#harmonic oscillation

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Harmonic Oscillation

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Hermite polynomial normalization constant

Hermite polynomial normalization constant

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Question: Hermite polynomial normalization constant ([1] pr. 2.21)

Derive the normalization constant \( c_n \) for the Harmonic oscillator solution

\begin{equation}\label{eqn:hermiteOrtho:20} u_n(x) = c_n H_n\lr{ x \sqrt{\frac{m\omega}{\Hbar}} } e^{-m \omega x^2/2 \Hbar}, \end{equation}

by deriving the orthogonality relationship using…

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Harmonic Oscillation

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     The sine wave or sinusoid is a mathematical curve that describes a smooth repetitive oscillation. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:

where:

A, the amplitude, is the peak deviation of the function from zero.

f, the ordinary frequency, is the number of oscillations (cycles) that occur each second of time.

ω= 2πf, the angular frequency, is the rate of change of the function argument in units of radians per second

φ, the phase, specifies (in radians) where in its cycle the oscillation is at t= 0.

When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ/ω seconds. A negative value represents a delay, and a positive value represents an advance.

The sine wave is important in physics because it retains its waveshape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.

The oscillation of an undamped spring-mass system around the equilibrium is a sine wave.