User:EfimovSP

Harmonical tensors

Formula

As a rule, harmonic functions are useful in theoretical physics to consider fields in far-zone when distance from charges is much further than size of their location. In that case, radius R is constant and coordinates (θ,φ) are convenient to use. Theoretical physics considers many problems when solution of Laplace's equation is needed as a function of Сartesian coordinates. At the same time, it is important to get invariant form of solutions relatively to rotation of space or generally speaking, relatively to group transformations. The simplest tensor solutions- dipole , quadrupole and octupole potentials are fundamental concepts of general physics:

$T^{(1)}{_{i}}=x_{i}$, $\quad T_{ik}^{(2)}=3x{_{i}}x{_{k}}-\delta _{ik}r^{2}$,$\quad T_{ikn}^{(3)}=15x{_{i}}x{_{k}}x{_{n}}-3\delta _{ik}{r^{2}}x{_{n}}-3\delta _{kn}{r^{2}}x{_{i}}-3\delta _{ni}{r^{2}}x{_{k}}$.

It is easy to verify that they are the harmonic functions. Total set of tensors is defined by Taylor series of point charge field potential for $r_{0}<r$:

${\frac {1}{\left|{\boldsymbol {r-r}}{_{0}}\right|}}=\sum _{l}(-1)^{l}{\frac {{({\boldsymbol {r_{0}}}\nabla )}^{l}}{l!}}{\frac {1}{r}}=\sum _{l}{\frac {x_{0i}\ldots x_{0k}}{l!\,r^{2l+1}}}T_{i\ldots k}^{(l)}({\boldsymbol {r}})=\sum _{l}{\frac {\left[\otimes {\boldsymbol {{r_{0}}^{l}T^{(l)}}}\right]}{l!\,r^{2l+1}}}$,

where tensor $x_{0i}\ldots x_{0k}$ is denoted by symbol $\otimes {\boldsymbol {r_{0}}}^{l}$ and convolution of the tensors is in the brackets [...]. Therefore, the tensor ${\boldsymbol {T}}^{(l)}$ is defined by l-th tensor derivative:

${\frac {\boldsymbol {T^{(l)}}}{r^{(2l+1)}}}=(-\otimes {\boldsymbol {\nabla }})^{l}{\frac {1}{r}}$

James Clerk Maxwell used similar considerations without tensors naturally. E. W. Hobson analysed Maxwell's method as well. One can see from the equation following properties that repeat mainly those of solid and spherical functions.

• Tensor is the harmonic polynomial i. e. $\Delta {\boldsymbol {T}}^{(l)}=0$.
• Trace over each two indices is zero, as far as $\Delta {\frac {1}{r}}=0$.
• Tensor is homogeneous polynomial of degree $l$ i.e. summed degree of variables x, y, z of each item is equal to $l$.
• Tensor has invariant form under rotations of variables x,y,z i.e. of vector $\mathbf {r}$.
• Total set of potentials $\mathbf {T} ^{(l)}$ is complete.
• Convolution of $\mathbf {T} ^{(l)}(\mathbf {r} )$ with tensor $\otimes {\boldsymbol {\rho }}^{l}$ is proportional to convolution of two harmonic potentials:

$\left[\mathbf {T} ^{(l)}(\mathbf {r} )\otimes {\boldsymbol {\rho }}^{l}\right]={\frac {1}{(2l-1)!!}}\left[\mathbf {T} ^{(l)}(\mathbf {r} )\mathbf {T} ^{(l)}({\boldsymbol {\rho }})\right]$

Formula for harmonical invariant tensor was found in paper . Detailed description is given in monography . Formula contains products of tensors $x_{i}\ldots x_{k}=\mathbf {\otimes } r^{m}$ and Kronecker symbols $\delta _{ik}$:

$\mathbf {T} ^{(l)}=(2l-1)!!\ \mathbf {(} \otimes r^{l})-(2l-3)!!\ r^{2}\left\langle \otimes \mathbf {r} ^{(l-2)}\otimes {\mathsf {\boldsymbol {\delta }}}^{1}\right\rangle +(2l-5)!!\ r^{4}\left\langle \otimes \mathbf {r} ^{(l-4)}\otimes {\mathsf {\boldsymbol {\delta }}}^{2}\right\rangle -\ldots$.

