Mehler Kernel¶
The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator.
Core Idea¶
Mehler Kernel is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator. The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator. It was first discovered by Mehler in 1866, and since then, as Einar Hille remarked in 1932, "has been rediscovered by almost everybody who has worked in this field".
Scope of Application¶
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Mehler's formula. and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials H(\cdot) based on weight function \exp(-x^2) as.
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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.
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As a fundamental solution, the kernel is additive,. The eigenfunctions of N are the usual Hermite functions \psin(x) which are therefore also Eigenfunctions of \mathcal{F} .
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Probability version. 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 x,y having zero means and.
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Fractional Fourier transform. If \alpha is an integer multiple of \pi , then the above cotangent and cosecant functions diverge.
Clarity¶
A clear use of Mehler Kernel names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator.
Manages Complexity¶
Mehler Kernel compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—when t > \pi the i \sin t in the inverse square-root should be replaced by \left|\sin t\right| and K{H} should be multiplied by an extra Maslov phase factor.—and the practical consequence—it was first discovered by Mehler in 1866, and since then, as Einar Hille.
Abstract Reasoning¶
- Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator.
- Check operation and conditions. Hardy gave a simple proof by the Fourier integral representation of Hermite polynomials.
- Demand recognition evidence. This expansion is most easily derived by using the two-dimensional Fourier transform of p(x,y) , which is.
- Test variation.
Knowledge Transfer¶
Within the home domain. Knowledge about Mehler Kernel transfers literally when a new case preserves the same carrier type, relation, and recognition test. and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials H(\cdot) based on weight function \exp(-x^2) as. In physics, the fundamental solution, (Green's function), or propagator of the Hamiltonian for the quantum harmonic oscillator is called the Mehler kernel. Beyond the home domain. No canonical parent is asserted for Mehler Kernel.
Neighborhood in Abstraction Space¶
Mehler Kernel sits in a crowded region of the domain-specific corpus (26th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Named Analytic Theorems & Operators (39 abstractions)
Nearest neighbors
- p-Variation — 0.90
- Invariant factorization of LPDOs — 0.90
- Fourier Sine Transform — 0.89
- Filling radius — 0.89
- Laurent Polynomial — 0.89
Computed from structural-signature embeddings · 2026-10-08