The Quantum Limit

The most common traditional definition of the quantum/classical limit is the point at which Planck s constant h - 0. However, this is an unreasonable stipulation [33] because h is not dimensionless and its value can therefore not be varied. A possible operational condition could be formulated in terms of a dimensionless parameter of the form h/S 1, where S is the action quantity in a given situation. It could be argued that for S sufficiently large compared to h, measurement at the macroscopic level cannot detect quantum effects because of limited instrument resolution. This argument implies that the coarse-grained appearance of a classical world is simply a question of experimental accuracy and that every physical system ultimately displays quantum features and that there is no classical limit. 

Another, equally common statement of the classical limit is in terms of the principal quantum number of a system that tends to infinity, n — oo. 

1 An ontic theory refers to qualities having a real being in an objectively existing reality. 

Even though individual energy eigenstates may lead to the required classical result in the limit n — oo, there are many examples to show that their superpositions do not necessarily satisfy the requirement. 

2A pure state exists where all members of an ensemble are identically prepared (e.g. by an electron gun, polarizer, etc.)and correspond to a single wave function. An ensemble whose member systems are represented by different wave functions correspond to a mixed state. 

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Ainong the first TFIz mixers to be constructed were those based on room-temperature Schottky diodes [11]. Over the past decade, new mixers based on superconducting tunnel junctions have been developed that have effective noise levels only a few tunes the quantum limit of [12]. Flowever, certain conditions... 

The symmetry coefficient = —P d nk/dAE is usually close to j, in agreement with the Marcus formula. Turning to the quantum limit, one observes that the barrier transparency increases with increasing AE as a result of barrier lowering, as well as of a decrease of its width. Therefore, k grows faster than the Arrhenius rate constant. At 7 = 0... 

Here Tq are coordinates in a reference volume Vq and r = potential energy of Ar crystals has been computed [288] as well as lattice constants, thermal expansion coefficients, and isotope effects in other Lennard-Jones solids. In Fig. 4 we show the kinetic and potential energy of an Ar crystal in the canonical ensemble versus temperature for different values of P we note that in the classical hmit (P = 1) the low temperature specific heat does not decrease to zero however, with increasing P values the quantum limit is approached. In Fig. 5 the isotope effect on the lattice constant (at / = 0) in a Lennard-Jones system with parameters suitable for Ne atoms is presented, and a comparison with experimental data is made. Please note that in a classical system no isotope effect can be observed, x "" and the deviations between simulations and experiments are mainly caused by non-optimized potential parameters. 

With increasing values of P the molar volume is in progressively better agreement with the experimental values. Upon heating a phase transition takes place from the a phase to an orientationally disordered fee phase at the transition temperature where we find a jump in the molar volume (Fig. 6), the molecular energy, and in the order parameter. The transition temperature of our previous classical Monte Carlo study [290,291] is T = 42.5( 0.3) K, with increasing P, T is shifted to smaller values, and in the quantum limit we obtain = 38( 0.5) K, which represents a reduction of about 11% with respect to the classical value. 

As is obvious from the table, Tc is almost doubled upon deuteration. These isotope effects are one of the largest observed in any solid state system. The question arises about isotope effects in non-hydrogen-bonded ferro- and antiferroelectrics. As already mentioned in the Introduction, within a mean-field scheme and in a purely ionic model it was predicted that these systems should not exhibit any isotope effect in the classical limit, which has been verified experimentally. Correspondingly, there was not much effort to look for these effects here. However, using a nonlinear shell-model representation it was predicted that in the quantum limit an isotope effect should... 

To stress similarities and differences with thermal activation, we present the results in the form of Arrhenius-like plots. We plot log / in units of (G/Gq)4>o versus the dimensionless Ah/A- Thermal activation with the effective temperature given by (8) would give a straight line (dashed lines in the plot). By virtue of our approach, the rates should exceed the quantum limit logJA (G/G(.i)4>()- This means that the rates should saturate at this value provided If —> 0. For each choice of S(x) we plot two curves... 

