Heisenberg width

This metastable shape resonance state has a Heisenberg width (31) of... 

It is important to emphasise that y-ray energies cannot be measured with this accuracy on an absolute scale indeed these energies are seldom known to better than 1 part in 10. In order to use the precision of the Heisenberg width, it is customary to use as a reference a radioactive source in which all the emitting atoms have an identical chemical environment. This is then compared to an absorbing chemical matrix of the same element and only the minute difference between the two transition energies is measured. The means whereby this is accomplished will be discussed shortly. 

The recoil-free y-ray energy of a typical Mossbauer transition is so precisely defined that its Heisenberg width corresponds to the energy change produced by an applied Doppler velocity of the order of 1 mms. It is therefore possible to imagine a particular relative velocity between source and absorber at which the y-ray energy from the source will precisely match the nuclear... 

It has already been shown in Chapter 1 that the resonant absorption curve for an ideally thin source and absorber has a width at half-height F, which is twice the Heisenberg width of the emitted y-photon. The Doppler velocity v corresponding to this energy F is given by... 

Claims for the conclusive detection of a unique quadrupole splitting in Sn02 have been made several times [12, 13], and one can certainly be generated under high pressure [14], but if such a splitting does exist it is on the limit of experimental resolution as determined by the Heisenberg width. The discrepancies reported in the literature must be due in part to differences in the stoichiometry of the samples used. 

The spectrum of the femtosecond pulse provides some infonnation on whether the input pulse is chirped, however, causing the temporal width of I(t) to be broader than expected from the Heisenberg indetenninancy relationship. 

The frill width at half maximum of the autocorrelation signal, 21 fs, corresponds to a pulse width of 13.5 fs if a sech shape for the l(t) fiinction is assumed. The corresponding output spectrum shown in fignre B2.1.3(T)) exhibits a width at half maximum of approximately 700 cm The time-bandwidth product A i A v is close to 0.3. This result implies that the pulse was compressed nearly to the Heisenberg indetenninacy (or Fourier transfonn) limit [53] by the double-passed prism pair placed in the beam path prior to the autocorrelator. 

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. 

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... 

Fig. 2.2 Intensity distribution /( ) for the emission of y-rays with mean transition energy Eq. The Heisenberg natural line width of the distribution, F = S/t, is determined by the mean lifetime T of the excited state (e)...

Ultrafast time-resolved resonance Raman (TR ) spectroscopy experiments need to consider the relationship of the laser pulse bandwidth to its temporal pulse width since the bandwidth of the laser should not be broader than the bandwidth of the Raman bands of interest. The change in energy versus the change in time Heisenberg uncertainty principle relationship can be applied to ultrafast laser pulses and the relationship between the spectral and temporal widths of ultrafast transform-limited Gaussian laser pulse can be expressed as... 

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

The ultimate (minimum) linewidth of an optical band is due to the natural or lifetime broadening. This broadening arises from the Heisenberg s uncertainty principle, AvAt < U2jt, Av being the full frequency width at half maximum of the transition and the time available to measure the frequency of the transition (basically, the life-... 

The laser used to generate the pump and probe pulses must have appropriate characteristics in both the time and the frequency domains as well as suitable pulse power and repetition rates. The time and frequency domains are related through the Fourier transform relationship that hmits the shortness of the laser pulse time duration and the spectral resolution in reciprocal centimeters. The limitation has its basis in the Heisenberg uncertainty principle. The shorter pulse that has better time resolution has a broader band of wavelengths associated with it, and therefore a poorer spectral resolution. For a 1-ps, sech -shaped pulse, the minimum spectral width is 10.5 cm. The pulse width cannot be <10 ps for a spectral resolution of 1 cm . An optimal choice of time duration and spectral bandwidth are 3.2 ps and 3.5 cm. The pump pulse typically is in the UV region. The probe pulse may also be in the UV region if the signal/noise enhancements of resonance Raman... 

Natural broadening occurs because of the finite lifetime (x) of the atom in the excited state. Heisenberg s uncertainty principle states that if we know the state of the atom, we must have uncertainty in the energy level. We assume that x for the ground state is infinity and therefore for a resonance line the natural width Av = IAtxx. 

Let us now consider the increase in the spot size due to the effect of Heisenberg s uncertainty. When a particle is confined to pass through a small space of width Ay, at the tip, the uncertainty in the tangential component of the momentum of the particle is of the order of hi2 Ay, and the corresponding velocity component is h/2M Ayt. Thus the spread of the spot size at the screen by this uncertainty alone is... 

From Fig. 20 one sees that as the width of the basic wavelet Ax0 changes, all the measurement-accessible space is browsed. This space is limited only by Heisenberg s space. The smaller is Axo, the greater is the precision of the measurement of the position, that is, the smaller is the uncertainty Ax, for any value of the error in the momentum. Given that the new relation contains the usual as a particular case, it implies that the measurement space available to the... 

It is noteworthy that the width of an absorption line is inversely proportional to the lifetime of the excited state (Heisenberg s uncertainty principle). Hence, for gases, the lifetime is long and the absorption lines are sharp. However, the lifetime is short for compounds in the condensed phase and band broadening occurs. Except for very simple molecules, no instrument allows the observation of individual lines. 

The second factor involves the theory that defines the natural width of the lines. Radiations emitted by atoms are not totally monochromatic. With plasmas in particular, where the collision frequency is high (this greatly reduces the lifetime of the excited states), Heisenberg s uncertainty principle is fully operational (see Fig. 15.4). Moreover, elevated temperatures increase the speed of the atoms, enlarging line widths by the Doppler effect. The natural width of spectral lines at 6000 K is in the order of several picometres. 

Spectral line width varies inversely with the excited-state lifetime according to Heisenbergs principle, AT X A H = hi 2n, where AT is the lifetime of the excited spin state, h is Planck s constant, and AH is the effective width of the absorption signal. Excited-state lifetimes are subject to environmental (including chemical) influences. The resulting line-shape changes yield information about the chemical environment of the Mn atoms. Both spin-lattice and spin-spin relaxation mechanisms can contribute to the overall lifetime. 

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... 

The measured half-life of the state is 89.4 ps, which corresponds to a energy width, T, or AE, due to the Heisenberg uncertainty principle of ... 

If you look at the nmr spectra of many different kinds of organic compounds, you will notice that some resonances are sharp and others are broad. In a few spectra, all of the peaks may be broad as the result of poor spectrometer performance, but this is not true for the spectra of Figures 9-29 (p. 312) and 24-2 (p. 1173) where, within a given spectrum, some resonances will be seen to be sharp and others broad. We can understand these differences by consideration of the lifetimes of the magnetic states between which the nmr transitions occur.1 The lifetimes of the states can be related to the width of the lines by the Heisenberg uncertainty principle. 

Following Eq. 4 there are three different sources of line broadening adding to the natural line width AB, which reflects the lifetime of the final state (Heisenberg uncertainty principle) and in some cases an unresolved spin orbit splitting. In first approximation, 7)... 

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