Quantity of Kronecker symbols is increased by two in the product of each following item when rang of tensor $x_{i}\ldots x_{k}$ is reduced by two accordingly. Operation $\left\langle \ldots \right\rangle$ symmetrizes tensor by means of all independent permutations of indices with following summing of got items. Particularly, $\delta _{ik}$ don't need to be transformed into $\delta _{ki}$ and tensor $x_{i}x_{k}$ don't go into $x_{k}x_{i}$.

Regarded tensors are convenient to substitute to Laplace equation:

$\Delta \left\langle \otimes \mathbf {r} ^{(l-2k)}\otimes {\mathsf {\boldsymbol {\delta }}}^{k}\right\rangle =2\left\langle \otimes \mathbf {r} ^{(l-2k-2)}\otimes {\mathsf {\boldsymbol {\delta }}}^{(k+2)}\right\rangle ,\quad (\mathbf {r} \mathbf {\nabla } )\left\langle \otimes \mathbf {r} ^{(l-2k)}\otimes {\mathsf {\boldsymbol {\delta }}}^{(k)}\right\rangle =l\left\langle \otimes \mathbf {r} ^{(l-2k)}\otimes {\mathsf {\boldsymbol {\delta }}}^{(k)}\right\rangle$.

The last relation is Euler formula for homogeneous polynomials actually. Laplace operator $\Delta$ leaves the indices symmetry of tensors. The two relations allows to substitute found tensor into Laplace equation and to check straightly that tensor is the harmonical function:

$\Delta \mathbf {T} ^{(l)}=0$.

Simplified moments

The last property is important for theoretical physics for the following reason. Potential of charges outside of their location is integral to be equal to the sum of multipole potentials:

$\iiint {\frac {f({\boldsymbol {r}})}{\left||\mathbf {r-r} {_{0}}\right|}}\,dx\,dy\,dz=\sum _{l}\iiint f(\mathbf {r} )\left[\mathbf {T} ^{(l)}(\mathbf {r} )dx\,dy\,dz{\frac {\mathbf {T} ^{(l)}(\mathbf {r} _{0})}{(2l-1)!!\,l!\,r_{0}^{(2l+1)}}}\right]$,

where ${f}(\mathbf {r} )$ is the charge density. The convolution is applied to tensors in the formula naturally. Integrals in the sum are called in physics as multipole moments. Three of them are used actively while others applied less often as their structure (or that of spherical functions) is more complicated. Nevertheless, last property gives the way to simplify calculations in theoretical physics by using integrals with tensor ${\boldsymbol {r}}^{l}$ instead of harmonical tensor $\mathbf {T} ^{(l)}$. Therefore, simplified moments give the same result and there is no need to restrict calculations for dipole, quadrupole and octupole potentials only. It is the advantage of the tensor point of view and not the only that.

Efimov's ladder operator

Spherical functions have a few recurrent formulas.. In quantum mechanics recurrent formulas plays a role when they connect $\psi -$functions of quantum states by means of a ladder operator.The property is occured due to symmetry group of considered system. The vector ladder operator for the invariant harmonical states found in paper and detailed in .

For that purpose, transformation of 3-d space is applied that conserves form of Laplace equation:

${\boldsymbol {\rho }}={\frac {\ \mathbf {r} }{r^{2}}}$.

When operator ${\boldsymbol {\nabla }}$ is applied to the harmonical tensor potential in ${\boldsymbol {\rho }}$-space then Efimov's ladder operator acts on transformed tensor in $\mathbf {r}$-space:

$\mathbf {D} =(2{\hat {l}}-1)\mathbf {r} -r^{2}{\boldsymbol {\nabla }}$,

where ${\hat {l}}$ is operator of module of angular momentum:

${\hat {l}}=({\boldsymbol {\nabla }}\mathbf {r} )$.

