Trembling motion

The trembling motion is a theoretical, rapid motion of elementary particles , especially electrons , which obey the ( relativistic ) Dirac equation .

The existence of such a motion was postulated in 1928 by Gregory Breit and in 1930 by Erwin Schrödinger , as a result of his analysis of wave packet solutions of the Dirac equation for relativistic electrons in a vacuum . This produces an interference between the positive and negative energy state , a fluctuation of the position of the electron around the average value with an angular frequency of

${\displaystyle \omega =2m_{\mathrm {e} }c^{2}/\hbar \approx 1{,}6\cdot 10^{21}\,{\text{s}}^{-1}}$

With

• the electron mass ${\displaystyle m_{\mathrm {e} }}$
• the speed of light ${\displaystyle c}$
• the reduced Planck quantum of action . ${\displaystyle \hbar }$

The trembling motion of a free relativistic particle was never observed, but the behavior of such a particle was simulated with a trapped ion by placing it in an environment so that the non-relativistic Schrödinger equation for the ion has the same mathematical form as the Dirac- Equation (although the physical situation is different).

theory

From the time-dependent Schrödinger equation

${\displaystyle H\psi (\mathbf {x} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {x} ,t),}$

where is the Dirac Hamiltonian for an electron in a vacuum ${\displaystyle H}$

${\displaystyle H=\left(\alpha _{0}mc^{2}+\sum _{j=1}^{3}\alpha _{j}p_{j}\,c\right)}$

and the wave function , ${\displaystyle \psi (\mathbf {x} ,t)}$

it follows in the Heisenberg picture that every operator  Q obeys the following equation:

${\displaystyle -i\hbar {\frac {\mathrm {d} Q}{\mathrm {d} t}}(t)=\left[H,Q\right].}$

Specifically, the time-dependent position operator is given by

${\displaystyle \hbar {\frac {\mathrm {d} x_{k}}{\mathrm {d} t}}(t)=i\left[H,x_{k}\right]=c\hbar \alpha _{k}}$

with . ${\displaystyle \alpha _{k}\equiv \gamma _{0}\gamma _{k}}$

The above equation shows that the operator can be interpreted as the kth component of the "speed operator ". ${\displaystyle \alpha _{k}}$

The time dependency of the speed operator is given by

${\displaystyle \hbar {\frac {\mathrm {d} \alpha _{k}}{\mathrm {d} t}}(t)=i\left[H,\alpha _{k}\right]=2[i\gamma _{k}m-\sigma _{kl}p^{l}]=2i[cp_{k}-\alpha _{k}H],}$

where is and the momentum . ${\displaystyle \sigma _{kl}\equiv {\tfrac {i}{2}}[\gamma _{k},\gamma _{l}]}$ ${\displaystyle p}$

Because both and are time independent, the above equation can be integrated twice to get the explicit time dependence of the location operator. First: ${\displaystyle p_{k}}$ ${\displaystyle H}$ ${\displaystyle x_{k}(t)}$

${\displaystyle \alpha _{k}(t)=\alpha _{k}(0)e^{-2iHt/\hbar }+cp_{k}H^{-1}}$

Then:

${\displaystyle x_{k}(t)=x_{k}(0)+c^{2}p_{k}H^{-1}t+{\frac {1}{2}}i\hbar cH^{-1}(\alpha _{k}(0)-cp_{k}H^{-1})(e^{-2iHt/\hbar }-1).}$

The resulting expression consists of

• an initial position ${\displaystyle x_{k}(0)}$
• a movement proportion proportional to time and ${\displaystyle c^{2}p_{k}H^{-1}t}$
• an unexpected vibration component ("trembling movement") with an amplitude that corresponds to the Compton wavelength . ${\displaystyle {\tfrac {1}{2}}i\hbar cH^{-1}(\alpha _{k}(0)-cp_{k}H^{-1})(e^{-2iHt/\hbar }-1)}$

Interestingly, if one takes the expected values for wave packets that consist entirely of waves with positive energy (or entirely of waves with negative energy), the tremor term vanishes . This can be achieved through the Foldy-Wouthuysen transformation .

See also

• Casimir effect
• Lamb shift

literature

• Gregory Breit : An Interpretation of Dirac's Theory of the Electron . In: Proceedings of the National Academy of Sciences . tape 14 , no. 7 , 1928, pp. 553–559 , doi : 10.1073 / pnas.14.7.553 (English).
• Erwin Schrödinger : About the force-free movement in relativistic quantum mechanics . In: Special edition from the meeting reports of the Prussian Academy of Sciences Phys.-Math. Class . tape 24 , 1930, ZDB -ID 959457-7 , p. 418-428 .
• Erwin Schrödinger: On the quantum dynamics of the electron . In: Meeting reports of the Prussian Academy of Sciences. Physical and mathematical class . 1931, p. 63-72 .
• Albert Messiah : Quantum Mechanics . tape 2 . North-Holland, Amsterdam 1962, XX.37, p. 950-952 (English).
• George Sparling: trembling movement . In: Seminaires & Congrès . tape 4 , 2000, ZDB -ID 2045737-6 , p. 277–305 (English, emis.de [PDF; 337 kB ]).

Web links

• Tobias Brandes : Lecture Notes on Quantum Mechanics II, TU Berlin, WS 2011/12. (pdf) pp. 21-25 , accessed on September 3, 2018 . 
• Adrian Wüthrich: Feynman's Struggle and Dyson's Surprise: The Development and Early Application of a New Means of Representation . In: Shaul Katzir, Christoph Lehner and Jürgen Renn (Eds.): Traditions and Transformations in the History of Quantum Physics. Third International Conference on the History of Quantum Physics, Berlin, June 28 - July 2, 2010 . 2013, ISBN 978-3-8442-5134-0 , pp. 277–279 (English, edition-open-access.de - historical perspective).
• David Hestenes: The trembling interpretation of quantum mechanics . In: Found Phys . tape 20 , 1990, pp. 1213 , doi : 10.1007 / BF01889466 (English, an alternative explanation beyond the interference of the positive and negative energy states).
• Christoph Wunderlich: trembling in the trap . In: Physics Journal . tape 9 , no. 3 , 2010, p. 20–24 ( pro-physik.de [PDF]).
• Rainer Scharf: Atomic tremors. In: pro-physik.de. January 7, 2010, accessed September 3, 2018 (Trapped Ion Simulation Summary).