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".
Mehler's formula Mehler (1866) defined a function
and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials H ( ⋅ ) {\displaystyle H(\cdot )} based on weight function exp ( − x 2 ) {\displaystyle \exp(-x^{2})} as
E ( x , y ) = ∑ n = 0 ∞ ( ρ / 2 ) n n ! H n ( x ) H n ( y ) . {\displaystyle E(x,y)=\sum _{n=0}^{\infty }{\frac {(\rho /2)^{n}}{n!}}~{\mathit {H}}_{n}(x){\mathit {H}}_{n}(y)~.}
This result is useful, in modified form, in quantum physics, probability theory, and harmonic analysis. Equivalently, in the probabilist's Hermite polynomials: 1 1 − ρ 2 exp ( − ρ 2 ( x 2 + y 2 ) − 2 ρ x y 2 ( 1 − ρ 2 ) ) = ∑ n = 0 ∞ ρ n n ! He n ( x ) He n ( y ) {\displaystyle {\frac {1}{\sqrt {1-\rho ^{2}}}}\exp \left(-{\frac {\rho ^{2}\left(x^{2}+y^{2}\right)-2\rho xy}{2\left(1-\rho ^{2}\right)}}\right)=\sum _{n=0}^{\infty }{\frac {\rho ^{n}}{n!}}~\operatorname {He} _{n}(x)\operatorname {He} _{n}(y)}
Substituting ρ = e − t {\displaystyle \rho =e^{-t}} , and letting h n := He n / n ! {\displaystyle h_{n}:=\operatorname {He} _{n}/{\sqrt {n!}}} , we have
tanh ( t / 2 ) exp ( − e − t ( x 2 + y 2 ) − 2 x y 4 sinh t ) = ∑ n = 0 ∞ ( 1 − e − t ) e − n t h n ( x ) h n ( y ) {\displaystyle {\sqrt {\tanh(t/2)}}\exp \left(-{\frac {e^{-t}(x^{2}+y^{2})-2xy}{4\sinh t}}\right)=\sum _{n=0}^{\infty }(1-e^{-t})e^{-nt}~h_{n}(x)h_{n}(y)}
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 φ ( x , t ) {\displaystyle \varphi (x,t)} to
∂ φ ∂ t = ∂ 2 φ ∂ x 2 − x 2 φ ≡ D x φ . {\displaystyle {\frac {\partial \varphi }{\partial t}}={\frac {\partial ^{2}\varphi }{\partial x^{2}}}-x^{2}\varphi \equiv D_{x}\varphi ~.}
The orthonormal eigenfunctions of the operator D {\displaystyle D} are the Hermite functions,
ψ n = H n ( x ) e − x 2 / 2 2 n n ! π , {\displaystyle \psi _{n}={\frac {H_{n}(x)\,e^{-x^{2}/2}}{\sqrt {2^{n}n!{\sqrt {\pi }}}}},}
with corresponding eigenvalues ( − 2 n − 1 ) {\displaystyle (-2n-1)} , furnishing particular solutions
φ n ( x , t ) = e − ( 2 n + 1 ) t H n ( x ) e − x 2 / 2 . {\displaystyle \varphi _{n}(x,t)=e^{-(2n+1)t}~H_{n}(x)e^{-x^{2}/2}.}
The general solution is then a linear combination of these; when fitted to the initial condition φ ( x , 0 ) {\displaystyle \varphi (x,0)} , the general solution reduces to
φ ( x , t ) = ∫ K ( x , y ; t ) φ ( y , 0 ) d y , {\displaystyle \varphi (x,t)=\int K(x,y;t)\varphi (y,0)dy~,}
where the kernel K {\displaystyle K} has the separable representation
K ( x , y ; t ) ≡ ∑ n ≥ 0 e − ( 2 n + 1 ) t π 2 n n ! H n ( x ) H n ( y ) exp ( − x 2 + y 2 2 ) . {\displaystyle K(x,y;t)\equiv \sum _{n\geq 0}{\frac {e^{-(2n+1)t}}{{\sqrt {\pi }}2^{n}n!}}~H_{n}(x)H_{n}(y)\exp \left(-{\frac {x^{2}+y^{2}}{2}}\right)~.}
Utilizing Mehler's formula then yields