Operator ${\hat {l}}$ multiplies harmonic tensor by its degree i.e. by $l$ if to recall according spherical function for quantum numbers $l$, $m$. To check action of the ladder operator $\mathbf {\hat {D}}$, one can apply it to dipole and quadrupole tensors:

$\mathbf {\hat {D}} _{i}x_{k}=3x_{i}x_{k}-\delta _{ik}$,

$\mathbf {\hat {D}} _{i}x_{k}x_{n}=15x_{i}x_{k}x_{n}-3\delta _{ik}x_{n}-3\delta _{kn}x_{i}-3\delta _{ni}x_{k}$.

Applying successively $\mathbf {\hat {D}}$ to $\mathbf {1}$ we get general form of invariant harmonic tensors:

$\mathbf {T} ^{(l)}=(\otimes \mathbf {\hat {D}} )^{l}\mathbf {1}$.

The operator $\mathbf {\hat {D}}$ analogous to the oscillator ladder operator. To trace relation with a quantum operator it is useful to multiply it by $i\hbar$ to go to reversed space:

${\boldsymbol {\rho }}={\frac {\mathbf {r} }{r^{2}}}$.

As a result, operator goes in ${\boldsymbol {\rho }}$-space into the operator of momentum:

$\mathbf {\hat {D}} \Rightarrow \mathbf {\hat {p}}$.

It is useful to apply the following properties of $\mathbf {\hat {D}}$ . Commutator of the coordinate operators is zero:

${\hat {D}}_{i}{\hat {D}}_{k}-{\hat {D}}_{k}{\hat {D}}_{i}=0$.

The scalar operator product is zero in the space of harmonical functions:

${\hat {D}}_{i}{\hat {D}}_{i}={\hat {\mathbf {D} }}{\hat {\mathbf {D} }}=r^{4}\Delta$.

The property gives zero trace of the harmonical tensor $\mathbf {T} ^{l}$ over each two indices. The ladder operator is analogous for that in problem of the quantum oscillator. It generates Glauber states those are created in the quantum theory of electromagnetic radiation fields. It was shown later as theoretical result that the coherent states are intrinsic for any quantum system with a group symmetry to include the rotational group. .

Invariant form of spherical harmonics

Spherical harmonics accord with the system of coordinates. Let be $\mathbf {n} _{x},\mathbf {n} _{y},\mathbf {n} _{z}$ the unit vectors along axises X, Y, Z. Denote following unit vectors as $\mathbf {n} _{+}$ and $\ \mathbf {n} _{-}$:

$\mathbf {n} _{\pm }={\frac {(\mathbf {n} _{x}\pm i\mathbf {n} _{y})}{\sqrt {2}}}$.

Using the vectors, the solid harmonics are equal to:

$r^{l}Y_{(l\pm m)}=C_{l,m}(\mathbf {n} _{z}\mathbf {\hat {D)}} ^{(l-m)}(\mathbf {n} _{\pm }\mathbf {\hat {D)}} ^{m}\mathbf {1}$= $C_{l,m}\left[\mathbf {M} ^{(l)}\otimes \mathbf {n_{z}} ^{(l-m)}\otimes \mathbf {n_{\pm }} ^{m}\right]$

where $C_{l,m}$ is the constant:

$C_{l,m}=$ ${\frac {2^{m\setminus 2}{\sqrt {2l+1}}}{\sqrt {(l+m)!(l-m)!}}}$

Angular momentum $\mathbf {\hat {L}}$ is defined by the rotational group. The mechanical momentum $\mathbf {\hat {p}}$ is related to the translation group. The ladder operator is the mapping of momentum upon inversion 1/r of 3-d space. It is raising operator. Lowering operator here is the gradient naturally together with partial convolution over single indice $i$ to leave others:

