Uncertainty principle , definition

In Science, every concept, question, conclusion, experimental result, method, theory or relationship is always open to reexamination. Molecules do exist Nevertheless, there are serious questions about precise definition. Some of these questions lie at the foundations of modem physics, and some involve states of aggregation or extreme conditions such as intense radiation fields or the region of the continuum. There are some molecular properties that are definable only within limits, for example, the geometrical stmcture of non-rigid molecules, properties consistent with the uncertainty principle, or those limited by the negleet of quantum-field, relativistic or other effects. And there are properties which depend specifically on a state of aggregation, such as superconductivity, ferroelectric (and anti), ferromagnetic (and anti), superfluidity, excitons. polarons, etc. Thus, any molecular definition may need to be extended in a more complex situation. 

While for some purposes it may be necessary to have accurate frequency definition, for others good time discrimination is useful. These are opposite requirements. Because of the Fourier relationship between frequency and time, the more precisely the time of a signal is known, the greater bandwidth of frequencies is necessary (there is a close analogy here with Heisenberg s uncertainty principle). Approximately, the time resolution t is the reciprocal of the bandwidth Bw, so that their product Bwr 1. 

By analogy, the energy uncertainty associated with a given state, AE, through the Heisenberg uncertainty principle can be obtained from the lifetime contributed by each decay mode. If we use the definition AE = T, the level width, then we can express F in terms of the partial widths for each decay mode T, such that... 

Let us start with an analogy. An ideal crystal, in which all the atoms are exactly located at the nodes of a geometrically perfect space lattice, can be conceived only on classical grounds and at absolute zero. However, it is impossible to accept this somewhat naive concept because of the uncertainty principle and thermal agitation at T 0°K. This does not, however, mean that the idea of crystallinity loses all definiteness or that, for instance, a crystal can melt in a continuous process, as Frenkel [1] seems to suggest. 

In his uncertainty principle, Heisenberg expressed the fact that the orbitals have to be considered as electron clouds rather than the definite circular paths of electron orbit around the nucleus. These electron clouds form an electric field in the atom. There can be some divisions in these energy levels according to the effects of external electric fields (created by other electrons or external sources). As a result, each energy level has sub-energy shells. 

This illustration may be regarded as an example of Heisenberg s Uncertainty Principle, which can be stated in the following form The more accurately the velocity of an electron is defined, the less certainly is its position known and, conversely, the more accurate the definition of the position of an electron, the less precise is the value of its velocity Expressed mathematically this becomes... 

In the classical view of the world, a moving particle has a definite location at any instant, whereas a wave is spread out in space. If an electron has the properties of both a particle and a wave, what can we determine about its position in the atom In 1927, the German physicist Werner Heisenberg postulated the uncertainty principle, which states that it is impossible to know the exact position and momentum (mass times speed) of a particle simultaneously. For a particle with constant mass m, the principle is expressed mathematically as... 

The essential point is the complementary nature of the descriptions in the time and frequency domains, a complementarity most familiar to us in the form of the time energy uncertainty principle. For our purpose we want a somewhat more detailed statement, a statement whose physical content can be loosely stated as the overall shape of the spectrum is determined by very short time dynamics, higher resolution corresponds to longer time evolution. A fully resolved spectrum is equivalent to a complete knowledge of the dynamics. We now proceed to make this into a technical statement by an appeal to the convolution theorem for the Fourier transform (51). A preliminary requirement for this development is the definition of the operation of smoothing. To erase details in a function (in our case, the spectrum) we convolute it with a localized window function. A convolution operation is defined by... 

D8.5 If the wavefunction describing the linear momentum of a particle is precisely known, the particle has a definite state of linear momentum, but then, according to the uncertainty principle, the position of the particle is completely unknown as demonstrated in the derivation leading to eqn 8.21. Conversely, if the position of a particle is precisely known, its linear momentum cannot be described by a single wavefunction, but rather by a superposition of many wavefunctions. each corresponding to a different value for the linear momentum. Thus all knowledge of the linear momentum of the particle is lost. In the limit of an infinite number of superpo.sed wavefunctions. the wavepacket illastrated in Fig. 8.31 turns into the sharply spiked packet shown in Fig. 8.30. But the requirement of the superposition of an infinite number of momentum wavefunctions in order to locate the particle means a complete lack of knowledge of the momenium. 