The rate constant for hydrogen atom transfer (conversion II into III) spans six orders of magnitude in the range 290-80 K. The quantum limit of the rate constant and crossover temperature are 5xl0 3s 1 and 100 K, respectively. The ratio kH/ku increases from 10 to 5 x 103 as the temperature falls from 290 to 100 K. It is the H atom in position a that is transferred, since the substitution of deuterium atom at position b (R = H) does not change the rate constant. 

Note that the quantum limit on coherent receivers is 0.5hv/k or 0.024 K/GHz (Wright, 1999). This corresponds to 0.5 photons per mode. The 0.3 K/GHz for cryogenic HEMTs is about 7 photons/mode. At 72 GHz the CMB has only n = 0.4 photons per mode, where n = (exp[hv/kT] — l)-1 is the mean number of photons per mode. Thus a background-limited incoherent detector could be much more sensitive than a coherent radiometer using HEMTs. 

When first confronted with the oddities of quantum effects Bohr formulated a correspondence principle to elucidate the status of quantum mechanics relative to the conventional mechanics of macroscopic systems. To many minds this idea suggested the existence of some classical/quantum limit. Such a limit between classical and relativistic mechanics is generally defined as the point where the velocity of an object v —> c, approaches the velocity of light. By analogy, a popular definition of the quantum limit is formulated as h —> 0. However, this is nonsense. Planck s constant is not variable. 

The quantum potential can now be identified as a surface effect that exists close to any interface, in this case the vacuum interface. The non-local effects associated with the quantum potential also acquire a physical basis in the form of the vacuum interface, now recognized as the agent responsible for mediating the holistic entanglement of the universe. The causal interpretation of Bohmian mechanics finds immediate support in the postulate of a vacuum interface. There is no difference between classical and quantum entities, apart from size. Logically therefore, the quantum limit depends on... 

If zero-point vibration amplitudes of the dot are comparable with the Fermi length of the electrons, the shuttling takes place at small bias voltage. This is the case for cold dots. The constructive interference of electron waves in the tunnel gap center effectively charges the dot. In the quantum limit, this charging requires a justification of the tunnel-term concept based on the Schrodinger equation. In next section we address a more rigorous quantum mechanical picture based on the "ab-initio" SET model. 

The point is that the zero-point oscillations are responsible for the so-called shot noise [14,15], determining the quantum limit of uncertainty in different optical measurements. The preceding result shows that the presence of an atom causes the increase of shot noise and hence a deterioration of the quantum limit of precision of measurements, at least, in some vicinity of the atom [22,29]. We discuss this effect in more details in Section VI. 

As clusters become much larger in size, the collective mode can be tracked from the quantum limit to the bulk [712], so that the evolution of the many-body resonances is known over an enormous range. In the bulk limit, it has been shown that the resonances tend towards the surface plasmon of the solid, except that, since the solid does not have spherical symmetry, the -y/3 factor of the Mie-Drude theory does not appear and... 

An examination of the Fourier method, which is a special case of an orthogonal collocation representation, elucidates the main considerations of representation theory. It will be shown that by optimizing the representation the quantum limit of one point per unit phase space volume of h can be obtained. Moreover, the Fourier method has great numerical advantages because of the fast nature of the algorithm (22-26). This means that the numerical effort scales semilinearly with the represented volume of phase space (27). 

Figure 9.5. Logarithmic plot of the experimental rate constants obtained at room temperature in methyltetrahydrofurane versus free energy changes of the ET reactions indicated in the figure. Structures are the various acceptors. The solid line is computed from Eq. (9.13) in the quantum limit (hco k T) with the parameters listed in the figure. The parameters Aj, and CO correspond to X and m in our notation. (Reproduced from [8c] with permission. Copyright (1988) by the American Association for the Advancement of Science.)...

The most convincing analysis of the quantum limit relies on the quantum potential which is given in a system with wave function... 

This implies that one can increase P and reach the quantum limit without having to increase the number of samples. 

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