$\left[\partial x_{i}\mathbf {T} _{i}^{(l-1)}\right]=(2l+1)l\,\mathbf {T} ^{(l-1)}$

Black-body in the theory of diffraction

Macdonald's model

The concept of black-body was formulated initially for size of it to be much less than the wave length. To say about diffraction the method of geometrical optics is valid then. To apply the Plank law for emitting of black body one has to regard the restriction:

$\lambda \ll \ L$,

where $L$ is the characteristic size of object. In 20-th age, the series of attempts was taken to find approach that is valid for any wave length - similar to ideal reflecting surface. In that case, an according mathematical condition has to be formulated on surface of black-body. As to be known, impedance matching is effective for the only angle of incidence. In 1911, Macdonald H.M. proposed nearly self-evident approach. He used two well formulated problems in electrodynamics- that of reflection from ideal metal:

$\mathbf {E} _{\tau }=0$,

and that of reflection from ideal magnetic:

$\mathbf {H} _{\tau }=0$.

Half-sum of solutions is the field around the black-body in Macdonald's model. The approach is clear in the scope of the geometrical optics. Two reflected rays have equal amplitudes of opposite signs and cancel each other. Therefore, the convex surface does not reflect rays at all. At the same time, surface forming convex and concave parts of surface allows double reflections. The second reflections have equal amplitudes of two rays what does not accord to black-body concept. Consequently, Macdonald's model is reasonable for the convex surface only. Diagrams of scattered fields around black ball for Macdonald's model are calculated on the base of Maxwell's equations in monography .

Adjunct space

Sommerfeld proposed to consider black flat screen as surface of continued space what is analogous to procedure in the theory of complex analysis. Therefore, the problem is got to be spacious instead of surface one.

The idea to continue physical space was developed later. In 1978, Sergei P. Efimov from Bauman Moscow State Technical University found that Macdonald's model is equivalent to that with symmetrical adjunct space. The spaces are connected formally on the surface of black-body. Actually, two problems are considered outside of the surface. One is with charges and currents, other is without that. Boundary conditions on the surface equate tangential components of electric and magnetic fields of two problems with changing sign of the magnetic component. In such a way, electric field in physical space is equal to half-sum of solutions of two problems for ideal reflecting surfaces:

$\mathbf {E} ={\frac {(\,\mathbf {E} _{+}+\mathbf {E} _{-})}{2}}\qquad$ (in physical space),

where $\mathbf {E} _{+}$ is field from problem for ideal metal and $\mathbf {E} _{-}$ is the field from problem for ideal magnetic. In the adjunct space, where no charges and currents, the sought electric field is equal to the difference of the same fields:

$\mathbf {E} ={\frac {(\,\mathbf {E} _{+}-\mathbf {E} _{-})}{2}}\qquad$ (in adjunct space).

The concept of adjunct space proves that Macdonald's model is physically correct for all frequencies. The causality holds in the approach and considerations of scatter of wave packs is acceptable. From symmetry of physical anf adjunct spaces follows two electrodynamical theorems:

• In state of the heat equilibrium, heat fluxes from surface in physical and adjunct spaces are equal to each other.
• Scattered field from thin black disc is equal to that from hole in flat thin screen (Babine's principle).

Macdonald's model and Efimov's consideration are valid for equations of acoustics, to equations of hydrodynamics, to diffusion equation. It should be noticed that half-sum of two subsidiary solutions is valid for linear equations only. It is clear that theoretical model needs a way for realizations. The concept of adjunct space can be applied to the black hole in theory of gravitation. The famous Schwarzschild metric looks mathematically simple:

$\mathbf {-} \,{\frac {\left(1-{\frac {r_{s}}{4R}}\right)^{2}}{\left(1+{\frac {r_{s}}{4R}}\right)^{2}}}dt^{2}+\left(1+{\frac {r_{s}}{4R}}\right)^{4}(dx^{2}+dy^{2}+dz^{2})$,

where $\mathbf {R} =(x,y,z)$ is radius-vector, $r_{s}$ is the Schwarzschild radius i.e. radius of black-hole. From point of view of concept based on the adjunct space , it is useful to apply the following transformation of physical space:

${\boldsymbol {\rho }}={\frac {a^{2}\mathbf {R} }{R^{2}}}$,

where $a$ is radius of sphere that adjunct space is attached to. As to be known, it is inversion in sphere. Radius $a$ is taken to give the Schwarzschild metrics again:

$a={\frac {r^{s}}{4}}$.

