Hypotheses of Quantum Mechanics
The underlying structure of the theory is based on the representation of "states" of a system which are to be represented by (i.e., to be in one-to-one correspondence with) vectors in some suitable chosen vector space. The messurements of the attributes of these states ("dynamical varibles" or "observables") are described in terms of operations on these vectors. These operations are assumed to be linear. The operation of an operator on a vector is intended to describe a physical operation (measurment) on the system. The relation between the mathematical operation and the physical "measurement" is assumed to be the following: that the result of a "measurment" of a dynamical variable must be an eigenvalue of the linear operator representing that dynamical variable. The state in which the dynamical variable has that value is represented by the corresponding eigenvector. The above statement is, of corse, an imprecise one, and raises many questions. For instance, what is the vector space used to describe a given physical system? How are the operators representing given dynamical variables to be chosen? How does one describe the time evolution of a dynamical system? These, and many other questions must be answered if we are to formulate a complete working theory. What we have done here, however is to specify the framework within which the theory is to be developed; a framework having its origin in the historical evolution of the subject. The historical process was, however, a tortuos one, marked by contradictions (experimental data in conflict with then existing theoretical concepts), hypothesis (often ad hoc, sometimes inspired), synthesis and generalization. The end product is much clearer and more coherent than the process by which it was constructed. We, therefore, forego historical analysis and build our structure inductively but with the wisdom of hindsight, in order to avoid wasting time on transitional steps and misleading sidetracks.
Let us, therefore, set about to construct, within our specified general framework, a detailed theory.
Consider first the question of the vector space by which a physical system can be represented. The major point to be made here is that this space is determined, not by mathematical prescription, but with physics. Since states of systems are to be represented by vectors, the vector space must be determined by the totality of attributes of the system. This is, of course,a function of our understanding, and must be modified from time to time in the light of discovery. This point may be readily illustrated by reference to the phenomenon of electron spin. If (to keep matters simple) we consider as our system the hydrogen atom, prior to the discovery of spin the vector space needed to describe the system would have been one based on the independent states of the atom as described by three "quantum numbers" specifying the energy of the atom, its total angular momentum, and one component thereof. However, once it was recognized that the electron possessed not only these characteristic but alos intrinsic angular momentum ("spin"), corresponding to each previously know state it was now necessary to recognize two, corresponding to the two states of electron spin which were found to exist. Thus, the recognition of a new physical attribute made it necessary to double the "dimensionality" of the vector space used to decribe the system. That is, at each point of the original "space" it was now necessary to construct a two-dimensional mainfold. Such a situation is of course well known to mathemticans, in whose language one says that the new vector space was the "direct product" of the original one and the new two-dimensional one. From every combination of one vector of the original one space and one of the "spin space" we construct a new vector of the product space.
The next question to be answered is, how does one choose the operators to represent the various dynamical variables?
Let use reformulate the problem in a more meaningful way. If we knew the complete spectrum of dynamical variable, that is, the array of values whihc could result from a measurment of the variable, we could easily set up a set of vectors and a corresponding operator. Consider, for example, vectord represented by column matrices. If kn where the possible results of measurement of the dynamic variable K, the matrix could be used to represent the dynamical variable K.
The eigenvalues would then be the results of measurement, as required; eigenvectors, representing the corresponding states, would be those vectors which have all but one component equal to zero. Conjugate vators could be represented by row matrices. The magnitude of such vectors are in each case the square of the nonzero element. If the nonzero element is chosen to be unity, the magnitude is unity, and the vectors are normilized.
These eigenvectors can be used as a basis for a vector space, in which the general vector is a row vector with element, a1, a2, ... ,an [These vectors in general have an infinit number of components.] If the a's are complex numbers, we have a complex vector space. This is required to formulate the quantum theory.
We have interpreted the "unit vectors" as representing states in which the dynamical variable K has specific values. What, then is the meaning of of a general vector; what sort of state does it represent? If we designate the eigenvectors (eigenstates) of K as "pure states" with respect to this dynamical variable, a general state is a superposition of pure states. It is necessary to interpret the components of the vectore in realtion to the measurement of the dynamical variable. For this purpose, we introduce a new and fundamental hypothesis. If we normalize the vector, that is to say, multiply by such constant value that the square of the magnitude of the resulting vector is unity, this hypothesis may be stated in the following terms: that the squares of the magnitudes of the components (that is, of the coefficients of the eigenvectors) represent the probability that a measurement of the dynamical variable K on the system in question will yield the corresponding eigenvalue. Thus, the probability that the result of a measurement yields the vaule kn is given by |an|2. The normalization of the vector assures that the probability adds up to unity.