![]() ![]() “The position-representation of a physical state is the Fourier transform, denoted by FT, of its momentum-representation” Generalizing for all other states, we propose the following primordial hypotheses. Is then somehow a Fourier transformation which, as we know, transforms the function Is equal to unity at any position in space. As the particle can take any place in space its position-representation Which is equal to zero anywhere except at the point This state corresponds to a wave function Of a universe fulfilled with identical particles having the same momentum Fourier Relationships between Different Representations , the Shrödinger equations, the creation of a photon from deexcitation of an atom following Bohr, the Heisenberg uncertainty principle.Ģ. , the Dirac fundamental commutation relation ![]() The principles of quantum mechanics which are the Planck relation We obtain then from the hypothesis saying that The aim of this work is to study the relationships between different modes of representation of a state. Lastly, it is important to remark that each set of eigenketsĬonstitutes a basis for spanning the whole Hilbert space of states, so that a ket may be represented by its components on each of these bases, for examples Represent respectively the probability for finding the particle in a state The eigenvalues of these operators are real numbers so that the state space is nondenumerably indefinite, called Hilbert space.įor example is a function of x, depending on the parameters containing inĪnd is called position-representation or wave function of The set of eigenkets of these operators are denoted byįor simplicity whenever we work in one dimensional (1D) space ![]() , the lap of time t counted from some origine of time and the energy E of a particle. Is an eigenket of A corresponding to the eigenvalue aįor simplicity we consider particles moving in a one-dimensional space and will generalize for three-dimensional space whenever it is necessary.īe respectively the operators representing the measurements of the x-component of the position The principle of measurement stipulates that if in a stateĪn observable A has a definite value then applying the operator A on Similar to the scalar product of two functionsĪn observable, such as momentum components, or more precisely the measure of an observable, may be represented by an operator which acts on a ket from the left or on a bra from the right. An inner product between a bra and a ket is then defined and denoted by the bracket In the similar manner as in Hermitian conjugation a row vector is formed with the complex conjugates of elements of a column vector. To the kets space is associated a dual space containing vectors called bras and denoted by This complex vector is denominated ket following Dirac and is denoted by the symbol Nevertheless, the general meaning of the energy-time principle is that a quantum state that exists for only a short time cannot have a definite energy.In Quantum Mechanics, it is postulated that a physical state is represented by a state vector containing all the information about it. For technical reasons beyond this discussion.
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