# Energy-time uncertainty relation

The energy-time uncertainty relation describes a boundary condition for the achievable measurement accuracy of energy and time in quantum mechanics .

In preliminary form it was in 1927 by Werner Heisenberg found and with the same time found uncertainty for city and impulse placed first on one level. Like the position-momentum uncertainty relation, the energy-time uncertainty relation is of a principle nature and not a consequence of inadequate measurements . However, the two relations show fundamental differences, each of which requires its own interpretation. While the position and momentum of a particle are observable quantities at any point in time, as they are represented in quantum mechanics by position and momentum operators , time is not an observable quantity in the same sense and cannot be represented consistently by a time operator .

Heisenberg led the estimate for the product of the inaccuracy of an energy measurement and the minimum duration that this measurement requires ${\ displaystyle \ Delta E}$${\ displaystyle \ Delta t}$

${\ displaystyle \ Delta E \; \ Delta t \ sim \ hbar}$

here, where is the reduced Planck constant . This shape is still widely used today. This can lead to a real inequality ${\ displaystyle \ hbar}$

${\ displaystyle \ Delta E \; \ Delta t \ geq {\ frac {\ hbar} {2}},}$

be tightened, in which the standard deviation of the energy values ​​represented in the system is and the smallest possible time span in which the expected value of an observable changes by one standard deviation. ${\ displaystyle \ Delta E}$${\ displaystyle \ Delta t}$

Even without reference to the term measurement accuracy, it is fundamentally true in quantum mechanics that a system whose state does not remain constant over time cannot have a clearly defined energy. Because only the eigen-states of the energy operator are stationary states. Depending on the case under consideration, different estimates can result for the smallest possible product from the range of the energy values ​​involved and a time period characteristic of the change in the system.

In popular scientific representations it is sometimes said that the energy-time uncertainty relation allows for a short time to violate the conservation of energy by the amount ; this explains the "virtual states" and vacuum fluctuations in perturbation theory in quantum mechanics and quantum field theory . This is not correct. The conservation of energy is always strictly guaranteed, and the terms mentioned from perturbation theory refer to mathematical constructs which as such cannot be observed. ${\ displaystyle \ Delta t}$${\ displaystyle \ Delta E}$

Because of the quantum mechanical relationship between energy and angular frequency , the energy-time uncertainty relation can also be written as a frequency-time uncertainty relation : ${\ displaystyle E = \ hbar \ omega}$ ${\ displaystyle \ omega}$

${\ displaystyle \ Leftrightarrow \ Delta \ omega \; \ Delta t \ geq {\ frac {1} {2}}}$.

This relation is z. B. used in high-frequency technology to calculate the time that is practically required to determine a frequency with the inaccuracy . ${\ displaystyle \ Delta t}$${\ displaystyle \ Delta \ omega}$

## Derivations

### A general formal derivation

The formal derivation is based on the generalized form of the uncertainty relation for the Hamilton operator and any other operator : ${\ displaystyle {\ hat {H}}}$${\ displaystyle {\ hat {A}}}$

${\ displaystyle \ Delta H \; \ Delta A \ geq {\ frac {1} {2}} \ left | \ left \ langle \ left [{\ hat {H}}, {\ hat {A}} \ \ right] \ right \ rangle \ right |}$

This is the standard deviation of the energy and the standard deviation of the observables in the state under consideration. However, since the time cannot be selected for, because this, in contrast to location, momentum, angular momentum, energy etc., cannot be represented by an operator, the detour via the change in the expected value over time is taken. The rate of change is according to the Ehrenfest theorem${\ displaystyle \ Delta {\ hat {H}} = \ Delta E}$${\ displaystyle \ Delta A}$${\ displaystyle A}$${\ displaystyle {\ hat {A}}}$${\ displaystyle {\ hat {A}}}$

${\ displaystyle \ left | {\ frac {d} {dt}} \ left \ langle {\ hat {A}} \ right \ rangle \ right | = {\ frac {1} {\ hbar}} \ left | \ left \ langle \ left [{\ hat {H}}, {\ hat {A}} \ right] \ right \ rangle \ right |}$,

therefore applies

${\ displaystyle \ Delta H \; \ Delta A \ geq {\ frac {\ hbar} {2}} \ left | {\ frac {d} {dt}} \ left \ langle {\ hat {A}} \ right \ rangle \ right |}$.

