Principle uncertainty

Heisenberg mathematically related the uncertainty in position. Ax, and the uncertainty in momentum, A(mv), to a quantity involving Planck s constant  

A brief calculation illustrates the dramatic implications of the uncertainty principle. The electron has a mass of 9.11 X 10 kg and moves at an average speed of about 5X10 m/s in a hydrogen atom. Let s assume that we know the speed to an uncertainty of 1% [that is, an uncertainty of (0.01 )(5 X 10 m/s) = 5 X 10 m/s] and that this is the only important source of uncertainty in the momentum, so that A(mv) = m Av. We can use Equation 6.9 to calculate the uncertainty in the position of the electron  

Because the diameter of a hydrogen atom is about 1 X 10 m, the uncertainty in the position of the electron in the atom is an order of magnitude greater than the size of the atom. Thus, we have essentially no idea where the electron is located in the atom. On the other hand, if we were to repeat the calculation with an object of ordinary mass, such as a tennis ball, the uncertainty would be so small that it would be inconsequential. In that case, tn is large and Ax is out of the realm of measurement and therefore of no practical consequence. 

De Broglie s hypothesis and Heisenberg s uncertainty principle set the stage for a new and more broadly applicable theory of atomic structure. In this approach, any attempt to define precisely the instantaneous location and momentum of the electron is abandoned. The wave nature of the electron is recognized, and its behavior is described in terms appropriate to waves. The result is a model that precisely describes the energy of the electron while describing its location not precisely but rather in terms of probabilities. 

What is the principal reason we must consider the uncertainty principle when discussing electrons and other subatomic particles but not when discussing our macroscopic world  

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Heisenburg uncertainty principle For small particles which possess both wave and particle. properties, it is impossible to determine accurately both the position and momentum of the particle simultaneously. Mathematically the uncertainty in the position A.v and momentum Ap are related by the equation... 

The Heisenberg uncertainty principle offers a rigorous treatment of the qualitative picture sketched above. If several measurements of andfi are made for a system in a particular quantum state, then quantitative uncertainties are provided by standard deviations in tlie corresponding measurements. Denoting these as and a, respectively, it can be shown that... 

One feature of this inequality warrants special attention. In the previous paragraph it was shown that the precise measurement of A made possible when v is an eigenfiinction of A necessarily results in some uncertainty in a simultaneous measurement of B when the operators /land fido not conmuite. However, the mathematical statement of the uncertainty principle tells us that measurement of B is in fact completely uncertain one can say nothing at all about B apart from the fact that any and all values of B are equally probable A specific example is provided by associating A and B with the position and momentum of a particle moving along the v-axis. It is rather easy to demonstrate that [p, x]=- ih, so that If... 

Figure Al.6.15. Schematic diagram, showing the time-energy uncertainty principle operative in resonance Raman scattering. If the incident light is detuned from resonance by an amount Aco, the effective lifetime on the excited-state is i 1/Aco (adapted from [15]).

For each degree of freedom, classical states within a small volume A/ij Aq- h merge into a single quantum state which cannot be fiirther distinguished on account of the uncertainty principle. For a system with /... 

Main J, Mandelshtam V A and Taylor H S 1997 High resolution quantum recurrence spectra beyond the uncertainty principle Phys. Rev. Lett. 78 4351... 

The force F which has to be applied to a molecular lever requires accurate knowledge of its position x if reversible work is to be perfonned. Specifying the positional accuracy as Ax, the uncertainty principle gives the energy requirement as... 

The flash lamp teclmology first used to photolyse samples has since been superseded by successive generations of increasingly faster pulsed laser teclmologies, leading to a time resolution for optical perturbation metliods tliat now extends to femtoseconds. This time scale approaches tlie ultimate limit on time resolution (At) available to flash photolysis studies, tlie limit imposed by chemical bond energies (AA) tlirough tlie uncertainty principle, AAAt > 2/j. 

Because of the quantum mechanical Uncertainty Principle, quantum m echanics methods treat electrons as indistinguishable particles, This leads to the Paiili Exclusion Pnn ciple, which states that the many-electron wave function—which depends on the coordinates of all the electrons—must change sign whenever two electrons interchange positions. That IS, the wave function must be antisymmetric with respect to pair-wise permutations of the electron coordinates. 

