Time, uncertainty relation

A light pulse of a center frequency Q impinges on an interface. Raman-active modes of nuclear motion are coherently excited via impulsive stimulated Raman scattering, when the time width of the pulse is shorter than the period of the vibration. The ultrashort light pulse has a finite frequency width related to the Fourier transformation of the time width, according to the energy-time uncertainty relation. 

Kobe DH, Aguilera-Navarro VC. Derivation of the energy-time uncertainty relation. Phys. Rev. A. 1994 50 933-938. DOI 10.1103/PhysRevA.50.933... 

Dodonov W, Dodonov AV. Energy-time and frequency-time uncertainty relations ... 

The other principal radiative correction is the vacuum polarization (Fig. 3 b)). It describes the interaction of a fermion with virtual electron-positron pairs which can be thought present in the vacuum for short times without violating the energy-time uncertainty relation. If external fields are present, these virtual pairs are influenced and act like a polarizable medium. Therefore the Coulomb interaction of the nucleus with the electrons is modified which leads to an energy shift compared to the pure Coulomb potential energy eigenvalue. 

The quantity bjy in (9.124) is sharply peaked at w = energy-time uncertainty relation (5.14). States with a finite lifetime have an uncertainty in their energy. 

Ivanov, A. L. (2006,) Energy-time uncertainty relations and time operators. J. Math. Chem. 43,1-11. 

In this section, the notation in the models is introduced, classified into several categories including static input variables, decision variables, state variables, time uncertainty related variables, quantity uncertainty related variables, dynamic input or derived variables, cost coefficients and performances indicators. 

The energy q of a nuclear or electronic excited state of mean lifetime t cannot be determined exactly because of the limited time interval At available for the measurement. Instead, q can only be established with an inherent uncertainty, AE, which is given by the Heisenberg uncertainty relation in the form of the conjugate variables energy and time,... 

Another Heisenberg uncertainty relation exists for the energy E ofa particle and the time t at which the particle has that value for the energy. The uncertainty Am in the angular frequency of the wave packet is related to the uncertainty A in the energy of the particle by Am = h.E/h, so that the relation (1.25) when applied to a free particle becomes... 

We see that the energy and time obey an uncertainty relation when At is defined as the period of time required for the expectation value of S to change by one standard deviation. This definition depends on the choice of the dynamical variable S so that At is relatively larger or smaller depending on that choice. If d(S)/dt is small so that S changes slowly with time, then the period At will be long and the uncertainty in the energy will be small. 

Quantum mechanics enters here with a statement of uncertainty relating energy and time. If you know the lifetime of the excited state in a transition then you cannot know exactly the energy of the transition. This uncertainty principle is wrapped up in the following relation ... 

The first view is based on the uncertainty relation between time and energy. When the electron is in the region of the barrier, its velocity, as determined by the kinetic energy, is ... 

The coherence time Axm relates to abscissa xm and resolution interval Ax as follows. By the Heisenberg uncertainty principle, Axm goes inversely with spectral purity, or... 

The usual uncertainty relations are a direct mathematical consequence of the nonlocal Fourier analysis therefore, because of this fact, they have necessarily nonlocal physical nature. In this picture, in order to have a particle with a well-defined velocity, it is necessary that the particle somehow occupy equally all space and time, meaning that the particle is potentially everywhere without beginning nor end. If, on the contrary, the particle is perfectly localized, all infinite harmonic plane waves interfere in such way that the interference is constructive in only one single region that is mathematically represented by a Dirac delta function. This implies that it is necessary to use all waves with velocities varying from minus infinity to plus infinity. Therefore it follows that a well-localized particle has all possible velocities. 

In the previous derivation of the new uncertainty relations, we were concerned only with conjugate observables space and momentum. The same process can be used step by step to derived the relations for the conjugate observables energy and time. It is sufficient to change the variables... 

Consider the following experiment. An electron source emits, at fixed known time t0, electrons with an uncertainty in position Ajcq 100A, which, according to Heisenberg uncertainty relations, corresponds to a minimum velocity dispersion Avo 10 6 cm. A millisecond later, the initial wavepacket will have spread to a size of about 10 m. Let us now suppose that from each electron wavepacket one cuts a piece of size 8xq 100 A (see Fig. 25). 

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