∑ n ≥ 0 ( ρ / 2 ) n n ! H n ( x ) H n ( y ) exp ( − x 2 + y 2 2 ) = 1 1 − ρ 2 exp ( 4 x y ρ − ( 1 + ρ 2 ) ( x 2 + y 2 ) 2 ( 1 − ρ 2 ) ) . {\displaystyle {\sum _{n\geq 0}{\frac {(\rho /2)^{n}}{n!}}H_{n}(x)H_{n}(y)\exp \left(-{\frac {x^{2}+y^{2}}{2}}\right)={\frac {1}{\sqrt {1-\rho ^{2}}}}\exp \left({\frac {4xy\rho -\left(1+\rho ^{2}\right)\left(x^{2}+y^{2}\right)}{2\left(1-\rho ^{2}\right)}}\right)}~.}
On substituting this in the expression for K {\displaystyle K} with the value e − 2 t {\displaystyle e^{-2t}} for ρ {\displaystyle \rho } , Mehler's kernel finally reads
When t = 0 {\displaystyle t=0} , variables x {\displaystyle x} and y {\displaystyle y} coincide, resulting in the limiting formula necessary by the initial condition,
K ( x , y ; 0 ) = δ ( x − y ) . {\displaystyle K(x,y;0)=\delta (x-y)~.}
As a fundamental solution, the kernel is additive,
∫ d y K ( x , y ; t ) K ( y , z ; t ′ ) = K ( x , z ; t + t ′ ) . {\displaystyle \int dy\,K(x,y;t)K(y,z;t')=K(x,z;t{+}t')~.}
This is further related to the symplectic rotation structure of the kernel K {\displaystyle K} . When using the usual physics conventions of defining the quantum harmonic oscillator instead via
i ∂ φ ∂ t = 1 2 ( − ∂ 2 ∂ x 2 + x 2 ) φ ≡ H φ , {\displaystyle i{\frac {\partial \varphi }{\partial t}}={\frac {1}{2}}\left(-{\frac {\partial ^{2}}{\partial x^{2}}}+x^{2}\right)\varphi \equiv H\varphi ,}
and assuming natural length and energy scales, then the Mehler kernel becomes the Feynman propagator K H {\displaystyle K_{H}} which reads
⟨ x | exp ( − i t H ) | y ⟩ ≡ K H ( x , y ; t ) = 1 2 π i sin t exp ( i 2 sin t ( ( x 2 + y 2 ) cos t − 2 x y ) ) , t < π , {\displaystyle {\begin{aligned}\left\langle x\right|\exp(-itH)\left|y\right\rangle &\equiv K_{H}(x,y;t)\\&={\frac {1}{\sqrt {2\pi i\sin t}}}\exp \left({\frac {i}{2\sin t}}\left(\left(x^{2}+y^{2}\right)\cos t-2xy\right)\right),\quad t<\pi ,\end{aligned}}}
i.e. K H ( x , y ; t ) = K ( x , y ; i t / 2 ) . {\displaystyle K_{H}(x,y;t)=K(x,y;it/2).}
When t > π {\displaystyle t>\pi } the i sin t {\displaystyle i\sin t} in the inverse square-root should be replaced by | sin t | {\displaystyle \left|\sin t\right|} and K H {\displaystyle K_{H}} should be multiplied by an extra Maslov phase factor
exp ( i θ Maslov ) = exp ( − i π 2 ( 1 2 + ⌊ t π ⌋ ) ) . {\displaystyle \exp \left(i\theta _{\text{Maslov}}\right)=\exp \left(-i{\frac {\pi }{2}}\left({\frac {1}{2}}+\left\lfloor {\frac {t}{\pi }}\right\rfloor \right)\right).}
When t = π / 2 {\displaystyle t=\pi /2} the general solution is proportional to the Fourier transform F {\displaystyle {\mathcal {F}}} of the initial conditions φ 0 ( y ) ≡ φ ( y , 0 ) {\displaystyle \varphi _{0}(y)\equiv \varphi (y,0)} since
φ ( x , t = π 2 ) = ∫ K H ( x , y ; π 2 ) φ ( y , 0 ) d y = 1 2 π i ∫ e − i x y φ ( y , 0 ) d y = e − i π / 4 F [ φ 0 ] ( x ) , {\displaystyle {\begin{aligned}\varphi (x,\,t{=}{\tfrac {\pi }{2}})&=\int K_{H}(x,y;{\tfrac {\pi }{2}})\varphi (y,0)\,dy\\[1ex]&={\frac {1}{\sqrt {2\pi i}}}\int e^{-ixy}\varphi (y,0)\,dy\\[1ex]&=e^{-i\pi /4}{\mathcal {F}}[\varphi _{0}](x)~,\end{aligned}}} and the exact Fourier transform is thus obtained from the quantum harmonic oscillator's number operator written as
