Chemical Engineering

please use the appendix data to solve. Good luck

Appendix B: Useful Relations for the One-Dimensional Harmonic Oscillator The eigenstates of the 1D hsrillonic oscillator described by the Hamiltonian 1-5) (B.1) 2m + m2 2.2 2 are given by 4.(9) (B.2) Ren! where n is an integer. Eros. and (B3 The corresponding eigenvalues are E = TE+ tie (B.4) In Equation (B.2), the quantities I. () are the Hermite polynomials given by (B.5) HAE) = 6 dn dea Using this last relation, the first few Hermite polynomials can be written explicitly as Ho(E) = 1. Hi() = 2 H7() = 42 - 2. H3 (5) = 882 12 (B.6) (B.7) {B.8) (B.9 The Hermite polynomials can also be obtained from the generating function (5) 12 +255 H () (B.10) n! En or (B.11) ie.. H.(9 = (dene 23). : 1.0 ==-1)^ ) (B.12) den Using the concept of unitary transformation, show that the polarizability a of a one-dimensional harmonic oscillator im* w 2 in a uniform external electric field is given by a Some of the basic properties of the one-diiinl honic oscillator are listed in Appendir B. Appendix B: Useful Relations for the One-Dimensional Harmonic Oscillator The eigenstates of the 1D hsrillonic oscillator described by the Hamiltonian 1-5) (B.1) 2m + m2 2.2 2 are given by 4.(9) (B.2) Ren! where n is an integer. Eros. and (B3 The corresponding eigenvalues are E = TE+ tie (B.4) In Equation (B.2), the quantities I. () are the Hermite polynomials given by (B.5) HAE) = 6 dn dea Using this last relation, the first few Hermite polynomials can be written explicitly as Ho(E) = 1. Hi() = 2 H7() = 42 - 2. H3 (5) = 882 12 (B.6) (B.7) {B.8) (B.9 The Hermite polynomials can also be obtained from the generating function (5) 12 +255 H () (B.10) n! En or (B.11) ie.. H.(9 = (dene 23). : 1.0 ==-1)^ ) (B.12) den Using the concept of unitary transformation, show that the polarizability a of a one-dimensional harmonic oscillator im* w 2 in a uniform external electric field is given by a Some of the basic properties of the one-diiinl honic oscillator are listed in Appendir B