It follows that the particle is equally likely to be found anywhere along the X axis, which is equivalent to stating that its position at any instant is unknown. From this we conclude that a particle with wavefunction yf -. 4e has a definite momentum but an undefined position. We will come back to this subject in Chapter 3, when the Heisenberg Uncertainty Principle is discussed. 

As an elementary particle, a few laws govern the electronic behavior. The uncertainty principle forbids the electron to be definitely located (on the nucleus) and have at the same time a definite energy level (or velocity). Therefore, the electron is delocalized in space and hovers over the nucleus. Thus, the uncertainty principle... 

There exist two major approaches to the theoretical description of the time and frequency gated spontaneous emission (TFG SE). In the first approach, the TFG SE spectrum is defined as the rate of emission of photons of a certain frequency within a definite time interval. The influence of the measuring device is not taken into account in this formulation. Starting from this definition, one obtains an ideal (bare) TFG SE spectrum, which is not guaranteed to be positive, however. For instance, for certain parameters of the Brownian oscillator model, the spectrum can attain negative values. Moreover, the time and frequency resolutions of this ideal spectrum are not limited by the fundamental time-frequency uncertainty principle. This underlines the necessity to develop a more comprehensive theory, in which both a spectrometer and a time-gating device enter the description from the outset. 

The quantity, a . is referred to as the Bohr radius (52.917726 pm) because it is identical to the radius of the orbit of the 1 s electron in the Bohr atom". The eariy Bohr theory of the atom invoked eiectron orbits of definite radii, but these are invaiid since they vioiate the Heisenberg uncertainty principle. The Bohr radius is used as the atomic unit of iength. 

Given the limitations indicated by the uncertainty principle, what then is the physical meaning of a wave function for an electron That is, what is an atomic orbital Although the wave function itself has no easily visualized meaning, the square of the function does have a definite physical significance. The square of the function... 

Another very important regularity appears in measurements of the position of the electron in hydrogen. Although the uncertainty principle tells us that we cannot measure this accurately, we can give a definite estimate of the likelihood that the electron will be found in some specified region of... 

Hence if some physical measurement is recorded of the electron s position or velocity, the outcome will always be a fuzzy or blurred picture. One of the important implications of Heisenberg s Uncertainty Principle shown by this calculation is that it rules out the existence of definite paths (trajectories) of electrons. 

There is no uncertainty principle comparable to the one for momentum and position that links the energy of a state with the state s lifetime. Indeed, there is no quantum mechanical operator for the lifetime of a state. There is, nevertheless, a relationship between the lifetime and our ability to assign the state a definite energy. One way to view this relationship is to recall that the full wavefunction for a system with energy is an oscillating function of time, and that the oscillation frequency is proportional to the energy (Eq. 2.16) ... 

We have seen that, according to the de Broglie relation, a wave of constant wavelength, the wavefunction sin(2nx/X), corresponds to a particle with a definite linear momentum p = h/L However, a wave does not have a definite location at a single point in space, so we cannot speak of the precise position of the particle if it has a definite momentum. Indeed, because a sine wave spreads throughout the whole of space, we cannot say anything about the location of the particle because the wave spreads everywhere, the particle maybe found anywhere in the whole of space. This statement is one half of the uncertainty principle, proposed by Werner Heisenberg in 1927, in one of the most celebrated results of quantum mechanics ... 

State can be used to re-excite another nucleus of the same kind. The y-ray vanishes in this process, it is absorbed. By definition resonance absorption can only occur if the emission energy exactly matched the absorption energy. This is the essential point. This requires that we look in more detail at the energy distribution N E) of the emitted y-ray and at the absorption cross section Lorentzian shape with a full width at half maximum F given by Heisenberg s uncertainty principle... 

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