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狄拉克《量子力学原理》书摘（11－20节）
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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 vertxi'> 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'vertalphavertxi''>. 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'_uvert 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"
https://blog.sciencenet.cn/blog-117333-345661.html
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