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Mathematical formulation of indeterminacy relations

Robertson–Schrödinger uncertainty relations

The most common general form of the uncertainty principle is the Robertson uncertainty relation.

For an arbitrary Hermitian operator ${\hat {\mathcal {O}}}$ we can associate a standard deviation

$\sigma _{\mathcal {O}}={\sqrt {\langle {\hat {\mathcal {O}}}^{2}\rangle -\langle {\hat {\mathcal {O}}}\rangle ^{2}}},$

where the brackets $\langle {\mathcal {O}}\rangle$ indicate an expectation value. For a pair of operators ${\hat {A}}$ and ${\hat {B}}$, we may define their commutator as

$[{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}},$

In this notation, the Robertson uncertainty relation is given by

$\sigma _{A}\sigma _{B}\geq \left|{\frac {1}{2i}}\langle [{\hat {A}},{\hat {B}}]\rangle \right|={\frac {1}{2}}\left|\langle [{\hat {A}},{\hat {B}}]\rangle \right|,$

The Robertson uncertainty relation immediately follows from a slightly stronger inequality, the Schrödinger uncertainty relation,

$\sigma _{A}^{2}\sigma _{B}^{2}\geq \left|{\frac {1}{2}}\langle \{{\hat {A}},{\hat {B}}\}\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle \right|^{2}+\left|{\frac {1}{2i}}\langle [{\hat {A}},{\hat {B}}]\rangle \right|^{2},$

where we have introduced the anticommutator,

$\{{\hat {A}},{\hat {B}}\}={\hat {A}}{\hat {B}}+{\hat {B}}{\hat {A}}.$

Uncertainty by the Pauli matrices

In 1976, an inequality that refines the Robertson relation by applying high-order commutators was found in Our 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. Found here 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 .

Fock's sphere in theory of hydrogen atom

The quantum Coulomb problem , which allows calculating the spectrum of a system of two opposite charges, is still fundamental in quantum theory . The names of the founders of twentieth physics are associated with it: N. Bohr, A. Sommerfeld, V. Pauly, E. Schrödinger and V. Fock. The introduction in the theory of atomic spectra begins with it, and it has been thoroughly studied using methods of the theory of special functions. Due to its simplicity and underlying symmetry- the group SO(4) of rotations in four dimensional space – it is an extremely useful and fine tool of theoretical physics for constructing various concepts .

Despite the apparent exhaustive treatment of the quantum Coulomb problem, there are still some questions that have not been fully studied. Fock’s result is surprising: why is the SO(4) symmetry realized in the momentum space wrapped into three –dimensional (3D) sphere, with an extension to the four-dimensional space ?

Essence of the problem

Let us recall the background proceeding Fock’s accomplishment. Two classical vector integrals, the angular momentum and the Laplace-Runge-Lenz vector, in quantum mechanics correspond to vector operators that commute with the energy operator, i.e.,with the Hamiltonian. An analysis of their commutators carried out in shows that they generate a Lie algebra (a linear space with a commutation operation) coinciding with a Lie algebra of operators of small (infinitesimal) rotations in 4D space .

For physics, the correspondence means that some transformation of variables and operators maps the original quantum Coulomb problem into the problem of free motion of a particle over a 3D sphere embedded in a 4D space. The energy operator is then invariant under rotations of the 3-D sphere . This is reminiscent of the remarkable effect of Lewis Carroll’s soaring grin of the Cheshire cat. .

Fock's approach struck contemporaries.The starting point in his theory is integral Schrödinger’s equation (SE) in momentum space. The space can be considered as 3D plane in a 4-D space. Fock then wraps 3D plane into a 3D sphere using stereographic projection, known since antiquity as convenient transformation of a globe into a flat map. (Fock’s globe is three-dimensional as is the map.) At the same time, additionally, Fock surmises the factor for wave functions such that original integral equation turns into an equation for spherical functions on 3D sphere (to be distinguished from functions on two-dimensional sphere.)

This equation, rarely used in physics but well-known in the theory of special functions, , is invariant under rotations in 4D space. Fock does not explain the physical meaning of transformation he found . As a result, the fundamental questions remain: why is the SO(4) symmetry realized in the wrapped momentum space rather than in the position space, and how has the electron 'learned about the stereographic projection?'

Recently, Fock's theory is further developed with the help of transformation of eigen-functions from the momentum space into 4-D position one. It was found that final transition of 4D spherical harmonics into physical space is algebraic and does not need an integration at all

Fock's theory

The Schrödinger equation for eigenfunctions (SE), using atomic units (the unit of energy beeing ${\frac {Z^{2}me^{4}}{\hbar ^{2}}}$ and the unit of length being Bohr's radius ${\frac {\hbar ^{2}}{Zme^{2}}}$ ) , has the form

$\left(-{\frac {1}{2}}\Delta -{\frac {1}{r}}\right)\Psi _{nlm}=-{\frac {1}{2n^{2}}}\Psi _{nlm}$.

