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Given a particle of mass m in the harmonic oscillator potential starts out in the state mwx (x, 0) = A 1-2 exp 2h a) Normalize the wave function (x, 0). b) The Hermite polynomials form an orthonormal basis for the set of square-integrable functions over the entire real line. One way of quickly obtaining the Hermite polynomials is to use Rodrigues' formula. Calculate the first five Hermite polynomials using the Rodrigues' formula dn Н, (Е) %3D (-1)" еxp(€?) -(ехp(-8?)) dɛ" c) Find the coefficients cn of y(x, 0) in the basis 1/4 .) ep (-) mw 1 Om(x) = (Th V2ni Hn (E) exp by expressing (1-2,7 mw in terms of the first three Hermite polynomials Ho (E), H, (E), and H2(E).

Given a particle of mass m in the harmonic oscillator potential starts out in the state mwx (x, 0) = A 1-2 exp 2h a) Normalize the wave function (x, 0). b) The Hermite polynomials form an orthonormal basis for the set of square-integrable functions over the entire real line. One way of quickly obtaining the Hermite polynomials is to use Rodrigues' formula. Calculate the first five Hermite polynomials using the Rodrigues' formula dn Н, (Е) %3D (-1)" еxp(€?) -(ехp(-8?)) dɛ" c) Find the coefficients cn of y(x, 0) in the basis 1/4 .) ep (-) mw 1 Om(x) = (Th V2ni Hn (E) exp by expressing (1-2,7 mw in terms of the first three Hermite polynomials Ho (E), H, (E), and H2(E).

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Given a particle of mass m in the harmonic oscillator potential starts out in the state mwx2 (x, 0) = A ( 1- 2 exp 2h a) Normalize the wave function (x, 0). b) The Hermite polynomials form an orthonormal basis for the set of square-integrable functions over the entire real line. One way of quickly obtaining the Hermite polynomials is to use Rodrigues' formula. Calculate the first five Hermite polynomials using the Rodrigues' formula Н, (€) %3D (-1)" еxp(€?) dn -(еxp(—8?)) dɛn c) Find the coefficients c, of y(x, 0) in the basis 1/4 Pn(x) = () mw mw Hn(E) exp V2"n! ; E = by expressing (1-2) mw in terms of the first three Hermite polynomials Ho (E), H1 (E), and H2(E).
Transcribed Image Text:Given a particle of mass m in the harmonic oscillator potential starts out in the state mwx2 (x, 0) = A ( 1- 2 exp 2h a) Normalize the wave function (x, 0). b) The Hermite polynomials form an orthonormal basis for the set of square-integrable functions over the entire real line. One way of quickly obtaining the Hermite polynomials is to use Rodrigues' formula. Calculate the first five Hermite polynomials using the Rodrigues' formula Н, (€) %3D (-1)" еxp(€?) dn -(еxp(—8?)) dɛn c) Find the coefficients c, of y(x, 0) in the basis 1/4 Pn(x) = () mw mw Hn(E) exp V2"n! ; E = by expressing (1-2) mw in terms of the first three Hermite polynomials Ho (E), H1 (E), and H2(E).
Assume that: Ja) is an arbitrary ket from the vector space V • operators Ä and B are linear operators acting on vectors from V • the set of all eigenvectors of  is given by |A1), A2), ... |AN), and form an orthonormal basis d; is eigenvalue of  that corresponds to the ket |A;) Aij and Bij are matrix elements of the matrix representations of the operators Ä and B
Transcribed Image Text:Assume that: Ja) is an arbitrary ket from the vector space V • operators Ä and B are linear operators acting on vectors from V • the set of all eigenvectors of  is given by |A1), A2), ... |AN), and form an orthonormal basis d; is eigenvalue of  that corresponds to the ket |A;) Aij and Bij are matrix elements of the matrix representations of the operators Ä and B
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