Using Weyl's geometrical interpretation of quantum mechanics the fundamental relations of the transformation theory can be treated without making use of Dirac's $\ensuremath{\delta}$ function. The principle of uncertainty is particularly clear in this system. It is shown that in the discrete representation for a definite axis of the coordinate $q$ all Eigenwerte of the conjugate momentum $p$ are equally probable. It is also shown how in the limit of very fine subdivision the results of Heisenberg and Bohr are obtained. The whole treatment receives an easy interpretation by applying it to the problem of the diffraction of waves by a grating. Weyl's method of discussing quantities with continuous ranges of values is found to be equivalent to investigating diffraction phenomena in the limiting case of an infinitesimal grating space. The quantum condition $i(PQ\ensuremath{-}QP)=1$ is shown to be correct in a purely operational sense in spite of the fact that as a matrix equation it is untrue.
Using Weyl's geometrical interpretation of quantum mechanics the fundamental relations of the transformation theory can be treated without making use of Dirac's \ensuremathδ function. The principle of uncertainty is particularly clear in this system. It is shown that in the discrete representation for a definite axis of the coordinate q all Eigenwerte of the conjugate momentum p are equally probable. It is also shown how in the limit of very fine subdivision the results of Heisenberg and Bohr are obtained. The whole treatment receives an easy interpretation by applying it to the problem of the diffraction of waves by a grating. Weyl's method of discussing quantities with continuous ranges of values is found to be equivalent to investigating diffraction phenomena in the limiting case of an infinitesimal grating space. The quantum condition i(PQ\ensuremath−QP)=1 is shown to be correct in a purely operational sense in spite of the fact that as a matrix equation it is untrue.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation: There are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be the subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. This latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system.
This paper is a small discussion about the uncertainty principle. With the great discoveries and researches made by predecessors, I try to answer some questions within the uncertainty principle. Using the definition of momentum, the momentum in de Broglie wave, the property of cosine function, the Euler’s formula, the basic property of plane wave, the law of derivative, the energy of quantum discovered by Max Planck, the photon energy discovered by Albert Einstein, the energy equation of a system and the Hamiltonian operator, I figure out the Schrodinger equation successfully. With the help of Schrodinger equation, the possibility density of a wave function, the concept of normalization, the average value of position and momentum, the Fourier transform, the integration by parts and the limit of normalized wave functions, I find the position operator and the momentum operator. Then using some knowledge of the standard deviation and the variance, the concept of inner product of vectors, some property of Dirac symbol, the law of calculation of complex number and the law of operator calculation, I obtain the general uncertainty relationship of abstract mathematical quantities. According to the general relationship, I get the inequality of uncertainty principle, and discuss the specific relationship between the position and the momentum. After getting the inequality, I use Heisenberg’s experiment of measuring an electron by using the γ ray to hit the electron to explain why we cannot measure the position and the momentum simultaneously. Based on Heisenberg’s experiment, I give my own explanation by discussing the earth and a spacecraft. Transforming the perspective, we can easily see how the accuracies of position and momentum change. In addition, I attempt to figure out why the uncertainty exists. However, I can only explain it by discussing the time and frequency in mathematical method and have no idea to work it out in a pure physics approach.
Based on the hypothesis that the position-representation of a physical state <x|α> is the Fourier transform of its momentum-representation <P|α> and that the time-representation <t|α> is the inverse Fourier transform of its energy-representation <E|α>, we are able to find the Planck relation E = hv, the de Broglie relation P = λh. Afterward, utilizing the Dirac delta function δ(x) and the property FTxf(x) = -iℏ∂, pFTxf(x) = -iℏ∂, pF(p) we obtain the links between operators in ordinary and Hilbert spaces leading to the Dirac fundamental commutation relation [p^,x^] = -iℏI^, the Schrödinger equations, the Heisenberg uncertainty principle in quantum mechanics, the annihilation & creation of a photon from excitation & de-excitation of an atom according to Bohr.
Although• quantum mechanics has achieved great success in the explanation of physical phenomena, the mathematics used in quantum mechanics is not rigorous.Unrigorous treatments occur in the case of continuous spectrum.In a Hilbert • space, let A be a self-adjoint operator.When we define the eigenvalue and the eigenelement by the ' ' relation Af=J.f, (1) ' ' h d -. -f(q)=).f(q)2m dq has solutions 2,ri f(q)=ce"Aq for any ;.., But these are not quadratically integrable.E. Hellinger and H. Weyi< 0 have already investigated the problem of continuous spectrum and introduced the " eigendifferential."But they are concerned with a special problem, i. e. bounded quadratic forms and solutions of differential equation.In respect of generality, they do not satisfy.< 2 > E. Fues< 3 > and some other writers<" have used this conception " eigendifferential " in quantum mechanics.But the treatments are either complicated or not rigorous.J. v. Neumann< 5 > has proved that in Hilbert space, for any selfadjoint operator A, there exists a resolution of identity E(;..) such that Af= J:and he has applied this theory to quantum mechanics.Although his theory gave a rigorous mathematical foundation to quantum mechanics, it did not give any interpretation to the representation theory of Dirac's form.Recently I investigated the "theory of vector-valued set functions."This theory removes the difficulties in the treatement of continuous spectrum, and gives a rigorous interpretation to the representation theory.< 5 > The vector-valued set function q(U) is a function of set U, and its functional value is an element (vector) in the Hilbert ,space, (1) E. Hellinger [l], 231-271.H. Weyl [1], 238-251.(2) T. Carlemann used this conception "eigendifferential" in the treatise of singular integral equations.Cf.T. Carlemann [1].(3) E. Fues [1], 295-303.(4) For example, L. de Broglie [2], 140-159.(5) J. v. Neumann [1], 53-121.(6) F. Maeda [2], [5], [6].A summary of the theory of vector-valued set functions is given in the first part Qf the present paper.(1) G. Temple has formulated a principle of quantum theory in which projective operators are given priority over self-adjoint operators.(C(.G. Temple [1)).But. the .whole discussion differs from that of my present paper.
Due to the requirements of the principle of causality in the theory of relativity, one cannot make a device for the simultaneous measuring of the canonical conjugate variables in the conjugate Fourier spaces. Instead of admitting that a particle’s position and its conjugate momentum cannot be accurately measured at the same time, we consider the only probabilities which can be determined when working at subatomic level to be valid. On the other hand, based on Schwinger's action principle and using the quadridimensional form of the unitary transformation generator function of the quantum operators in the paper, the general form of the evolution equation for these operators is established. In the nonrelativistic case one obtains the Heisenberg's type evolution equations which can be particularized to derive Heisenberg's uncertainty relations. The analysis of the uncertainty relations as implicit evolution equations allows us to put into evidence the intrinsic nature of the correlation expressed by these equations in straight relations with the measuring process. The independence of the quantisation postulate from the causal evolution postulate of quantum mechanics is also put into discussion.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation is that there are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be a chief subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. The latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system. Here we will formulate what we mean by vector spaces in general, and Hilbert space in particular, in a way that is independent of the coordinates we use to describe directions in these spaces.