In what follows, it is convenient to reduce the orbits of all radii $na_{B}$ to a single radius , i. e., change the radius vector for each eigen function as $\mathbf {r'} ={\frac {\mathbf {r} }{n}}$. The equation then takes a deceptively simple form.

$(-\Delta +1)\Psi _{nlm}(\mathbf {r} )={\frac {2n}{r}}\Psi _{nlm}(\mathbf {r} ),$

where $r$ is the modulus of the vector $\mathbf {r}$. The eigenfunctions in the momentum representation then have scaled argument $\mathbf {p'} =n\mathbf {p}$ .

The Schrödinger equation (with $(\hbar =1)$)when moving to the momentum representation:

$\Psi _{nlm}(\mathbf {r} )={\frac {1}{(2\pi )^{3}}}\int \ a_{nlm}(\mathbf {p} )e^{(i\mathbf {pr} )}d^{3}\mathbf {p}$,

contains a convolution with respect to momenta. Because the potential ${\frac {1}{r}}$ goes to ${\frac {4\pi }{\left|\mathbf {p^{2}} \right|}}$, the SE in the momentum space is nonlocal:

$(\left|\mathbf {p^{2}} \right\|+1)a_{nlm}(\mathbf {p} )-{\frac {2n}{2\pi ^{2}}}\int {\frac {a_{nlm}(\mathbf {p'} )}{\left|\mathbf {p-p'} \right|^{2}}}d^{3}\mathbf {p'} .$

Fock's sphere

The first step of the theory is to multiply the function $a_{nlm}(\mathbf {p} )$ (done without an explanation) by factor $(1+\mathbf {p^{2}} )^{2}$ . The second step is to wrap the 3D plane into 3-D sphere (with four coordinates $({\boldsymbol {\xi }},{\mathit {\xi _{0}}})$; see Fig.1). Figure 1 shows that the tangent of the slope of the projecting (red) straight line is $\tan \varphi ={\frac {1}{\left|\mathbf {p} \right|}}.$

Fig.1 Stereographic projection of $3D$ $\mathbf {p}$-plane on $3D$ sphere

Hence follow the formulas

$\left|{\boldsymbol {\xi }}\right|=\sin 2\varphi ={\frac {2\left|\mathbf {p} \right|}{(1+\mathbf {p^{2}} )}},$

${\boldsymbol {\xi }}={\frac {2\mathbf {p} }{(1+\mathbf {p^{2}} )}},$

${\mathit {\xi _{0}}}=\cos 2\varphi ={\frac {(\mathbf {p^{2}} -1)}{(\mathbf {p^{2}} +1)}},$

${\boldsymbol {\xi }}^{2}+\xi _{0}^{2}=1.$

The stereographic projection doubles the tilt angle $\varphi$ and this is the effect it produces. The flat drawing correctly reflects the 4-D transformation.

In the new variables, with Fock’s factor taken into account, the eigen-function becomes

$b_{nlm}({\boldsymbol {\xi }},\xi _{0})=(\mathbf {p^{2}} +1)^{2}a_{nlm}(\mathbf {p} ).$

It is essential that the projection be given by a conformal transformation. In this case, the angles between intersecting curves are preserved. The metric on the sphere in the momentum space coordinates ( of the p-plane) is expressed as

${\frac {4}{\mathbf {(p^{2}+1)^{2}} }}(d\mathbf {p} )^{2}.$

Hence, the contraction coefficient for elements of the p-space is ${\frac {(\mathbf {p^{2}} +1)}{2}}$ . The volume element in Fock's equation can be expressed in terms of the 3D surface element of sphere:

$d^{3}\mathbf {p} ={\frac {1}{8}}(\mathbf {p^{2}} +1)^{3}dS_{\xi }$

Additionally, the kernel of the integral can be (very fortunately but not obviously) transformed as

${\frac {1}{\mathbf {(p-p')^{2}} }}={\frac {2}{\mathbf {(p^{2}+1)} }}{\frac {1}{[{\boldsymbol {(\xi -\xi ')^{2}}}+(\xi _{0}-\xi '_{0})^{2}]}}{\frac {2}{\mathbf {(p'^{2}+1)} }},$

which doesn’t follow from the conformal property. This relations allow Fock to obtain the integral equation

$b_{nlm}({\boldsymbol {\xi }},\xi _{0})-{\frac {n}{2\pi ^{2}\ }}\int {\frac {b_{nlm}({\boldsymbol {\xi '}},\xi '_{0})}{[{\boldsymbol {(\xi -\xi ')^{2}}}+(\xi _{0}-\xi '_{0})^{2}]}}dS_{\xi '},$

where as can be seen from the figure, the surface element of the unit 4D sphere with the volume $2\pi ^{2}$ is

$dS_{\xi }={\frac {d{\xi _{1}}d\xi _{2}d{\xi _{3}}}{\xi _{0}}}={\frac {dV_{\xi }}{\xi _{0}}}$.

