ; see Section And systems mean either objects or collections of objects.

Postulate I: To every system corresponds a Hilbert space H whose vectors state vectors, wave functions completely describe the state of the system. Comments: i By definition, A is a constant of the motion if it satisfies conditions 44 and Comments : iii This tendency should not become actualized if the projections it induces results in a violation of Postulate III or lead the state vector outside the Hilbert space. For simplicity, instead of dealing with the general case, here we shall refer to the following one: Let H, A and B be three operators representing respectively the energy, the physical quantity A and the physical quantity B of the system.

On the other hand, as the relations. Then we state:. Definition : The preferential states of the system are the common eigenstates of H, A and B.

The condition of uniqueness of the set of preferential states remains valid in the general case. Consider, for instance, the determination of the energy levels of the Hg atom in the Franck-Hertz experiment, where a curve of electrical current versus the applied voltage is obtained; this curve presents peaks of the current at regular intervals of voltage.

But once the way the device works has been understood, the Franck-Hertz experiment provides a direct measurement of the energy difference between the quantum states of the atom: it appears on the dial of a voltmeter! It is worth stressing that no entanglement is invoked in the analysis of this experiment and the same is true of many others related, e. By contrast, in the ideal measurement scheme entanglements are unavoidable.

This is for instance the case of the system photon meeting the device tourmaline crystal mentioned by Dirac; see Section 7. In the following we shall address the conceptual problem of the ideal measurement scheme in the framework of SPA.

We are going to analyze the measurement of the physical quantity A S pertaining to the system S; for simplicity we shall deal with the discrete non-degenerate case. On the one hand, the inclusion in the formalism of two laws irreducible to one another has been a source of dissatisfaction from quantum mechanics birth. On the other hand, OQM which includes the Projection Postulate has been extremely successful in the area of experimental predictions; and even if the Projection Postulate should be applied only in cases where measurements are performed, in the present work we have shown that it is also used ad-hoc, when needed to explain processes which supposedly are spontaneous.

### 1. Introduction

Some authors have suggested that measurement processes could be a particular kind of natural processes. But, then, we confront the problem pointed out by Bell [ 60 ]: there is nothing in OQM formalism to tell which natural processes have the special status of measurements, i. Looking for a solution to this problem, we have proposed a Spontaneous Projection Approach SPA to quantum mechanics, a theory where spontaneous and measurement processes are treated on the same footing and the behavior of macroscopic and microscopic objects are ruled by the same laws.

The first step to achieve this objective is to admit that projections can occur spontaneously in nature, even in closed systems, without being acted by any external perturbation. But, then, the theory must say in which situations and to which vectors the state vector can collapse, and which are the corresponding probabilities. These goals have been achieved in the framework of SPA.

So, in general its predictions coincide with those of OQM.

Our treatment, however, has the merit of predicting results which completely agree with those obtained in the framework of OQM, without having recourse to the observer consciousness, to the macroscopic character of the measurement device, or to interactions with the environment producing decoherence, something that in the long term looks and smells like a collapse. In SPA projections occurring in spontaneous processes such as those involved in radioactivity, interactions between light and matter, etc.

Differing from OQM, SPA yields an expression for the probability of transitions to the continuum which is valid for every time and, except for some minimal restrictions, for every added potential. We have pointed out in [ 61 ] that these predictions could be experimentally tested. I am grateful to Professors D.

Bes, J. Centeno and F. Criscuolo for fruitful discussions. Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3. Help us write another book on this subject and reach those readers. Login to your personal dashboard for more detailed statistics on your publications. Edited by Mohammad Reza Pahlavani. We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals.

Downloaded: The primitive undefined notions of OQM are system, physical quantity and state ; and the formalism can be summarized in the following way [ 2 ]: To every system corresponds a Hilbert space H whose vectors state vectors, wave functions completely describe the states of the system. The measurement problem and the statistical interpretation of quantum mechanics The problem pointed out by Einstein at the Fifth Solvay Congress has been considered one of the most serious flaws that quantum mechanics confronts.

Controversies about the projection postulate and the theory of measurement Most authors agree on the following point: neither the primitive notions nor the first four postulates of OQM are controversial. The many worlds interpretation In [ 24 ] H. Decoherence Decoherence is a process which prevents different elements in the quantum superposition of the total system's wave function from interfering with each other.

## Control of Quantum Systems | SpringerLink

Validity of conservation laws in spontaneous processes and in measurement processes In the framework of OQM, in general, physical quantities are not sharp. Learn more. Volume , Issue 4. The full text of this article hosted at iucr. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account.

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