Look at the general formulae for the wave functions (eigenfunctions) and energie

Question

Look at the general formulae for the wave functions (eigenfunctions) and energies (eigenvalues) of the quantum harmonic oscillator, including the Hermite polynomials, and then for the quantum numbers v = 0, 1, and 2: (a) obtain the energy value, (b) write explicitly in terms of the variable x (not of y) the wave function, (c) obtain the positions of the wavefunctions nodes from the roots of the corresponding Hermite polynomial, and (d) from your answers to (b) and (c), sketch the wave functions, indicating nodes and boundary conditions behaviors at +/- infinite in the drawings.

Explanation / Answer

Use this theory and formulas in it to get your answer.... try yourself and if any difficulty then post comment i will post detail answer There are two different ways of standardizing the Hermite polynomials: (the "probabilists' Hermite polynomials"), and (the "physicists' Hermite polynomials"). These two definitions are not exactly equivalent; either is a rescaling of the other, to wit These are Hermite polynomial sequences of different variances; see the material on variances below. The notation He and H is that used in the standard references Tom H. Koornwinder, Roderick S. C. Wong, and Roelof Koekoek et al. (2010) and Abramowitz & Stegun. The polynomials Hen are sometimes denoted by Hn, especially in probability theory, because is the probability density function for the normal distribution with expected value 0 and standard deviation 1. The first six (probabilists') Hermite polynomials Hen(x). The first eleven probabilists' Hermite polynomials are: The first six (physicists') Hermite polynomials Hn(x). and the first eleven physicists' Hermite polynomials are: [edit]Properties Hn is a polynomial of degree n. The probabilists' version He has leading coefficient 1, while the physicists' version H has leading coefficient 2n. [edit]Orthogonality Hn(x) and Hen(x) are nth-degree polynomials for n = 0, 1, 2, 3, .... These polynomials are orthogonal with respect to the weight function (measure) (He) or (H) i.e., we have when m ? n. Furthermore, (probabilist) or (physicist). The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function. [edit]Completeness The Hermite polynomials (probabilist or physicist) form an orthogonal basis of the Hilbert space of functions satisfying in which the inner product is given by the integral including the Gaussian weight function w(x) defined in the preceding section, An orthogonal basis for L2(R, w(x) dx) is a complete orthogonal system. For an orthogonal system, completeness is equivalent to the fact that the 0 function is the only function

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