Next, V. Fock refers to the theory of spherical functions in 4-D space . These functions contain product of 3D spherical functions by the Gegenbauer polynomials of the argument $\xi _{0}$.

Found equation is rather complicated while Gegenbauer polynomials are very simple and useful for physicists.

See also

• Second quantization
• Fock state
• Hartree-Fock method
• Fock-Lorentz symmetry
• Fock-Schwinger gauge

Further reading

Fock, V.A. (2004) V.A. Fock-Selected Works:Quantum Mechanics and Quantum Field Theory. CRC Press.https://doi.org/10.1201/9780203643204

Category: Quantum mechanics

Category: Atomic, molecular, and optical physics

Category: Quantum chemistry

Category: Quantum field theory

Category:Group theory

Category: Theoretical physics

Anomalous magnetic moment of the electron

Here, it is worth pointing out, the empiric formula is valid for the anomalous magnetic moment $\delta \mu _{e}$

$\delta \mu _{e}={\frac {e\hbar }{2c}}\left[{\frac {3}{{\sqrt {2}}\left(m_{p}-\exp \left(1\right)\delta m\right)}}\right]$,

where $m_{p}$ is the proton mass and $\delta m$ is difference between neutron and proton masses $\left(m_{p}-m_{n}\right)$. In terms of the Bohr's magneton $\mu _{B}={\frac {e\hbar }{2m_{e}c}}$, the accuracy of the formula as follows

$\delta \mu _{e}=0.0011596523\times \mu _{B}$.

Harmonic tensors

In this article spherical functions are replaced by polynomials that have been well known in electrostatics since the time of Maxwell and associated with multipole moments . In physics, dipole and quadrupole moments typically appear because fundamental concepts of physics are associated precisely with them ; dipole moment and quadrupole moments are:

$\int x_{i}\rho (\mathbf {x} )dV$,

$\int (3x_{i}x_{k}-\delta _{ik})\rho (\mathbf {x} )dV$,

where $\rho (\mathbf {x} )$ is density of charges (or other quantity).

Octupole moment

$\int 3(5x_{i}x_{k}x_{l}-x_{l}\delta _{ik}-x_{k}\delta _{il}-x_{i}\delta _{lk})\rho (\mathbf {x} )dV$

is used rather seldom. As a rule, high-rank moments are calculated with the help of spherical functions. Spherical functions are convenient in scattering problems. Polynomials are preferable in calculations with differential operators. Here, properties of tensors, including high-rank moments as well, are considered to repeat basically features of solid spherical functions but having their own specifics.

Using of invariant polynomial tensors in Cartesian coordinates, as shown in a number of recent studies, is preferable and simplifies the fundamental scheme of calculations . The spherical coordinates are not involved here. The rules for using harmonic symmetric tensors are demonstrated that directly follow from their properties. These rules are naturally reflected in the theory of special functions, but are not always obvious, even though the group properties are general . At any rate, let us recall the main property of harmonic tensors: the trace over any pair of indices vanishes . Here, those properties of tensors are selected that not only make analytic calculations more compact and reduce 'the number of factorials' but also allow correctly formulating some fundamental questions of the theoretical physics .

General properties

Four properties of symmetric tensor $\mathbf {M} _{i...k}$ lead to the use of it in physics.

A. Tensor is homogeneous polynomial:

$\mathbf {M} _{i...k}(k\mathbf {x} )=k^{l}\mathbf {M} _{i...k}(\mathbf {x} )$,

where $l$ is the number of indices,i.e., tensor rank ;

B. Tensor is symmetric with respect to indices;

C. Tensor is harmonic, i.e., it is a solution of the Laplace equation:

$\Delta \mathbf {M} _{i...k}(\mathbf {x} )=0$;

D. Trace over any two indices vanishes:

$\mathbf {M} _{ii[...]}(\mathbf {x} )=0$,

where symbol $[...]$ denotes remaining $(l-2)$ indices after equating $i=i$.

Components of tensor are solid spherical functions. Tensor can be divided by factor $r^{l}$ to acquire components in the form of spherical functions.

Multipole tensors in electrostatics

The multipole potentials arise when the potential of a point charge is expanded in powers of coordinates $x_{oi}$ of the radius vector $\mathbf {r} _{o}$ ('Maxwell poles') . For potential

${\frac {1}{\left|\mathbf {r} -\mathbf {r} _{o}\right|}}$,

there is well known formula:

${\frac {1}{\left|\mathbf {r} -\mathbf {r} _{o}\right|}}=\sum _{l}(-1)^{l}{\frac {(\mathbf {r} _{0}\nabla )^{l}}{l!}}{\frac {1}{r}}=\sum _{l}{\frac {x_{0i}...x_{0k}}{l!r^{2l+1}}}\mathbf {M} _{i...k}^{(l)}(\mathbf {r} )=\sum _{l}{\frac {\mathbf {r} _{0}^{\otimes l}\mathbf {M} _{[i]}^{(l)}}{l!r^{2l+1}}}$,

where the following notation is used. For the $l$th tensor power of the radius vector

$x_{0i}...x_{0k}=\mathbf {r} _{0}^{\otimes l}$,

and for a symmetric harmonic tensor of rank $l$,

$\mathbf {M} _{i...k}^{(l)}(\mathbf {r} )=\mathbf {M} _{[i]}^{(l)}(\mathbf {r} )$.

