Function of observables we are able to give a meaning to any function f of an observable, provided only that the domain of existence of the function of a real variable f(x) includes all the eigenvalues of the observable. If the domain of existence contains other pionts besides these eigenvalues, then the values of f(x) for these other points will not affect the function of the observable. Commuting variables in the special case when the two observables commute, the observations are to be considered as non-interfering or compatible, in such a way that one can give a meaning to the two observations being made simultaneously and can discuss the probability of any particular results being obtained. The two observations may, in fact, be considered as a single observation of a more complicated type , the results of which is expressible by two numbers instead of a single number. From the point of view of general theory, any two or more commuting observations may be counted as a single observable, the result of a measurement of which consists of two or more numbers. The states for which this measurement is certain to lead to one particular result are the simultaneous eigenstates. Representation The way in which the abstract quantities are to be replaced by numbers is not unique, there being many possible ways corresponding to the many systems of coordinates one can have in geometry. Each of these ways is called a representation and the set of numbers that replace an abstract quantity is called the representative of that abstract quantity corresponds to the coordinates of a geometrical object. When one has a particular problem to work out in quantum mechanics, one can minimize the labour by using a representation in which the representative of the more important abstract quantities occuring in that problem are as simple as possible. A complete set of commuting observables Let us define a complete set of commuting observables to be a set of observables which all commute with one another and for which there is only one simultaneous eigenstate belonging to any set of eigenvalues. \delta function \delta(x) is not a function of x according to the usual mathematical definition of a function, which requires a function to have a definite value for each point in its domain, but is something more general, which we may call an 'improper function' to show up its difference from a function defined by the usual definition. Thus \delta(x) is not a quantity which can be generally used in mathematical analysis like an ordinary function, but its use must be confined to certain simple types of expression for which it is obvious that no inconsistency can arise. Unit operator if \vert\xi' is multipulied on the right by \xi'\vert the resulting linear operator, summed for all \xi', equals the unit operator. The representative of a linear operator Take first the case when there is only one \xi, forming a complete commuting set by itself, and suppose that it has discrete eigenvalues \xi'. The representative of \alpha is then the discrete set of numbers \xi'\vert\alpha\vert\xi''. If one had to write out these numbers explicitly, the natural way of arraing them would be as a two-dimensional array, Matrices and linear operators the matrices are subject to the same algebraic relations as the linear operators. If any algebraic equation holds between certain linear operators, the same equation must hold between the matrices representing those operators. Relative probability amplitude We may be interested in a state whose corresponding ket \vert x cannot be normalized. This occurs, for example, if the state is an eigenstate of some observable belonging to an eigenvalue lying in a range of eigenvalues. it will give correctly the ratios of the probabilities for different \xi' 's. The numbers \xi'_1...\xi'_u\vert x may then be called relative probability amplitudes. Representation in practice To introduce a representation in practice (i) We look for observables which we would like to have diagonal, either because we are interested in their probabilities or for reasons of mathematical simplicity; (ii) We must see that they all commute -- a necessary condition since diagonal matrices always conmmute; (iii) We then see that they form a complete commuting set, and if not we add some more commuting observables to them to make them into a complete commuting set; (iv) We set up an orthogonal representation with this complete commuting set diagonal. Theorem of reciprocity the probability of the \xi's having the values \xi' for the state for which the \eta's certainly have the value \eta' is equal to the probability of the \eta's having the values \eta' for the state for which the \xi's certainly have the values \xi'. Theorem of observables Theorem 1. A linear operator that commutes with an observable \xi commutes also with any function of \xi. Theorem 2. A linear operator that commutes with each of a comlete set of commuting observables is a function of those observables. Theorem 3. If an observable \xi and a linear operator g are such that any linear operator that commutes with \xi also commutes with g, then g is a function of \xi. Standard ket The ket \psi(\xi) may be considered as the product of the linear operator \psi(\xi) with a ket which is denoted by without a label. We call the ket the standard ket. Any ket whatever can be expressed as a function of the \xi's multiplied into the standard ket. Wave function A further contraction may be made in the notation, namely to leave the symbol for the standard ket understood. A ket is then written simply as \psi(\xi), a function of the observables \xi. A function of the \xi's used in this way to denote a ket is called a wave function. The system of notation provided by wave functions is the one usually used by most authors for calculations in quantum mechanics. In using it one should remember that each wave function is understood to have the standard ket multiplied into it on the left, which prevents one from multiplying the wave function by any operator on the right. Wave functions can be multiplied by operators only on the left. Dirac The Principles of Quantum Mechanics
Causality Causality applies only to a system which is left undisturbed. If a system is small, we cannot observe it without producing a serious disturbance and hence we cannot expect to find any causal connextion between the results of our observations. Wave and particles The essential point is the association of each of the translational states of a photon with one of the wave functions of ordinary wave optics. The nature of this association cannot be pictured on a basis of classical mechanics, but is something entirely new. It would be quite wrong to picture the photon and its associated wave as interacting in the way in which particles and waves can interact in classical mechanics. The association can be interpreted only statistically, the wave function giving us information about the probability of our finding the photon in any particular place when we make an observation of where it is. Physical pictures the main object of physical science is not the provision of picutres, but is the formulation of laws governing phenomena and the application of these laws to the discovery of new phenomena. If a picture exists, so much the better; but whether a picture exists or not is a matter of only secondary importance. Superposition of states (简单说就是态叠加只是改变每个结果被测得的概率，结果本身不会变化。) The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states. State of rest Again, while there exists a classical state with zero amplitude of oscillation everywhere, namely the state of rest, there does not exist any corresponding state for a quantum system, the zero ket vector corresponding to no state at all. Properties of real operator , it should be noted that, if \xi and \eta are real, in general \xi\eta is not real. This is an important difference from classical mechanics. However, \xi\eta+\eta\xi is real, and so is i(\xi\eta-\eta\xi). Only when \xi and \eta commute is \xi\eta itself also real. Dynamical Variable When we make an observation we measure some dynamical variables. It is obvious physically that the result of such a measurement must always be a real number, so we should expect that any dynamical variable that we can measure must be a real dynamical variable. One might think one could measure a complex dynamical variable by measuring separately its real and pure imaginary parts. But this would involve two measurements or two observations, which would be all right in classical mechanics, but would not do in quantum mechanics, where two observations in general interfere with one another--it is not in general permissible to consider that two observations can be made exactly simultaneously, and if they are made in quick succession the first will usually disturb the state of the system and introduce an indeterminacy that will affect the second. Complete Set Now these states into which the system may jump are all eigenstates of \xi, and hence the original state is dependent on eigenstates of \xi. But the original state may be any state, so we can conclude that any state is dependent on eigenstates of \xi. If we define a complete set of states to be a set such that any state is dependent on them, then our conclusion can be formulated-the eigenstates of \xi form a complete set. Observable Not every real dynamical variable has sufficient eigenstates to form a complete set. Those whose eigenstates do not form complete sets are not quantities that can be measured. We obtain in this way a further condition that a dynamical variable has to satisfy in order that it shall be susceptible to measurement, in addition to the condition that it shall be real. We call a real dynamical variable whose eigenstates form a complete set an observable. Thus any quantitiy that can be measured is an observable. Hilbert space The space of bra and ket vectors when the vectors are restricted to be of finite length and to have finite scalar products is called by mathematicians Hilbert space.
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