March 2018
Intermediate to advanced
864 pages
22h 49m
English
Content preview from Mathematical Methods in Science and Engineering, 2nd Edition
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Chapter 3Hermite Polynomials
Hermite polynomials appear in many different branches of science like the probability theory, combinatorics, and numerical analysis. We also encounter them in quantum mechanics in conjunction with the harmonic oscillator problem and in systems theory in connection with Gaussian noise. They are named after the French mathematician Charles Hermite (1822–1901).
3.1 Harmonic Oscillator in Quantum Mechanics
In terms of the Hamiltonian operator, 
the time-independent Schrödinger equation is written as
3.1 
where 
is the wave function and 
stands for the energy eigenvalues. In general, 
is obtained from the classical Hamiltonian by replacing the position, 
and the momentum, 
, with their operator counterparts:
3.2
In one-dimension, the Hamiltonian operator for the ...
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