The tensor is a homogeneous harmonic polynomial with described the general properties. Contraction over any two indices (when the two gradients become the $\Delta$ operator) is null. If tensor is divided by $r^{2l+1}$, then a multipole harmonic tensor arises

${\frac {\mathbf {M} _{i...k}^{(l)}(\mathbf {r} )}{r^{2l+1}}}={\frac {\mathbf {M} _{[i]}^{(l)}(\mathbf {r} )}{r^{2l+1}}}$,

which is also a homogeneous harmonic function with homogeneity degree $-(l+1)$.

From the formula for potential follows that

${\frac {\mathbf {M} _{im...k}^{(l+1)}(\mathbf {r} )}{r^{2l+3}}}=-\nabla _{i}{\frac {\mathbf {M} _{m...k}^{(l)}(\mathbf {r} )}{r^{2l+1}}}$,

which allows to construct a ladder operator.

Theorem on power-law equivalent moments in electrostatics

There is an obvious property of contraction

$(2l-1)!!(\mathbf {r} _{0}^{\otimes l},\mathbf {M} _{[i]}^{(l)}(\mathbf {r} ))=(\mathbf {M} _{[i]}^{(l)}(\mathbf {r} _{0}),\mathbf {M} _{[i]}^{(l)}(\mathbf {r} ))$,

that give rise to a theorem simplifying essentially the calculation of moments in theoretical physics.

Theorem

Let $\rho (x)$ be a distribution of charge. When calculating a multipole potential, power-law moments can be used instead of harmonic tensors (or instead of spherical functions ):

$(\int \rho (\mathbf {r} _{0})\mathbf {r} _{0}^{\otimes l}dV_{\mathbf {r} _{0}},{\frac {\mathbf {M} _{[i]}^{(l)}(\mathbf {r} )}{l!r^{2l+1}}})=(\int \rho (\mathbf {r} _{0})\mathbf {M} _{[i]}^{(l)}(\mathbf {r} _{0})dV_{\mathbf {r} _{0}},{\frac {\mathbf {M} _{[i]}^{(l)}(\mathbf {r} )}{(2l-1)!!l!r^{2l+1}}})$.

It is an advantage in comparing with using of spherical functions.

Example 1.

For the quadrupole moment, instead of the integral

$\int \rho (\mathbf {r} )(3x_{i}x_{k}-r^{2}\delta _{ik})dV$,

one can use 'short' integral

$3\int \rho (\mathbf {r} )x_{i}x_{k}dV$.

Moments are different but potentials are equal each other.

Formula for a harmonic tensor

Formula for the tensor was considered in using a ladder operator. It can be derived using the Laplace operator . Similar approach is known in the theory of special functions . The first term in the formula, as is easy to see from expansion of a point charge potential, is equal to

$\mathbf {M} _{[i]}^{(l)}(\mathbf {r} )=(2l-1)!!x_{i_{1}}...x_{i_{l}}+...=(2l-1)!!\mathbf {r} ^{\otimes l}+...$.

The remaining terms can be obtained by repeatedly applying the Laplace operator and multiplying by an even power of the modulus $r$. The coefficients are easy to determine by substituting expansion in the Laplace equation . As a result, formula is following:

$\mathbf {M} _{[i]}^{(l)}(\mathbf {r} )=(2l-1)!!\mathbf {r} ^{\otimes l}-{\frac {(2l-3)!!}{1!2^{1}}}r^{2}\Delta \mathbf {r} ^{\otimes l}+{\frac {(2l-5)!!}{2!2^{2}}}r^{4}\Delta ^{2}\mathbf {r} ^{\otimes l}-{\frac {(2l-7)!!}{3!2^{3}}}r^{6}\Delta ^{3}\mathbf {r} ^{\otimes l}+...$

.

This form is useful for applying differential operators of quantum mechanics and electrostatics to it. The differentiation generates product of the Kronecker symbols.

Example 2

$\Delta x_{i}x_{k}=2\delta _{ik}$,

$\Delta x_{i}x_{k}x_{m}=2(\delta _{ik}x_{m}+\delta _{km}x_{i}+\delta _{im}x_{k})$,

$\Delta \Delta x_{i}x_{k}x_{m}x_{n}=8(\delta _{ik}\delta _{mn}+\delta _{km}\delta _{in}+\delta _{im}\delta _{kn})$.

