Mehler kernel

The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator. defined a function and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials H(.) based on weight function exp(−x2) as This result is useful, in modified form, in quantum physics, probability theory, and harmonic analysis. 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 φ(x,t) to The orthonormal eigenfunctions of the operator D are the Hermite functions, with corresponding eigenvalues (2n+1), furnishing particular solutions The general solution is then a linear combination of these; when fitted to the initial condition φ(x,0), the general solution reduces to where the kernel K has the separable representation Utilizing Mehler's formula then yields On substituting this in the expression for K with the value exp(−2t) for ρ, Mehler's kernel finally reads When t = 0, variables x and y coincide, resulting in the limiting formula necessary by the initial condition, As a fundamental solution, the kernel is additive, This is further related to the symplectic rotation structure of the kernel K. When using the usual physics conventions of defining the quantum harmonic oscillator instead via and assuming natural length and energy scales, then the Mehler kernel becomes the Feynman propagator which reads i.e. When the in the inverse square-root should be replaced by and should be multiplied by an extra Maslov phase factor When the general solution is proportional to the Fourier transform of the initial conditions since and the exact Fourier transform is thus obtained from the quantum harmonic oscillator's number operator written as since the resulting kernel also compensates for the phase factor still arising in and , i.e.