Fradkin tensor

The Fradkin tensor , also Jauch-Hill-Fradkin tensor , after Josef-Maria Jauch and Edward Lee Hill as well as David M. Fradkin , is a conserved quantity in the treatment of the isotropic multi-dimensional harmonic oscillator in classical mechanics . In the treatment of the quantum mechanical harmonic oscillator in quantum mechanics , the tensor-valued Fradkin operator appears in its place .

The Fradkin tensor delivers enough conserved quantities for a three-dimensional harmonic oscillator, so that the solution of the oscillator's equations of motion are maximally superintegrable . That means, in order to determine the trajectories , no differential equation has to be solved.

Like the Laplace-Runge-Lenz vector in the Kepler problem , the conservation of the Fradkin tensor is based on a hidden symmetry of the harmonic oscillator.

definition

Let be the Hamilton function of a harmonic oscillator ${\displaystyle H}$

${\displaystyle H={\frac {{\vec {p}}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}{\vec {x}}^{2}}$

With

• the pulse , ${\displaystyle {\vec {p}}}$
• the crowd , ${\displaystyle m}$
• the angular frequency and ${\displaystyle \omega }$
• the place , ${\displaystyle {\vec {x}}}$

then the Fradkin tensor (apart from an arbitrary normalization) is defined as:

${\displaystyle F_{ij}={\frac {p_{i}p_{j}}{2m}}+{\frac {1}{2}}m\omega ^{2}x_{i}x_{j}}$

In particular, with the trace operator . The Fradkin tensor is therefore a symmetric matrix and has independent entries in -dimensions for a harmonic oscillator, i.e. five in three dimensions. ${\displaystyle H=\operatorname {Tr} (F)}$ ${\displaystyle \operatorname {Tr} }$ ${\displaystyle n}$ ${\displaystyle {\tfrac {n(n+1)}{2}}-1}$

properties

• The Fradkin tensor is orthogonal to the angular momentum : ${\displaystyle {\vec {L}}={\vec {x}}\times {\vec {p}}}$

  ${\displaystyle F_{ij}L_{j}=0}$

• Contraction of the Fradkin tensor with the position vector gives the connection

  ${\displaystyle x_{i}F_{ij}x_{j}=E{\vec {x}}^{2}-{\frac {{\vec {L}}^{2}}{2m}}}$.

• The five independent components of the Fradkin tensor and the three of the angular momentum form the eight generators of one , the special unitary group in 3 dimensions, with the relations ${\displaystyle SU(3)}$

  ${\displaystyle {\begin{aligned}\{L_{i},L_{j}\}&=\varepsilon _{ijk}L_{k}\\\{L_{i},F_{jk}\}&=\varepsilon _{ijn}F_{nk}+\varepsilon _{ikn}F_{jn}\\\{F_{ij},F_{kl}\}&={\frac {\omega ^{2}}{4}}\left(\delta _{ik}\varepsilon _{jln}+\delta _{il}\varepsilon _{jkn}+\delta _{jk}\varepsilon _{iln}+\delta _{jl}\varepsilon _{ikn}\right)L_{n}\,,\end{aligned}}}$

where are the Poisson brackets , the Kronecker delta and the Levi-Civita symbol . ${\displaystyle \{\cdot ,\cdot \}}$ ${\displaystyle \delta }$ ${\displaystyle \varepsilon }$

Proof of conservation

In Hamiltonian mechanics applies to any function that is defined on the phase space ${\displaystyle A}$

${\displaystyle {\frac {\mathrm {d} A}{\mathrm {d} t}}=\{A,H\}=\sum _{k}\left({\frac {\partial A}{\partial x_{k}}}{\frac {\partial H}{\partial p_{k}}}-{\frac {\partial A}{\partial p_{k}}}{\frac {\partial H}{\partial x_{k}}}\right)+{\frac {\partial A}{\partial t}}}$,

thus for the Fradkin tensor in the harmonic oscillator

${\displaystyle {\frac {\mathrm {d} F_{ij}}{\mathrm {d} t}}={\frac {1}{2}}\omega ^{2}\sum _{k}{\Big (}(x_{j}\delta _{ik}+x_{i}\delta _{jk})p_{k}-(p_{j}\delta _{ik}+p_{i}\delta _{jk})x_{k}{\Big )}=0}$.

The Fradkin tensor is that according to Noether's theorem for transformation

${\displaystyle x_{i}\to x_{i}'=x_{i}+{\frac {1}{2}}\omega ^{-1}\varepsilon _{jk}\left({\dot {x}}_{j}\delta _{ik}+{\dot {x}}_{k}\delta _{ij}\right)}$

belonging conservation size.

Quantum mechanics

In quantum mechanics, position and momentum vectors are to be replaced by the position and momentum operator and the Poisson brackets by the commutator . Accordingly, the Hamilton function becomes the Hamilton operator, the angular momentum the angular momentum operator and the Fradkin tensor the Fradkin operator. The further statements remain valid under these substitutions.

Individual evidence

1. ^ Josef-Maria Jauch, Edward Lee Hill: On the Problem of Degeneracy in Quantum Mechanics . In: Physical Review . tape 57 , no. 7 , April 1, 1940, p. 641-645 , doi : 10.1103 / PhysRev.57.641 .
2. ^ David M. Fradkin: Existence of the Dynamic Symmetries and for All Classical Central Potential Problems ${\displaystyle O_{4}}$ ${\displaystyle SU_{3}}$ . In: Progress of Theoretical Physics . tape 37 , no. 5 , May 1, 1967, pp. 798-812 , doi : 10.1143 / PTP.37.798 .
3. ^ Jean-Marc Lévy-Leblond: Conservation Laws for Gauge-Variant Lagrangians in Classical Mechanics . In: American Journal of Physics . tape 39 , no. 5 , May 1, 1971, p. 502-506 , doi : 10.1119 / 1.1986202 .