The last quality can be verified using the contraction with $i=k$ . It is convenient to write the differentiation formula in terms of the symmetrization operation. A symbol for it was proposed in , with the help of sum taken over all independent permutations of indices:

${\frac {\Delta ^{k}\mathbf {r} ^{\otimes l}}{k!2^{k}}}=\left\langle \left\langle \delta _{[..]}^{\otimes (l-2k)}\mathbf {r} ^{\otimes (l-2k)}\right\rangle \right\rangle$.

As a result, the following formula is obtained:

$\mathbf {M} _{[i]}^{(l)}(\mathbf {r} )=(2l-1)!!\mathbf {r} ^{\otimes l}-(2l-3)!!r^{2}\left\langle \left\langle \delta _{[..]}^{\otimes 1}\mathbf {r} ^{\otimes (l-2)}\right\rangle \right\rangle +(2l-5)!!r^{4}\left\langle \left\langle \delta _{[..]}^{\otimes 2}\mathbf {r} ^{\otimes (l-4)}\right\rangle \right\rangle -...$

,

where the symbol $\otimes k$ is used for a tensor power of the Kronecker symbol $\delta _{im}$ and conventional symbol [..] is used for the two subscripts that are being changed under symmetrization.

Following one can find the relation between the tensor and solid spherical functions. Two unit vectors are needed: vector $\mathbf {n} _{z}$ directed along the $z$-axis and complex vector $\mathbf {n} _{x}\pm i\mathbf {n} _{y}=\mathbf {n} _{\pm }$. Contraction with their powers gives the required relation

$(\mathbf {M} _{[i]}^{(l)}(\mathbf {r} ),\mathbf {n} _{z}^{\otimes (l-m)}\mathbf {n} _{\pm }^{\otimes m})=(l-m)!(x+iy)^{m}r^{(l-m)}{\frac {d^{m}}{dt^{m}}}P_{l}(t)\mid _{t={\frac {z}{r}}}$,

where $P_{l}(t)$ is a Legendre polynomial .

Special contractions

In perturbation theory, it is necessary to expand the source in terms of spherical functions. If the source is a polynomial, for example, when calculating the Stark effect, then the integrals are standard, but cumbersome. When calculating with the help of invariant tensors, the expansion coefficients are simplified, and there is then no need to integrals. It suffices, as shown in , to calculate contractions that lower the rank of the tensors under consideration. Instead of integrals, the operation of calculating the trace ${\hat {T}}r$ of a tensor over two indices is used. The following rank reduction formula is useful:

${\hat {T}}r\left\langle \left\langle \delta _{ik}\mathbf {M} _{ik[m]}^{(l)}\right\rangle \right\rangle =(2l+3)\mathbf {M} _{[m]}^{(l-2)}$,

where symbol [m] denotes all left (l-2) indices.

If the brackets contain several factors with the Kronecker delta, the following relation formula holds:

${\hat {T}}r\left\langle \left\langle \delta _{i_{1}p_{1}}...\delta _{i_{k}p_{k}}\mathbf {M} _{i_{1}p_{1}[m]}^{(l)}\right\rangle \right\rangle =(2l+2k+1)\left\langle \left\langle \delta _{i_{2}p_{2}}...\delta _{i_{k}p_{k}}\mathbf {M} _{[m]}^{(l-2)}\right\rangle \right\rangle$.

Calculating the trace reduces the number of the Kronecker symbols by one, and the rank of the harmonic tensor on the right-hand side of the equation decreases by two. Repeating the calculation of the trace k times eliminates all the Kronecker symbols:

${\hat {T}}r_{1}...{\hat {T}}r_{k}\left\langle \left\langle \delta _{i_{1}p_{1}}...\delta _{i_{k}p_{k}}\mathbf {M} _{[m]}^{(l)}\right\rangle \right\rangle =(2l+2k+1)!!\left\langle \left\langle \mathbf {M} _{[m]}^{(l-2k)}\right\rangle \right\rangle$.

Harmonic 4D tensors

The Laplace equation in four-dimensional 4D space has its own specifics. The potential of a point charge in 4D space is equal to ${\frac {1}{r^{2}}}$ . From the expansion of the point-charge potential ${\frac {1}{{(\mathbf {r} -\mathbf {r} _{0})}^{2}}}$ with respect to powers $\mathbf {r} _{0}^{\otimes n}$ the multipole 4D potential arises:

${\frac {{\mathfrak {M}}_{i...k}^{(n)}}{r^{2n+2}}}={(-1)}^{n}\nabla _{i}...\nabla _{k}{\frac {1}{r^{2}}}$.