N ≡ 1 2 ( x − ∂ ∂ x ) ( x + ∂ ∂ x ) = H − 1 2 = 1 2 ( − ∂ 2 ∂ x 2 + x 2 − 1 ) {\displaystyle {\begin{aligned}N&\equiv {\frac {1}{2}}\left(x-{\frac {\partial }{\partial x}}\right)\left(x+{\frac {\partial }{\partial x}}\right)\\&=H-{\frac {1}{2}}={\frac {1}{2}}\left(-{\frac {\partial ^{2}}{\partial x^{2}}}+x^{2}-1\right)~\end{aligned}}} since the resulting kernel
⟨ x | exp ( − i t N ) | y ⟩ ≡ K N ( x , y ; t ) = e i t / 2 K H ( x , y ; t ) = e i t / 2 K ( x , y ; i t / 2 ) {\displaystyle {\begin{aligned}\left\langle x\right|\exp(-itN)\left|y\right\rangle &\equiv K_{N}(x,y;t)\\&=e^{it/2}K_{H}(x,y;t)\\&=e^{it/2}K(x,y;it/2)\end{aligned}}} also compensates for the phase factor still arising in K H {\displaystyle K_{H}} and K {\displaystyle K} , i.e.
φ ( x , t = π 2 ) = ∫ K N ( x , y ; π / 2 ) φ ( y , 0 ) d y = F [ φ 0 ] ( x ) , {\displaystyle \varphi (x,\,t{=}{\tfrac {\pi }{2}})=\int K_{N}(x,y;\pi /2)\varphi (y,0)dy={\mathcal {F}}[\varphi _{0}](x)~,}
which shows that the number operator can be interpreted via the Mehler kernel as the generator of fractional Fourier transforms for arbitrary values of t {\displaystyle t} , and of the conventional Fourier transform F {\displaystyle {\mathcal {F}}} for the particular value t = π / 2 {\displaystyle t=\pi /2} , with the Mehler kernel providing an active transform, while the corresponding passive transform is already embedded in the basis change from position to momentum space. The eigenfunctions of N {\displaystyle N} are the usual Hermite functions ψ n ( x ) {\displaystyle \psi _{n}(x)} which are therefore also Eigenfunctions of F {\displaystyle {\mathcal {F}}} .
Proofs There are many proofs of the formula. The formula is a special case of the Hardy–Hille formula, using the fact that the Hermite polynomials are a special case of the associated Laguerre polynomials: H 2 n ( x ) = ( − 1 ) n 2 2 n n ! L n ( − 1 / 2 ) ( x 2 ) H 2 n + 1 ( x ) = ( − 1 ) n 2 2 n + 1 n ! x L n ( 1 / 2 ) ( x 2 ) {\displaystyle {\begin{aligned}H_{2n}(x)&=\left(-1\right)^{n}2^{2n}n!L_{n}^{(-1/2)}(x^{2})\\[4pt]H_{2n+1}(x)&=\left(-1\right)^{n}2^{2n+1}n!xL_{n}^{(1/2)}(x^{2})\end{aligned}}} The formula is a special case of the Kibble–Slepian formula, so any proof of it immediately yields of proof of the Mehler formula. Foata gave a combinatorial proof of the formula. Hardy gave a simple proof by the Fourier integral representation of Hermite polynomials. Using the Fourier transform of the Gaussian e − x 2 = 1 π ∫ e − t 2 + 2 i x t d t {\textstyle e^{-x^{2}}={\frac {1}{\sqrt {\pi }}}\int e^{-t^{2}+2ixt}dt} , we have H n ( x ) = ( − 1 ) n e x 2 d n d x n e − x 2 = e x 2 π ∫ ( − 2 i t ) n e − t