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Zero point energy

The Zero point energy (also Ground state energy or Vacuum energy or Quantum vacuum) is the difference between the energy that a quantum mechanical system has in its ground state and the minimum energy that the system would have if it were described in a classical way. In thermodynamic systems that exchange energy with their environment, the zero point energy is therefore also the same as the energy of the system at absolute temperature zero.[1][2]

Potential function of the harmonic oscillator (red).
A particle in this potential can only take on certain energies (blue), the lowest possible of these energies is above the potential minimum

One-dimensional single-particle systems

The zero point energy is usually introduced using one-dimensional systems of a particle in a potential. In classical (that is, non-quantum mechanical) physics, the lowest energy state is that in which the particle is at rest in the potential minimum. In quantum mechanics, the smallest achievable energy above the value of the minimum potential. For given example systems, this can be verified by explicitly determining the energy eigenstates.

Alternatively one can get this result by using the uncertainty relation:[3] A finite position uncertainty that z. B. present in bound states, generally requires a pulse uncertainty greater than zero. Therefore the momentum and the kinetic energy cannot be exactly zero. Since the kinetic energy cannot become negative:

$ E_ \ mathrm {kin}> 0 $

the total energy, the sum of potential energy and kinetic energy, must therefore be greater than the minimum of the potential energy:

$ \ Rightarrow E_ \ mathrm {ges} = E_ \ mathrm {pot} + E_ \ mathrm {kin}> E_ \ mathrm {pot} $

Harmonic oscillator

The standard example of zero point energy is the quantum physical harmonic oscillator. This one has the potential

$ V (x) = \ frac {1} {2} \ cdot m \ cdot \ omega ^ 2 \ cdot x ^ 2 $


thus a potential minimum $ V (x = 0) = 0 $,

and the energy spectrum

$ E_ {n} = \ left (n + \ frac {1} {2} \ right) \ cdot \ hbar \ cdot \ omega $,


  • the reduced Planckian constant $ \ hbar $
  • $ n $: natural number including 0.

Even in the energetically lowest state, the ground state with $ n = 0 $, there is an energy different from zero:

$ E_ {0} = \ frac {1} {2} \ cdot \ hbar \ cdot \ omega. $

In the classical case, the state of lowest energy is that in which the particle is at rest at $ x = 0 $, i.e. $ p = 0 $. In quantum mechanics, however, the uncertainty relation between position and momentum prohibits both quantities from having exact values. The more precisely the location is known, the less precisely the impulse is known and vice versa. The zero point energy clearly results as the mean value of these fluctuations.


The Casimir effect and the Lamb shift are often interpreted as an experimental proof for the zero point energy and the vacuum fluctuations caused by it. However, it is possible to derive the Casimir effect without resorting to vacuum fluctuations.[4] The Lamb shift is a phenomenon in an interacting quantum field theory, which accordingly Not can be attributed to the vacuum energy; The misunderstanding arises from the fact that although it is a result of virtual particle-antiparticle pair formation, which does not take place from the vacuum, but is based on interactions between fields.

The vacuum energy is considered to be a possible candidate for the dark energy, which in astronomy would offer an explanation for the observed accelerated expansion of the universe. In this context, the amount of vacuum energy represents one of the greatest problems of modern physics, since the experimentally found and the theoretically predicted values ​​for the vacuum energy as dark energy differ from one another: Based on observations, the energy density of the vacuum is reduced to a value of the order of 10−9 until 10−11 J / m3 estimated,[5][6][7] it is around a factor of 10120 lower than in the theoretical calculations.

Historical development

After abandoning the aether filling empty space as a medium for the propagation of waves and a frame of reference for the movement of bodies, the idea of ​​a vacuum containing neither matter nor any form of energy prevailed in classical physics.

However, the radiation law of Max Planck's “second theory” found by Max Planck in 1911 suggested a zero-point energy of the electromagnetic field in a vacuum, since a variable ½ $ h \ nu $ occurred which was independent of temperature. However, Planck initially did not attach any importance to this in terms of experimental evidence.[8][9]

With similar considerations, Albert Einstein and Otto Stern came to the conclusion in 1913 that the zero point fluctuations of the electromagnetic field at the absolute zero point of temperature were $ h \ nu $.[9]

Building on Planck's work, Walther Nernst proposed zero point fluctuations for the electromagnetic field around the value ½ $ h \ nu $[9] and on the other hand, that the entire universe is filled with zero-point energy.[10]

In 1927 Werner Heisenberg formulated his uncertainty relation, which is considered the basis of the zero point energy in every quantum mechanical system.[11]

Georges Lemaître, who had done pioneering theoretical work on the big bang and the expansion of the universe, found in 1934 a match between the vacuum energy and Einstein's cosmological constant (1917), the introduction of which Einstein later described as the "greatest donkey" of his life.[12]

During an investigation of the Van der Waals forces in colloid solutions, Hendrik Casimir and Dirk Polder used a quantum mechanical approach in 1947, which led to a discussion with Niels Bohr. Bohr commented on this, "that must have something to do with zero fluctuations".[13] Casimir pursued the idea that the attraction between neutral atoms might only be based on vacuum fluctuations and published his fundamental work in 1948 About the attraction between two perfectly conductive plates.[14] In it he described a theoretical test arrangement with two metal plates in a vacuum, which according to his calculations should attract each other due to the vacuum energy of the electromagnetic quantum field (Casimir effect).