The harmonic tensor in the numinator has a structure similar to 3D harmonic tensor. Its contraction with respect to any two indices must vanish. The dipole and quadruple 4-D tensors, as follows from here, are expressed as

${\mathfrak {M}}_{i}^{(1)}(\mathbf {r} )=2x_{i}$,

${\mathfrak {M}}_{ik}^{(2)}(\mathbf {r} )=2(4x_{i}x_{k}-\delta _{ik})$,

The leading term of the expansion, as can be seen, is equal to

${\mathfrak {M}}_{[i]}^{(n)}(\mathbf {r} )=(2n)!!x_{i_{1}}...x_{i_{n}}-...=(2n)!!\mathbf {r} ^{\otimes n}-...$

The method described for 3D tensor, gives relations

${\mathfrak {M}}_{[i]}^{(n)}(\mathbf {r} )=(2n)!!\mathbf {r} ^{\otimes n}-{\frac {(2n-2)!!}{1!2^{1}}}r^{2}\Delta \mathbf {r} ^{\otimes n}+{\frac {(2n-4)!!}{2!2^{2}}}r^{4}\Delta ^{2}\mathbf {r} ^{\otimes n}-{\frac {(2n-6)!!}{3!2^{3}}}r^{6}\Delta ^{3}\mathbf {r} ^{\otimes n}+...$

,

${\mathfrak {M}}_{[i]}^{(l)}(\mathbf {r} )=(2n)!!\mathbf {r} ^{\otimes n}-(2n-2)!!r^{2}\left\langle \left\langle \delta _{[..]}^{\otimes 1}\mathbf {r} ^{\otimes (n-2)}\right\rangle \right\rangle +(2n-4)!!r^{4}\left\langle \left\langle \delta _{[..]}^{\otimes 2}\mathbf {r} ^{\otimes (n-4)}\right\rangle \right\rangle -...$

.

Four-dimensional tensors are structurally simpler than 3D tensors.

Decomposition of polynomials in terms of harmonic functions

Applying the contraction rules allows decomposing the tensor with respect to the harmonic ones. In the perturbation theory, even the third approximation often considered good. Here, the decomposition of the tensor power up to the rank l=6 is presented:

$\bullet \ l=2$, $\qquad 3x_{i}x_{k}=\mathbf {M} _{ik}^{(2)}+r^{2}\delta _{ik}$,

$\bullet \ l=3$, $\qquad 5!!x_{i}x_{k}x_{m}=\mathbf {M} _{ikm}^{(3)}+3r^{2}\left\langle \left\langle \delta _{ik}x_{m}\right\rangle \right\rangle$,

$\bullet \ l=4$, $\qquad 7!!x_{i}x_{k}x_{l}x_{m}=\mathbf {M} _{iklm}^{(4)}+5r^{2}\left\langle \left\langle \mathbf {M} _{ik}^{(2)}\delta _{lm}\right\rangle \right\rangle +7r^{4}\left\langle \left\langle \delta _{ik}\delta _{lm}\right\rangle \right\rangle$,

$\bullet \ l=5$, $\qquad 9!!x_{i}x_{k}x_{l}x_{m}x_{n}=\mathbf {M} _{iklmn}^{(5)}+7r^{2}\left\langle \left\langle \mathbf {M} _{ikl}^{(3)}\delta _{mn}\right\rangle \right\rangle +27r^{4}\left\langle \left\langle \delta _{ik}\delta _{lm}x_{n}\right\rangle \right\rangle$ ,

$\bullet \ l=6$, $\qquad 11!!x_{i}x_{k}x_{l}x_{m}x_{n}x_{p}=$:$\qquad \qquad \qquad =\mathbf {M} _{iklmnp}^{(6)}+9r^{2}\left\langle \left\langle \mathbf {M} _{iklm}^{(4)}\delta _{np}\right\rangle \right\rangle +55r^{4}\left\langle \left\langle \mathbf {M} _{ik}^{(2)}\delta _{lm}\delta _{np}\right\rangle \right\rangle +99r^{6}\left\langle \left\langle \delta _{ik}\delta _{lm}\delta _{np}\right\rangle \right\rangle$.

To derive the formulas, it is useful to calculate the contraction with respect two indices,i.e., the trace. The formula for $l=6$ then implies the formula for $l=4$ . Applying the trace, there is convenient to use rules of previous section. Particular, the last term of the relations for even values of $l$ has the form

${\frac {2l-1)!!}{(l+1)!!}}$.

Also useful is the frequently occuring contraction over all indices,

$(\mathbf {M} _{[i]}^{(l)},\mathbf {M} _{[i]}^{(l)})=(\mathbf {M} _{i...k}^{(l)}(\mathbf {x} ),\mathbf {M} _{i...k}^{(l)}(\mathbf {x} ))={\frac {(2l)!}{2^{l}}}r^{2l}$

which arises when normalizing the states.