Therefore, the black hole can be considered as the connection of two symmetrical spaces on the surface of ball of radius $a$. Adjunct space has the same Schwarzschild metrics. In that case, well known singularity $R=0$ is disappeared.

Black-body of arbitrary form

Non-reflecting chamber has absolutely absorbing walls. Regarding physical picture, adjunct space now is simply the surrounding space as far as the walls are missed. Therefore, adjunct spaces for the totally convex surface and concave one are identical. The adjunct space is continued along normal directed into side of convexity of surface. Details are described in the paper. The equivalent electrodynamical problem can be formulated on the base of boundary condition. It analogous to the impedance matching. Nevertheless, the boundary condition binds tangential components of electric and magnet fields not in the point but on all surface. The condition is based on Stratton - Chu formula.

To demonstrate approach, it is useful to deduce boundary condition for scalar problem when Helmholtz equation is valid. Fields on the surface are bound by Green's function in two points $\mathbf {x}$ and $\mathbf {y}$:

$G(\mathbf {x,y} )={\frac {\exp(\mathbf {\mid x-y\mid } )}{4\pi \mathbf {\mid x-y\mid } }}.$

Let be charges (or radiation sources) are placed in non-reflecting chamber i.e. in free space. Green's formula defines field $u(\mathbf {x} )$ in adjunct space by boundary values on the surface of non-reflecting chamber. Upon sending argument $\mathbf {x}$ on the surface, formula gives boundary condition:

${\frac {u(\mathbf {x} )}{2}}=\iint \limits _{S,\,y\neq x}\left[{\mathbf {-} \,G(\mathbf {x,y} ){\frac {\partial u(\mathbf {y} )}{\partial n_{y}}}+u(\mathbf {y} ){\frac {\partial G(\mathbf {(x,y)} }{\partial n_{y}}}}\right]\,dS_{y}.$

The surface integral is calculated in the sense of Hadamard regularization. Normal $n_{y}$ is directed outside of chamber i.e. in side of convexity of surface.

Boundary condition for convex black-body (for example ball) differes by sign of first item in the integral as far as derivative changes sign on the surface for going from physical space to adjunct one. Normal is directed outside of black-body. At last, boundary condition for arbitrary surface, containing convex and concave parts, conserves its form under requirement that first sign is (+) for convex part and sign (-) for concave that.

Runge-Lenz Operator in Momentum Space

Runge-Lenz operator in the momentum space was found recently in . Formula for the operator is simplier than in the position space:

${\hat {\mathbf {A} }}_{\mathbf {p} }=\imath ({\hat {l}}_{\mathbf {p} }+1)\mathbf {p} -{\frac {(p^{2}+1)}{2}}\imath \mathbf {\nabla } _{\mathbf {p} },$

where "degree operator"

${\hat {l}}_{\mathbf {p} }=(\mathbf {p} \mathbf {\nabla } _{\mathbf {p} })$

multiplies a homogeneous polynomial by its degree.

The Efimov inequality by Pauli matrices

In 1976, Sergei P. Efimov deduced an inequality that refines the Robertson relation by applying high-order commutators. His approach is based on the Pauli matrices. Later V.V. Dodonov used the method to derive relations for a several observables by using Clifford algebra.