First corresponding attempts to prove the Casimirkraft in a vacuum were carried out in 1958 by Marcus Sparnaay,[15] however with a measurement error of about 100%.[16] Gradually the measurements reached the Casimirkraft (Value for two mirrors of 1 cm² area at a distance of 1 µm: 10−7 N[13]) higher accuracy, e.g. B. the measurement error in van Bloklands and Oveerbeeks 1978 was 25%[17] and with Steven Lamoreaux in 1996 only 5%.[18]

In recent years the cosmological constant, which is closely related to the curvature of space-time, has again received more attention, especially since it is now viewed as a small positive energy density of the vacuum.[19] A more recent explanation for the cosmological constant provides, for example, a cyclic universe.[20]

Popular culture

In various cinematic works, zero point energy is described as a source or form of energy:

  • In various episodes of the Star-Gate franchise, zero-point energy appears as the most powerful known energy source;[21]
  • In the animated film The Incredibles, the villain uses an anti-gravity beam, which he describes as powered by zero-point energy
  • In the first episode of the sequel to the 2016 X-Files television series, zero-point energy is mentioned as an energy source for a spaceship.

Individual evidence

  1. ↑ Harald Lesch: Zero point energy (also vacuum energy). Retrieved January 20, 2017.
  2. ↑ Martin Bäker: The vacuum energy and the Casimir effect.
  3. ↑ F. Schwabl: Quantum mechanics, 6th edition, chapter 3.1.3, ISBN 3-540-43106-3 (google books)
  4. ↑ R. L. Jaffe: The Casimir Effect and the Quantum Vacuum. In: Physical Review D.. Volume 72, 2005 (arxiv: hep-th / 0503158)
  5. ↑ J. Baez. What's the energy density of the vacuum ?, 2006.
  6. ↑ p. M. Carroll: The Cosmological Constant, 2001.
  7. ↑ A. Tillemans. Plato's allegory of the cave and the vacuum energy of the universe, 2002.
  8. ↑ Max Planck: A new radiation hypothesis. In: Negotiations of the German Physical Society. Volume 13, 1911, pp. 138-148.
  9. 9,09,19,2B. Haisch, A. Rueda and Y. Dobyns: Inertial mass and the quantum vacuum fields. In: Annals of Physics. Volume 10, 2000, pp. 393-414. arxiv: gr-qc / 0009036.
  10. ↑ Walther Nernst: To return to the assumption of constant changes in energy through an attempt from quantum theoretical considerations. In: Negotiations of the German Physical Society. Volume 4, 1916, p. 83.
  11. ↑ Werner Heisenberg: About the descriptive content of quantum theoretical kinematics and mechanics. In: magazine for physics. Volume 43, 1927, pp. 172-198. Summary/
  12. ↑ J.-P. Luminet: The Rise of Big Bang Models, from Myth to Theory and Observations. 2007, arxiv: 0704.3579.
  13. 13,013,1A. Lambrecht: The vacuum gains strength. In: Physics in our time. Volume 2, 2005, pp. 85–91, bibcode: 2005PhuZ ... 36 ... 85L.
  14. ↑ Hendrik Casimir: On the attraction between two perfectly conducting plates. In: Proc. Con. Ned. Akad. Van Wetensch. B51 (7), 1948, pp. 793-796, reprint online
  15. ↑ M.J. Sparnaay: Measurements of attractive forces between flat plates. In: Physica. 24, 1958, p. 751, doi: 10.1016 / S0031-8914 (58) 80090-7.
  16. ↑ R. Onofrio: Casimir forces and non-Newtonian gravitation. In: New Journal of Physics. Volume 8, 2006, p. 237 Abstract.
  17. ↑ P. H. G. M van Blokland and J. T. G. Oveerbeek: The measurement of the van der Waals dispersion forces in the range 1.5 to 130 nm. In: Journal of the Chemical Society Faraday Transactions. Volume I74, 1978, p. 2637.
  18. ↑ p. K. Lamoreaux: Demonstration of the Casimir force in the 0.6 to 6 mm range. Physical Review Letters. Volume 78, No. 1, 1997, pp. 5-8 Abstract.
  19. ↑ p. M. Carroll: The Cosmological Constant. 2001.
  20. ↑ Paul Steinhardt and N. Turok: A Cyclic Model of the Universe. In: Science. Volume 296, No. 5572, 2002, pp. 1436-1439 abstract.
  21. Zero point module. The German-language Stargate Lexicon. In: Stargate Wiki. Retrieved February 9, 2016.