Decomposition of polynomials in 4D space

The decomposition of tensor powers of a vector is also compact in four dimensions:

$\bullet \ n=2$, $\qquad 4!!x_{i}x_{k}={\mathfrak {M}}_{ik}^{(2)}+2r^{2}\delta _{ik}$,

$\bullet \ n=3$, $\qquad 6!!x_{i}x_{k}x_{m}={\mathfrak {M}}_{ikm}^{(3)}+8r^{2}\left\langle \left\langle \delta _{ik}x_{m}\right\rangle \right\rangle$,

$\bullet \ n=4$, $\qquad 8!!x_{i}x_{k}x_{l}x_{m}={\mathfrak {M}}_{iklm}^{(4)}+6r^{2}\left\langle \left\langle {\mathfrak {M}}_{ik}^{(2)}\delta _{lm}\right\rangle \right\rangle +16r^{4}\left\langle \left\langle \delta _{ik}\delta _{lm}\right\rangle \right\rangle$,

$\bullet \ n=5$, $\qquad 10!!x_{i}x_{k}x_{l}x_{m}x_{n}={\mathfrak {M}}_{iklmn}^{(5)}+8r^{2}\left\langle \left\langle {\mathfrak {M}}_{ikl}^{(3)}\delta _{mn}\right\rangle \right\rangle +80r^{4}\left\langle \left\langle \delta _{ik}\delta _{lm}x_{n}\right\rangle \right\rangle$ ,

$\bullet \ n=6$, $\qquad 12!!x_{i}x_{k}x_{l}x_{m}x_{n}x_{p}=$:$\qquad \qquad \qquad ={\mathfrak {M}}_{iklmnp}^{(6)}+10r^{2}\left\langle \left\langle {\mathfrak {M}}_{iklm}^{(4)}\delta _{np}\right\rangle \right\rangle +72r^{4}\left\langle \left\langle {\mathfrak {M}}_{ik}^{(2)}\delta _{lm}\delta _{np}\right\rangle \right\rangle +240r^{6}\left\langle \left\langle \delta _{ik}\delta _{lm}\delta _{np}\right\rangle \right\rangle$.

When using the tensor notation with indices suppressed, the last equality becomes

$\bullet \ n=6$, $\qquad 12!!\mathbf {x} ^{\otimes (6)}={\mathfrak {M}}_{[i]}^{(6)}+10r^{2}\left\langle \left\langle {\mathfrak {\mathfrak {M}}}_{[i]}^{(4)}\delta _{[..]}\right\rangle \right\rangle +72r^{4}\left\langle \left\langle {\mathfrak {M}}_{[i]}^{(2)}\delta _{[..]}^{\otimes 2}\right\rangle \right\rangle +240r^{6}\left\langle \left\langle \delta _{[..]}^{\otimes 3}\right\rangle \right\rangle$.

Decomposition of higher powers is not more difficult using contractions over two indices.

Ladder operator

Ladder operators are useful for representing eigen functions in a compact form. They are a basis for constructing coherent states . Operators considered here, in mani respects close to the 'creation' and 'annihilation' operators of an oscillator.

Efimov's operator $\mathbf {\hat {D}}$ that increases the value of rank by one was introduced in . It can be obtained from expansion of point-charge potential:

$\mathbf {\nabla } _{i}{\frac {\mathbf {M} _{k...m}^{(l)}(\mathbf {r} )}{r^{2l+1}}}=-{\frac {\mathbf {M} _{ik...m}^{(l+1)}(\mathbf {r} )}{r^{2l+3}}}$.

Straightforward differentiation on the left-hand side of the equation yields a vector operator acting on a harmonic tensor:

${\hat {D}}_{i}M_{k...m}^{(l)}(\mathbf {r} )=M_{ik...m}^{(l+1)}(\mathbf {r} )$,

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

where operator

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

multiplies homogeneous polynomial by degree of homogeneity $l$. In particular,

${\hat {D}}_{i}x_{k}=3x_{i}x_{k}-r^{2}\delta {ik}$,

${\hat {D}}_{i}{\hat {D}}_{k}{\hat {D}}_{m}1=3[5x_{i}x_{k}x_{m}-r^{2}(\delta _{ik}x_{m}+\delta _{km}x_{i}+\delta _{im}x_{k})]$.

As a result of an $l$- fold application to unity, the harmonic tensor arises:

${\hat {D}}_{i}{\hat {D}}_{k}...{\hat {D}}_{m}\mathbf {1} =M_{ik...m}^{(l)}=\mathbf {D} _{[i]}^{\otimes l}=\mathbf {M} _{[i]}^{(l)}$,

written here in different forms.

The relation of this tensor to the angular momentum operator ${\hat {\mathbf {L} }}$ $(\hbar =1)$ is as follows:

$\mathbf {\hat {D}} ={\hat {l}}\mathbf {x} +i[\mathbf {x} \times \mathbf {\hat {L}} ]$.

Some useful properties of the operator in vector form given below. Scalar product

${\hat {D}}_{i}{\hat {D}}_{i}=r^{2}\Delta$

yields a vanishing trace over any two indices. The scalar product of vectors ${\hat {\mathbf {D} }}$ and $\mathbf {x}$ is

$\mathbf {x} {\hat {\mathbf {D} }}=r^{2}({\hat {l}}+1)$,

${\hat {\mathbf {D} }}\mathbf {x} =r^{2}{\hat {l}}$,

and, hence, the contraction of the tensor with the vector $\mathbf {x}$ can be expressed as

$x_{i}M_{ik...m}^{(l)}=2lr^{2}M_{k...m}^{(l-1)}$,

where $l$ is a number.