Another feature of the spectrum shown in Figure 10.19 is the narrow width of the absorption lines, which is a consequence of the fixed difference in energy between the ground and excited states. Natural line widths for atomic absorption, which are governed by the uncertainty principle, are approximately 10 nm. Other contributions to broadening increase this line width to approximately 10 nm. 

The electromagnetic spectrum is a quantum effect and the width of a spectral feature is traceable to the Heisenberg uncertainty principle. The mechanical spectrum is a classical resonance effect and the width of a feature indicates a range of closely related r values for the model elements. 

From the Heisenberg uncertainty principle as stated in Equation (1.16) estimate, in cm and Hz, the wavenumber and frequency spread of pulsed radiation with a pulse length of 30 fs, typical of a very short pulse from a visible laser, and of 6 ps, typical of pulsed radiofrequency radiation used in a pulsed Fourier transform NMR experiment. 

If the radiofrequency spectmm is due to emission of radiation between pairs of states - for example nuclear spin states in NMR spectroscopy - the width of a line is a consequence of the lifetime, t, of the upper, emitting state. The lifetime and the energy spread, AE, of the upper state are related through the uncertainty principle (see Equation 1.16) by... 

An important consequence of shortening a laser pulse is that the line width is increased as a result of the uncertainty principle as stated in Equation (1.16). When the width of the pulse is very small there is difficulty in measuring the energy precisely because of the rather small number of wavelengths in the pulse. For example, for a pulse width of 40 ps there is a frequency spread of the laser, given approximately by (2 iAt), of about 4.0 GFIz (0.13 cm ). 

The uncertainty principle, according to which either the position of a confined microscopic particle or its momentum, but not both, can be precisely measured, requires an increase in the carrier energy. In quantum wells having abmpt barriers (square wells) the carrier energy increases in inverse proportion to its effective mass (the mass of a carrier in a semiconductor is not the same as that of the free carrier) and the square of the well width. The confined carriers are allowed only a few discrete energy levels (confined states), each described by a quantum number, as is illustrated in Eigure 5. Stimulated emission is allowed to occur only as transitions between the confined electron and hole states described by the same quantum number. 

Early in the twentieth century physicists established that molecules are composed of positively charged nuclei and negatively charged electrons. Given their tiny size and nonclassical behavior, exemplified by the Heisenberg uncertainty principle, it is remarkable (at least to me) that Eq. (1) can be considered exact as a description of the electrostatic forces acting between the atomic nuclei and electrons making up molecules and molecular systems. Eor those readers who are skeptical, and perhaps you should be skeptical of such a claim, I recommend the very readable introduction to Jackson s electrodynamics book [1]. 

The uncertainty principle necessitates that any extremal trajectory should be spread , and the next step in our calculation is to find the prefactor by incorporating the small fluctuations around... 

Equation (4.24) indicates that the quantum number of the transverse x-vibration is an adiabatic invariant of the trajectory. At T=0 becomes the instantaneous zero-point spread of the transverse vibration (2co,) in agreement with the uncertainty principle. 

Of the variety of quantum effects which are present at low temperatures we focus here mainly on delocalization effects due to the position-momentum uncertainty principle. Compared to purely classical systems, the quantum delocalization introduces fluctuations in addition to the thermal fluctuations. This may result in a decrease of phase transition temperatures as compared to a purely classical system under otherwise unchanged conditions. The ground state order may decrease as well. From the experimental point of view it is rather difficult to extract the amount of quantumness of the system. The delocahzation can become so pronounced that certain phases are stable in contrast to the case in classical systems. We analyze these effects in Sec. V, in particular the phase transitions in adsorbed N2, H2 and D2 layers. 

In the earlier treatment we reached the conclusion that resonance absorption occurs at the Larmor precessional frequency, a conclusion implying that the absorption line has infinitesimal width. Actually NMR absorption bands have finite widths for several reasons, one of which is spin-lattice relaxation. According to the Heisenberg uncertainty principle, which can be stated... 

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