Hermitian function

Plot of the first five Hermitian functions h n and ψ n

The Hermitian functions are obtained from the Hermitian polynomials by multiplying them by the density of the Gaussian normal distribution . ${\ displaystyle h_ {n} (x)}$ ${\ displaystyle H_ {n} (x)}$

${\ displaystyle h_ {n} (x) = {\ frac {(-1) ^ {n}} {\ sqrt {2 ^ {n} n! {\ sqrt {\ pi}}}}} e ^ {x ^ {2} / 2} {\ frac {\ mathrm {d} ^ {n}} {\ mathrm {d} x ^ {n}}} e ^ {- x ^ {2}} = {\ frac {1 } {\ sqrt {2 ^ {n} n! {\ sqrt {\ pi}}}}} H_ {n} (x) e ^ {- {\ frac {1} {2}} x ^ {2}} ,}$
${\ displaystyle \ int _ {- \ infty} ^ {\ infty} h_ {n} (x) h_ {m} (x) \, \ mathrm {d} x = \ delta _ {n, m} \ qquad \ qquad n, m = 0,1,2, \ ldots}$

They are a very good example of the definition (creation) of an orthonormal basis , similar to the sine / cosine functions . While the latter are able to break down a periodic signal into a frequency spectrum by means of spectral analysis ( Fourier analysis ), the Hermitian functions allow the description of singular events.

They have an important meaning in physics for the construction of the orthonormal solution functions of the quantum mechanical harmonic oscillator . Motivated by the creation and annihilation operators of quantum mechanics, the following recursive representation of the Hermitian functions is obtained

${\ displaystyle h_ {n} (x) = n ^ {- {\ frac {1} {2}}} a ^ {\ dagger} h_ {n-1} (x), \ qquad h_ {0} (x ) = \ pi ^ {- {\ frac {1} {4}}} e ^ {- {\ frac {1} {2}} x ^ {2}},}$

where the operator is defined by ${\ displaystyle a ^ {\ dagger}}$

${\ displaystyle a ^ {\ dagger} = {\ frac {1} {\ sqrt {2}}} {\ Big (} x - {\ frac {\ mathrm {d}} {\ mathrm {d} x}} {\ Big)}.}$

Singular events are usually characterized by intensity , mean and standard deviation . However, these characteristic values ​​can be identical for different, very different events, so that they are not sufficient for characterization. Therefore, the so-called “higher statistical moments ” are determined as further comparative variables. However, these are very sensitive to noise and drift of the zero line and therefore only suitable to a limited extent. If one develops a distribution in Hermitian functions, the coefficients are very stable, since the functions only live in the central area and thus suitably attenuate measurement data located further outside.

The development of a function representing an event according to Hermitian functions has a certain similarity to the wavelet transform .

Hermitian functions as eigenfunctions of the Fourier transformation

The Hermitian functions are eigenfunctions of the Fourier transformation in the one-dimensional to the eigenvalues : ${\ displaystyle \ left (- \ mathrm {i} \ right) ^ {n}}$

${\ displaystyle {\ mathcal {F}} \, h_ {n} = \ left (- \ mathrm {i} \ right) ^ {n} \, h_ {n} \ qquad \ left (n \ in \ mathbb { N} _ {0} \ right).}$

What is more, they also form a complete orthonormal system of eigenfunctions in space${\ displaystyle L ^ {2} \ left (\ mathbb {R} \ right)}$ .

Individual evidence

1. Helmut Fischer, Helmut Kaul: Mathematics for Physicists, Volume 2: Ordinary and partial differential equations, mathematical foundations of quantum mechanics . 2nd ed., BG Teubner, Wiesbaden 2004. ISBN 3-519-12080-1 , §12 section 4.2, pp. 300-301.