The time span is introduced by ${\ displaystyle \ Delta t}$

${\ displaystyle \ Delta A = \ left | {\ frac {d} {dt}} \ left \ langle {\ hat {A}} \ right \ rangle \ right | \ Delta t}$.

is set. is therefore not a measure of a scatter, but the time that must pass for the observable to change by one standard deviation. If the last inequality is multiplied by, can be removed. What remains is the uncertainty relation we are looking for: ${\ displaystyle \ Delta t}$${\ displaystyle \ Delta A}$${\ displaystyle \ Delta t}$${\ displaystyle \ Delta A}$

${\ displaystyle \ Delta H \; \ Delta t \ geq {\ frac {\ hbar} {2}}}$.

Since the right-hand side of the inequality does not depend on the choice of , it applies in general and cannot be undercut for any observable. ${\ displaystyle {\ hat {A}}}$

### Energy uncertainty and lifespan

With a different physical interpretation, a similar energy-time uncertainty relation applies to the disintegration of a system in a metastable state into two particles, i.e. also to any type of emission. Here the relation is given by an equation:

${\ displaystyle \ Delta E \; \ tau = \ hbar}$.

This does not stand for the standard deviation, but for the half-width of the kinetic energy of the decay products, and not for a time uncertainty, but for the well-determined mean life of the metastable state. However, it can also be viewed as a time uncertainty because in an ensemble of identical systems the individual decay times show an exponential distribution for which the mean value also indicates the standard deviation. The derivation takes place within the framework of the resonance theory for the scattering at a potential well. It applies to every decaying system, because this can be understood as a resonance in the reverse reaction, i.e. when the decay products are scattered together. ${\ displaystyle \ Delta E}$${\ displaystyle \ tau}$${\ displaystyle \ tau}$

### Further individual examples

There are a number of other individual examples in which a similar energy-time uncertainty relation can be found. Amongst other things:

• Refers to the uncertainty in the determination of the point in time at which a particle passes a location, then the wave function of the particle is modeled as a wave packet with a certain expansion and thus also energy uncertainty. At a given (average) speed is proportional to the length of the wave packet, which in turn is inversely proportional to the energy uncertainty. It turns out .${\ displaystyle \ Delta t}$${\ displaystyle \ Delta t}$${\ displaystyle \ Delta E \; \ Delta t \ sim \ hbar}$
• Derived from this, it follows that an energy measurement with the accuracy requires at least the time .${\ displaystyle \ Delta E}$${\ displaystyle \ Delta t = \ hbar / \ Delta E}$
• If the system is in a superposition of two levels with energy spacing , then the wave function oscillates with the period between the symmetrical and the antisymmetrical form.${\ displaystyle \ Delta E}$${\ displaystyle \ tau = h / \ Delta E}$

## Individual evidence

1. ^ W. Heisenberg: About the descriptive content of quantum theoretical kinematics and mechanics . In: Journal of Physics . tape 43 , no. 3 , 1927, pp. 172–198 , doi : 10.1007 / BF01397280 ( original article [PDF; 2.7 MB ]).
2. [1]
3. Eckhard Rebhan: Theoretical Physics II . 1st edition. Spectrum Academic Publishing House, Munich 2005, p. 96 ff .
4. Wolfgang Nolting : Basic Course Theoretical Physics 5/1; Quantum Mechanics - Basics. 7th edition, Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-68868-6 , page 220ff
5. John M. Blatt, Viktor Weisskopf : Theoretische Kernphysik . 1st edition. BGTeubner, Leipzig 1959, p. 354 ff .
6. Franz Schwabl: Quantum Mechanics - An Introduction . 7th edition. Springer, Heidelberg 2007, p. 101 ff .
7. Albert Messiah : Quantum Mechanics I . North Holland Publishing, Amsterdam 1970, ISBN 0-7204-0044-9 , pp. 136 ff .