heisenberg uncertainty principle fourier transform

#Heisenberg uncertainty principle fourier transform pdf

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 Quantum level.In Quantum Mechanics, it is postulated that a physical state is represented by a state vector containing all the information about it. Limitations when interpreting the impact of the HUP on the nature of the That we need to discern between measurement limitations and inherent ByĬomparison with other fields where Fourier Transform theory is used, we propose Manifestation of the Nyquist-Shannon Sampling Theorem at the quantum level. We show that Brillouin zones in Solid State physics are a It is a better principle to describe the measurement limitations of Quantum The Nyquist-Shannon Sampling Theorem of Fourier Transform theory, and show that We introduce the Quantum Mechanical equivalent of We note that the HUP is not a quantum mechanical It also establishes that the HUP is purely a relationshipīetween the effective widths of Fourier transform pairs of variables (i.e.Ĭonjugate variables). Uncertainty Theorem of Fourier Transform theory demonstrates that the HUPĪrises from the dependency of momentum on wave number that exists at the

#Heisenberg uncertainty principle fourier transform pdf

Millette Download PDF Abstract: The derivation of the Heisenberg Uncertainty Principle (HUP) from the Download a PDF of the paper titled The Heisenberg Uncertainty Principle and the Nyquist-Shannon Sampling Theorem, by Pierre A.