Uncertainty principle time-energy

We now wish to derive the energy-time uncertainty principle, which is discussed in Section 1.5 and expressed in equation (1.45). We show in Section 1.5 that for a wave packet associated with a free particle moving in the x-direction the product A A/ is equal to the product AxApx if AE and At are defined appropriately. However, this derivation does not apply to a particle in a potential field. 

To obtain the energy-time uncertainty principle for a particle in a time-independent potential field, we setyf equal to H in equation (3.81)... 

Thermodynamics, quantization, energy-time uncertainty principle, negative Kelvin... 

The energy-time uncertainty principle a purely dynamic (not thermodynamic) Third-Law limitation under quantum mechanics... 

We re-emphasize (recall Sect. 3.2, especially the second-to-last paragraph thereof) that the attainment of absolute zero requires perfect certainty that our entire new n-oscillator Subsystem B is in its ground state — that all n oscillators of Subsystem B are in the ground state. But the energy-time uncertainty principle may contravene [29-41]. 

Dr. Bernard L. Cohen [29] employs the energy-time uncertainty principle in discussing quantum fluctuations. Dr. Robert Comer [30] (cited by Dr. Cohen [29]) shows how the position-momentum uncertainty principle can be employed in more limited circumstances. 

Note the qualitative — not merely quantitative — distinction between the thermodynamic (Boltzmann-distribution) probability discussed in Sect. 3.2. as opposed to the purely dynamic (quantum-mechanical) probability Pg discussed in this Sect. 3.3. Even if thermodynamically, exact attainment of 0 K and perfect verification [22] that precisely 0 K has been attained could be achieved for Subsystem B, the pure dynamics of quantum mechanics, specifically the energy-time uncertainty principle, seems to impose the requirement that infinite time must elapse first. [This distinction between thermodynamic probabilities as opposed to purely dynamic (quantum-mechanical) probabilities should not be confused with the distinction between the derivation of the thermodynamic Boltzmann distribution per se in classical as opposed to quantum statistical mechanics. The latter distinction, which we do not consider in this chapter, obtains largely owing to the postulate of random phases being required in quantum but not classical statistical mechanics [42,43].]... 

In summary, the thermodynamic difficulties in attaining precisely 0 K via TSRR [2-5] seem to be circumventable via CSRR. By contrast, the purely dynamic (quantum-mechanical) limitation imposed by the energy-time uncertainty principle as per Sects. 3.3. and 3.4. is, strictly, not circumventable via either TSRR or CSRR, but this limitation may not be crucial if we do... 

But the issue of maintenance [23] seems inseparable from that of verifiability [22]. For, as discussed in Sects. 3.3. and 3.4, the energy-time uncertainty principle requires Af —> oo for perfect verifiability that Tq OK has been attained, which is obviously incompatible with Tq = OK being maintained only for an instant, or even for any finite number of instants, in accordance with Eqs. (6) and (7) and the associated discussions. If Tq = 0 K can be maintained only for an instant, (or any finite number of instants), then the energy-time uncertainty principle seems to preclude verification even for all practical purposes that Tq 0 K... 

We now wish to derive the energy-time uncertainty principle, which is discussed in Section 1.5 and expressed in equation (1.45). We show in Section... 

To understand the physical mechanism undei lying these processes, it is helpful to first consider the broad significance of the energy-time Uncertainty Principle for photoabsorption processes. It is well-known that for a molecular excited state with an average lifetime St and an average energy displacement SE from the ground state, there exists the relation (Finkel 1987)... 

Other localized window functions will lead to somewhat different detailed smoothing but the essential point remains—smoothing is achieved by convoluting with a localized window function. The other point, illustrated in Fig. 5, is that the Fourier transform of a function localized in frequency is a function localized in time, where the two widths are inverse to one another A broad window function has a transform which is tightly localized about the origin of the time axis, and vice versa. This is a mathematical property of the Fourier transform relation between two functions, familiar in its implication as the energy-time uncertainty principle. 

Although discouraging for ion detection, the energy-time uncertainty principle opens up new perspectives for ion excitation. According to Av x At 1, a rectangular pulse of 1 ms duration covers a frequency bandwidth of 1000 Hz a pulse of even 1 ps should correspond to a bandwidth (frequency uncertainty) of 1 MHz, i.e., shorter pulses cover increasingly broader ranges. This has been employed in so-called pulse excitation. 

FT of a transient signal, i.e., a signal vanishing to zero amplitude at the end of the observation time span, delivers a Lorentzian curve in the frequency domain (Fig. 4.55c,d). In accordance with the energy-time uncertainty principle, the frequency domain output becomes increasingly sharper the longer the observation time span, because the frequency is more accurately determined the more cycles are recorded in the transient. 

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