Energy-time uncertainty relation

There is also an uncertainty relation involving energy and time ... 

An uncertainty relation for energy and time is suggested by applying Eq (4.26) to the time variable t and the energy operator E = ihdjdt. The result is... 

The energy of a photon can be determined in a photon-box experiment by measuring the energy of a photon box after emission of a photon. This conflicts with the uncertainty relation between energy and time. 

The mass of N is about 1237 MeV. Its half-life is about I0"- s and, in accordance with the uncertainty relation between energy and time (Section 3-13), the mass (energy) of the particle is not well defined, but has an uncertainty of about 60 MeV. (The half-life 10 s is in fact calculated from the observed distribution function for the mass of the N complex.)... 

Rodebush has also implied that the accuracy with which very low temperatures can be measured is restricted by the uncertainty principle and by the nature of the substance under investigation. However, the accuracy of a temperature measurement is not limited in a serious way by the uncertainty principle for energy, inasmuch as the relation between the uncertainty in temperature and the length of time involved in the measurement depends on the size of the thermometer, and the uncertainty in temperature can be made arbitrarily small by sufficiently increasing the size of the thermometer we assume as the temperature of the substance the temperature of the surrounding thermostat with which it is in either stable or metastable equilibrium, provided that thermal equilibrium effective for the time of the investigation is reached. 

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. 

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... 

The position, momentum, and energy are all dynamical quantities and consequently possess quantum-mechanical operators from which expectation values at any given time may be determined. Time, on the other hand, has a unique role in non-relativistic quantum theory as an independent variable dynamical quantities are functions of time. Thus, the uncertainty in time cannot be related to a range of expectation values. 

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 ... 

NMR techniques (65MI2 65MI3 77MI2). The uncertainty principle relates the uncertainty in the energy of a system, Se, with the uncertainty in the time, St, through... 

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... 

The amount of transferred energy A E can be regarded as a measure of the uncertainty in the energy of the charged particle and is related to the time of transfer by the uncertainty relation27 28... 

The wavefunction tot Ef) is a solution of the time-independent Schrodinger equation and therefore corresponds to a particular energy Ef. According to the uncertainty relation, it contains all times, i.e., the entire history of the dissociation process (see also Section 16.1). 

Fig. 7.13. Illustration of the interrelation between resolution in the energy domain and in the time domain.

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