“Mehler kernel”的意思、由来-开放百科全书

The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator.

Mehler's formula

and showed, in modernized notation,^[2] that it can be expanded in terms of Hermite polynomials {{mvar|H}}(.) based on weight function exp(−{{mvar|x}}²) as

This result is useful, in modified form, in quantum physics, probability theory, and harmonic analysis.

Physics version

In physics, the fundamental solution, (Green's function), or propagator of the Hamiltonian for the quantum harmonic oscillator is called the Mehler kernel. It provides the fundamental solution---the most general solution^[3] {{math|φ(x,t)}} to

The orthonormal eigenfunctions of the operator {{mvar|D}} are the Hermite functions,

The general solution is then a linear combination of these; when fitted to the initial condition {{math|φ(x,0)}}, the general solution reduces to

On substituting this in the expression for {{mvar|K}} with the value exp(−2{{mvar|t}}) for {{mvar|ρ}}, Mehler's kernel finally reads

When {{mvar|t}} = 0, variables {{mvar|x}} and {{mvar|y}} coincide, resulting in the limiting formula necessary by the initial condition,

This is further related to the symplectic rotation structure of the kernel {{mvar|K}}.^[4]

Probability version

The result of Mehler can also be linked to probability. For this, the variables should be rescaled as {{math|x → x/{{radic|2}}}}, {{math|y → y/{{radic|2}}}}, so as to change from the 'physicist's' Hermite polynomials {{mvar|H}}(.) (with weight function exp(−{{mvar|x}}²)) to "probabilist's" Hermite polynomials {{math|He}}(.) (with weight function exp(−{{mvar|x}}²/2)). Then, {{mvar|E}} becomes

The left-hand side here is p(x,y)/p(x)p(y) where p(x,y) is the bivariate Gaussian probability density function for variables {{math|x,y}} having zero means and unit variances:

and {{math|p(x), p(y)}} are the corresponding probability densities of {{mvar|x}} and {{mvar|y}}.

This expansion is most easily derived by using the two-dimensional Fourier transform of {{math|p(x,y)}}, which is

The Inverse Fourier transform then immediately yields the above expansion formula.

This result can be extended to the multidimensional case (Kibble 1945, Slepian 1972,^[6] Hörmander 1985 ^[7]).

Fractional Fourier transform

Since Hermite functions {{math|ψ_n}} are orthonormal eigenfunctions of the Fourier transform,

in harmonic analysis and signal processing, they diagonalize the Fourier operator,

Thus, the continuous generalization for real angle {{mvar|α}} can be readily defined (Wiener, 1929;^[8] Condon, 1937^[9]), the fractional Fourier transform (FrFT), with kernel

This is a continuous family of linear transforms generalizing the Fourier transform, such that, for {{math|α {{=}} π/2}}, it reduces to the standard Fourier transform, and for {{math|α {{=}} −π/2}} to the inverse Fourier transform.

The Mehler formula, for {{mvar|ρ}} = exp(−i{{mvar|α}}), thus directly provides

The square root is defined such that the argument of the result lies in the interval [−π /2, π /2].

If {{mvar|α}} is an integer multiple of {{mvar|π}}, then the above cotangent and cosecant functions diverge. In the limit, the kernel goes to a Dirac delta function in the integrand, {{mvar|δ(x−y)}} or {{mvar|δ(x+y)}}, for {{mvar|α}} an even or odd multiple of {{mvar|π}}, respectively. Since

[{{mvar|f}} ] = {{mvar|f}}(−{{mvar|x}}),

[{{mvar|f}} ] must be simply {{math|f(x)}} or {{math|f(−x)}} for {{mvar|α}} an even or odd multiple of {{mvar|π}}, respectively.

See also

References

1. ^{{Citation | last1=Mehler | first1=F. G. | title=Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002152975 | language=German |id={{ERAM|066.1720cj}} | year=1866 | journal=Journal für die Reine und Angewandte Mathematik | issn=0075-4102 | issue=66 | pages=161–176}} (cf. p 174, eqn (18) & p 173, eqn (13) )
2. ^{{citation|first=Arthur|last= Erdélyi|authorlink1=Arthur Erdélyi|first2= Wilhelm |last2=Magnus|authorlink2=Wilhelm Magnus|first3= Fritz|last3= Oberhettinger|first4=Francesco G.|last4= Tricomi|authorlink4=Francesco Tricomi|title= Higher transcendental functions. Vol. II|publisher= McGraw-Hill| year=1955}} (scan: p.194 10.13 (22))
3. ^Pauli, W., Wave Mechanics: Volume 5 of Pauli Lectures on Physics (Dover Books on Physics, 2000) {{ISBN|0486414620}} ; See section 44.
4. ^The quadratic form in its exponent, up to a factor of −1/2, involves the simplest (unimodular, symmetric) symplectic matrix in Sp(2,ℝ). That is, :

where:

so it preserves the symplectic metric,:

5. ^{{Citation | last1=Kibble | first1=W. F. | title=An extension of a theorem of Mehler's on Hermite polynomials | doi=10.1017/S0305004100022313 | mr=0012728 | year=1945 | journal=Proc. Cambridge Philos. Soc. | volume=41 | pages=12–15}}
6. ^{{Citation | last1=Slepian | first1=David | title=On the symmetrized Kronecker power of a matrix and extensions of Mehler's formula for Hermite polynomials | doi=10.1137/0503060 | mr=0315173 | year=1972 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=3 | pages=606–616}}
7. ^{{Cite journal|journal=Mathematische Zeitschrift |date=1995 |volume =219 |pages=413–449 |title=Symplectic classification of quadratic forms, and general Mehler formulas |author= Hörmander, Lars | doi = 10.1007/BF02572374}}
8. ^Wiener, N (1929), "Hermitian Polynomials and Fourier Analysis", Journal of Mathematics and Physics 8: 70-73.
9. ^Condon, E. U. (1937). "Immersion of the Fourier transform in a continuous group of functional transformations", Proc. Natl. Acad. Sci. USA 23, 158–164. [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076889/pdf/pnas01779-0028.pdf online]

volume =2 | date= 1981| pages= 239–249| title=Extension of the Kibble-Slepian formula for Hermite polynomials using boson operator methods | doi = 10.1016/0196-8858(81)90005-1}}