According to Jackiw, the Robertson uncertainty is valid only when the commutator is C-number. The Efimov method is effective for variables that have commutators of high-order - for example for the kinetic energy operator and for coordinate one. Consider two operators ${\hat {A}}$ and ${\hat {B}}$ that have commutator ${\hat {C}}$:

$\left[{\hat {A}},{\hat {B}}\right]={\hat {C}}.$

To shorten formulas we use the operator deviations:

$\delta {\hat {A}}={\hat {A}}-\left\langle {\hat {A}}\right\rangle$,

when new operators have the zero mean deviation. To use the Pauli matrices we can consider the operator:

${\hat {F}}=\gamma _{1}\,\delta {\hat {A}}\,\sigma _{1}+\gamma _{2}\,\delta {\hat {B}}\,\sigma _{2}+\gamma _{3}\,\delta {\hat {C}}\,\sigma _{3},$

where 2×2 spin matrices $\sigma _{i}$ have commutators:

$\left[\sigma _{i},\sigma _{k}\right]=\,i\,e_{ikl}\sigma _{l},$

where $e_{ikl}$ antisymmetric symbol. They act in the spin space independently from $\delta {\hat {A}}{,}\,\delta {\hat {B}}{,}\,\delta {\hat {C}}$. Pauli matrices define the Clifford algebra. We take arbitrary numbers $\gamma _{i}$ in operator ${\hat {F}}$ to be real.

Physical square of the operator is equal to:

${\hat {F}}{\hat {F}}^{+}=\gamma _{1}^{2}(\,\delta {\hat {A}}\delta {\hat {A}}^{+})+\gamma _{2}^{2}(\,\delta {\hat {B}}\delta {\hat {B}}^{+})+\gamma _{3}^{2}(\,\delta {\hat {C}}\delta {\hat {C}}^{+})+\gamma _{1}\gamma _{2}\,{\hat {C}}\,\sigma _{3}+\gamma _{2}\gamma _{3}\,{\hat {C}}_{2}\,\sigma _{1}-\gamma _{1}\gamma _{3}\,{\hat {C}}_{3}\,\sigma _{2},$

where ${\hat {F}}^{+}$ is adjoint operator and commutators ${\hat {C}}_{2}$ and ${\hat {C}}_{3}$ are following:

${\hat {C}}_{2}=i\left[\delta {\hat {B}},{\hat {C}}\right],\qquad {\hat {C}}_{3}=i\left[\delta {\hat {A}},{\hat {C}}\right].$

Operator ${\hat {F}}{\hat {F}}^{+}$ is positive-definite, what is essential to get an inequality below . Taking average value of it over state $\left|\psi \right\rangle$, we get positive-definite matrix 2×2:

${\begin{aligned}\left\langle \psi \right|{\hat {F}}{\hat {F}}^{+}\left|\psi \right\rangle &=\gamma _{1}^{2}\,\left\langle (\delta {\hat {A}})^{2}\right\rangle +\gamma _{2}^{2}\,\left\langle (\delta {\hat {B}})^{2}\right\rangle +\gamma _{3}^{2}\,\left\langle (\delta {\hat {C}})^{2}\right\rangle +\\&+\gamma _{1}\gamma _{2}\,\left\langle {\hat {C}}\right\rangle \,\sigma _{3}+\gamma _{2}\gamma _{3}\,\left\langle {\hat {C}}_{2}\right\rangle \,\sigma _{1}-\gamma _{1}\gamma _{3}\,\left\langle {\hat {C}}_{3}\right\rangle \,\sigma _{2},\end{aligned}}$

where used the notion:

$\left\langle (\delta {\hat {A}})^{2}\right\rangle =\left\langle (\delta {\hat {A}}\delta A^{+})\right\rangle$

and analogous one for operators $B,\,C$. Regarding that coefficients $\gamma _{i}$ are arbitrary in the equation, we get the positive-definite matrix 6×6. Its leading principal minors are non-negative. The Robertson uncertainty follows from minor of forth degree. To strengthen result we calculate determinant of sixth order:

$\left\langle {(\delta {\hat {A}})^{2}}\right\rangle \left\langle {(\delta {\hat {B}})^{2}}\right\rangle \left\langle {(\delta {\hat {C}})^{2}}\right\rangle \geq {\frac {1}{4}}\left\langle {\hat {C}}\right\rangle ^{2}\left\langle {(\delta {\hat {C}})^{2}}\right\rangle +{\frac {1}{4}}\left\langle (\delta {\hat {A}})^{2}\right\rangle \left\langle {\hat {C}}_{2}\right\rangle ^{2}+{\frac {1}{4}}\left\langle (\delta {\hat {B}})^{2}\right\rangle \left\langle {\hat {C}}_{3}\right\rangle ^{2}$

The equality is observed only when the state is an eigenstate for the operator ${\hat {F}}$ and likewise for the spin variables:

${\hat {F}}\,{\left|\psi \right\rangle }{\left|{\hat {s}}\right\rangle }$ = 0.

Found relation we may apply to the kinetic energy operator ${\hat {E}}_{kin}={\frac {\mathbf {\hat {p}} ^{2}}{2}}$ and for operator of the coordinate $\mathbf {\hat {x}}$:

$\left\langle (\delta {\hat {E}})^{2}\right\rangle \left\langle (\delta {\hat {\mathbf {x} }})^{2}\right\rangle \geq {\frac {\hbar ^{2}}{4}}\left\langle \,\mathbf {\hat {p}} \,\right\rangle ^{2}+{\frac {\hbar ^{2}}{2}}\left\langle (\delta {\hat {E}})^{2}\right\rangle \left\langle (\delta \mathbf {\hat {p}} )^{2}\right\rangle ^{-1}$

In particular, equality in the formula is observed for the ground state of the oscillator, whereas the right item of the Robertson uncertainty vanishes:

$\left\langle \,\mathbf {\hat {p}} \,\right\rangle =0$.

Physical sence of the relation is more clear if to divide it by the squared nonzero average impulse what yields:

$\left\langle (\delta {\hat {E}})^{2}\right\rangle (\delta t)^{2}\geq {\frac {\hbar ^{2}}{4}}+{\frac {\hbar ^{2}}{2}}\left\langle (\delta {\hat {E}})^{2}\right\rangle \left\langle (\delta \mathbf {\hat {p}} )^{2}\right\rangle ^{-1}\left\langle \,\mathbf {\hat {p}} \,\right\rangle ^{-2},$

where $(\delta t)^{2}=\left\langle (\delta \mathbf {\hat {x}} )^{2}\right\rangle \left\langle \mathbf {\,} {\hat {p}}\,\right\rangle ^{-2}$ is squared effective time within which a particle moves near the mean trajectory.

The method can be applied for three noncomuting operators of angular momentum $\mathbf {\hat {L}}$. We compile the operator:

${\hat {F}}=\gamma _{1}\,\delta {\hat {L}}_{x}\,\sigma _{1}+\gamma _{2}\,\delta {\hat {L}}_{y}\,\sigma _{2}+\gamma _{3}\,\delta {\hat {L}}_{z}\,\sigma _{3}.$

We recall that the operators $\sigma _{i}$ are auxiliary and there is no relation between the spin variables of the particle. In such way, their commutative properties are of importance only. Squared and averaged operator ${\hat {F}}$ gives positive-definite matrix where we get following inequality from:

$\left\langle {(\delta {\hat {L}}_{x})}^{2}\right\rangle \left\langle {(\delta {\hat {L}}_{y})}^{2}\right\rangle \left\langle {(\delta {\hat {L}}_{z})}^{2})\right\rangle \geq {\frac {\hbar ^{2}}{4}}\sum _{i=1}^{3}\left\langle (\delta {\hat {L}}_{i})^{2}\right\rangle \left\langle {\hat {L}}_{i}\right\rangle ^{2}$

To develop method for a group of operators one may use the Clifford algebra instead of the Pauli matrices .