Heisenberg’s uncertainty relation

Notice that in this example, the speed of the packet is inversely proportional to the packet s spatial size. While there is certainly nothing unique about this particular representation, it is interesting to speculate, along with Minsky, whether it may be true that, just as the simultaneous information about position and momentum is fundamentally constrained by Heisenberg s uncertainty relation in the physical universe, so too, in a discrete CA universe, there might be a fundamental constraint between the volume of a given packet and the amount of information that can be encoded within it. 

The natural line width is determined by Heisenberg s uncertainty relation... 

F. A Measurement Process that Goes beyond Heisenberg s Uncertainty Relations... 

If we want to show that there are physical concrete situations not described by Heisenberg s uncertainty relations, it is necessary to predict the uncertainties, for the two conjugate noncommutative observables, for example, position, Ax, and the uncertainty in momentum, p, for the microparticle M, after the interaction with the photon, and then make their product and see whether they are contained in Heisenberg uncertainty measurement space. 

The uncertainty for the momentum of the particle M, after interaction with the photon, can be predicted in many different ways, as can bee seen in a variety of textbooks on quantum mechanics. Each author tries a slightly different approach, taking into account more or fewer factors, but at the end, of course, all of them unavoidably find the same formula. The main reason why all of these authors find the same final formula, even when they follow different approaches, results from the known fact that the uncertainty for the position is fixed and given by the microscope theoretical resolution. Therefore, since the uncertainty for the position is fixed, there is no liberty for the expression of the uncertainty in momentum if one whishes, as is always the case, to stay in agreement with Heisenberg s uncertainty relations. 

Z. Marie, K. Popper, and J. P. Vigier, Violation of Heisenberg s uncertainty relations on individual particles within subset of gamma photons in e+e = 2y pair creation, Found. Phys. Lett. 1(4) (1988). 

Alternatively, we can work in momentum-space with the momentum distribution given by the square of the modulus of the momentum wavefunc-tion. However, because of Heisenberg s uncertainty relation it is impossible to specify uniquely the coordinates and the momenta simultaneously. Either the coordinates or the momenta can be defined without uncertainty. In classical mechanics, on the other hand, the coordinates as well as the momenta are simultaneously measurable at each instant. In particular, both the coordinates and the momenta must be specified at t — 0 in order to start the trajectory. Thus, we have the problem of defining a distribution function in the classical phase-space which simultaneously weights coordinates and momenta and which, at the same time, should mimic the quantum mechanical distributions as closely as possible. 

The standard derivation of Heisenberg s uncertainty relation neglects the possibility that two operators A and B, say q and p,which fulfill the commutator relation... 

A truly mechanistic (in the sense of classical mechanics) description of a molecule s reaction is in fact prohibited by Heisenberg s uncertainty relations (Equation 2.1). Some reaction mechanisms of small molecules in the gas phase have been elucidated in the utmost detail, that is, reaction rate constants have been determined for individual rotational and vibrational quantum states of the reactant. We take a more modest view a reaction mechanism is the step-by-step sequence of elementary processes and reaction intermediates by which overall chemical change occurs. 

This relation is called Heisenberg s uncertainty relation. In our example therefore, it signifies that, as the result of the definition of the electron s position by means of the slit, which involves the uncertainty (or possible error) Aa , the particle acquires momentum parallel to the slit of the order of magnitude stated (i.e. with the indicated degree of uncertainty). Only subject to this uncertainty is its momentum known from the diffraction pattern. According to the uncertainty relation, therefore, h represents an absolute limit to the simultaneous mmsurement of co-ordinate and momentum, a limit which in the most favourable case we may get down to, but which we can never get beneath. In quantum mechanics, moreover, the uncertainty relation... 

We can now deduce inequalities referring to the mean values of two real operators A, B— inequalities which lead to Heisenberg s uncertainty relation. 

In his detailed analysis of Dirac s theory [6], de Broglie pointed out that, in spite of his equation being Lorentz invariant and its four-component wave function providing tensorial forms for all physical properties in space-time, it does not have space and time playing full symmetrical roles, in part because the condition of hermiticity for quantum operators is defined in the space domain while time appears only as a parameter. In addition, space-time relativistic symmetry requires that Heisenberg s uncertainty relations. 

Due to the contribution of various broadening mechanisms, the linewidths typically observed in atomic spectrometry are significantly broader than the natural width of a spectroscopic line which can be theoretically derived. The natural width of a spectral line is a consequence of the limited lifetime r of an excited state. Using Heisenberg s uncertainty relation, the corresponding half-width expressed as frequency is ... 

Assume that all other line-broadening effects except the natural linewidth have been eliminated by one of the methods discussed in the previous chapters. The question that arises is whether the natural linewidth represents an insurmountable natural limit to spectral resolution. At first, it might seem that Heisenberg s uncertainty relation does not allow outwit the natural linewidth (Vol. 1, Sect. 3.1). In order to demonstrate that this is not true, in this section we give some examples of techniques that do allow observation of structures within the natural linewidth. It is, however, not obvious that all of these methods may realty increase the amount of information about the molecular structure, since the inevitable loss in intensity may outweigh the gain in resolution. We discuss under what conditions spectroscopy within the natural linewidth may be a tool that really helps to improve the quality of spectral information. 

These fluctuations are illustrated in Fig. 9.95 in two different ways the time-dependent electric field E t) and its mean fluctuations of the amplitude 0 and phase (p are shown in an E t) diagram and in a polar phase diagram with the axes E and 2- In the latter, amplitude fluctuations cause an uncertainty of the radius r = o, whereas phase fluctuations cause an uncertainty of the phase angle (p (Fig. 9.95b). Because of Heisenberg s uncertainty relation it is not possible that both uncertainties of amplitude and phase become simultaneously zero. 

The final state density can be derived firom Heisenberg s uncertainty relation. The space coordinates shall be denoted by x, y, z, and the momentum coordinates by p, py, p. Then, according to the uncertainty relation (in the form used by Fermi (1934))... 

As is well known from atomic physics, the diffraction of light can be explained by Heisenberg s uncertainty relation. Photons passing through a slit of width Ax have the uncertainty A px of the x-component px of their momentum p, given by ApxAx>h (Fig. 2.25). 

In analogy to infrared spectroscopy, probe molecules can be used to study Lewis acid sites. In the case of NMR spectroscopy, however, due to the much smaller resonance frequencies (which correspond to a much longer time scale) exchange effects may lead to an average line instead of the expected separate lines due to molecules adsorbed on the various adsorption sites. From Heisenberg s uncertainty relation ... 

Heisenberg s uncertainty relation states that the product of the measurement uncertainties of two conjugate variables, such as position and momentum, is a number on the order of Planck s constant or larger. So if position were measurable with a small uncertainty, Heisenberg s principle would imply a quite large uncertainty for a measurement of momentum. This is another point of difference from classical mechanics in which we can know both the position and momentum exactly at any instant of time. Of course, there is a correspondence between the two pictures. Recall that /i is a very tiny value relative to... 

This means AT is zero. There is no uncertainty in the measurement of T in this case. Every measurement yields the same value, namely, t. Then, according to Heisenberg s uncertainty relation, the uncertainty in the variable conjugate to T is infinite. 

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