The commutator in the scalar product on the sphere is equal to unity:

$\mathbf {x} \mathbf {D} -\mathbf {D} \mathbf {x} =r^{2}$.

To calculate the divergence of a tensor, a useful formula is

$\mathbf {\nabla } {\hat {\mathbf {D} }}=({\hat {l}}+1)(2{\hat {l}}+3)$,

whence

$\nabla _{i}{\hat {M}}_{ik...m}^{(l)}=l(2l+1)M_{(k...m)}^{(l-1)}$

($l$ on the right-hand side is a number).

Four-dimensional ladder operator ${\mathfrak {D}}$

The raising operator in 4D space

${\hat {\mathfrak {D}}}_{i}{\mathfrak {M}}_{k...m}^{(n)}(\mathbf {r} ,\tau )={\mathfrak {M}}_{ik...m}^{(n+1)}(\mathbf {r} ,\tau )={\mathfrak {M}}_{ik...m}^{(n+1)}(\mathbf {y} )$

has largely similar properties. The main formula for it is

${\hat {\mathfrak {D}}}=2{\hat {n}}\mathbf {y} -\left|\mathbf {y} \right|^{2}\mathbf {\nabla } _{\mathbf {y} }$

, where $y_{i}$ is a 4D vector, $i=1,2,3,4$,

$\mathbf {y} =(\mathbf {r} ,\tau ),\quad \left|\mathbf {y} \right|^{2}=\rho _{\mathbf {y} }^{2}=\mathbf {x} ^{2}+\tau ^{2}$,

and the ${\hat {n}}$ operator multiplies a homogeneous polynomial by its degree. Separating the $\tau$ variable is convenient for physical problems:

${\hat {n}}=(\mathbf {x} \mathbf {\nabla } _{\mathbf {x} }+\tau {\frac {\partial }{\partial \tau }})=\mathbf {y} \mathbf {\nabla } _{\mathbf {y} }$.

In particular,

${\hat {\mathfrak {D}}}_{i}\mathbf {1} =2y_{i},\quad {\hat {\mathfrak {D}}}_{i}y_{k}=2(4y_{i}y_{k}-\rho _{\mathbf {y} }^{2}\delta _{ik})$,

${\hat {\mathfrak {D}}}_{i}{\hat {\mathfrak {D}}}_{k}{\hat {\mathfrak {D}}}_{m}=4!![6x_{i}x_{k}x_{m}-\rho _{\mathbf {y} }^{2}(\delta _{ik}x_{m}+\delta _{km}x_{i}+\delta _{im}x_{k})]$.

The scalar product of the ladder operator ${\hat {\mathfrak {D}}}$ and $y$ is as simple as in 3D space:

$\mathbf {y} {\hat {\mathfrak {D}}}={\hat {n}}_{\mathbf {y} }\rho _{\mathbf {y} }^{2},\quad {\hat {\mathfrak {D}}}\mathbf {y} =({\hat {n}}-2)\rho _{\mathbf {y} }^{2}$.

The scalar product of ${\hat {\mathfrak {D}}}$ and $\mathbf {\nabla }$ is

$\mathbf {\nabla } {\hat {\mathfrak {D}}}=2({\hat {n}}+2)^{2}$.

The ladder operator is now associated with the angular momentum operator and additional operator of rotations in 4D space ${\hat {\mathbf {A} }}$. They perform Lie algebra as the angular momentum and the Laplace-Runge-Lenz operators. Operator ${\hat {\mathbf {A} }}$ has the simple form

$\mathbf {\hat {A}} =i(\tau \mathbf {\nabla } -\mathbf {x} {\frac {\partial }{\partial \tau }})$.

Separately for the 3D $\mathbf {r}$ -component and the forth coordinate $\tau$ of the raising operator, formulas are

${\hat {\mathfrak {D}}}_{\mathbf {r} }=({\hat {n}}+1)\mathbf {r} +i[\mathbf {r} \times {\hat {\mathbf {L} }}]+i\tau {\hat {\mathbf {A} }}$,

${\hat {\mathfrak {D}}}_{\tau }=({\hat {n}}+1)\tau +i(\mathbf {r} {\hat {\mathbf {A} }})$.

See also

• Tensor
• Spherical harmonics
• Operator (physics)
• Laplace-Runge-Lenz vector
• Angular momentum operator
• Ladder operator

External links

• Feynman, Richard (1964–2013). "31.Tensors". The Feynman Lectures. California Institute of Technology.{{cite web}}: CS1 maint: date format (link)
• Edmonds, A.R. (1996). Angular Momentum in Quantum Mechanics. Princeton: Princeton University Press. p. 146. ISBN 0691025894.

Category: Harmonic analysis Category: Rotational symmetry Category: Quantum mechanics Category: Operator theory Category: Theoretical physics