The Hamiltonian operator of a system is H = -(dI dx) + x. Show that Nx exp (-x/2) is an eigenfunction of H and determine the eigenvalue. Also evaluate N by normalization of the function.

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A: Here. We have to find the relationship between eigenvalues and characteristics polynomial.

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A: Step 1:Step 2: Step 3:

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A: The solution is given as

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A: Since you have posted multiple question, but according to guidelines we will solve only first…

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Q: Explain the connection between straight-line solutions, eigenvectors, and the general solution

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Q: How and why are eigenvectors used to describe solution sets of linear systems of ODE's?

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Q: What is the eigenvalue for the eigenfunction e^4ix of the operator ((d^2)/(dx^2))+(d/dx)−4i?

A: The solution is given as

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Q: Find e^(tA). Use e^(tA) = P e^(tD) P^(-1). The matrix is \begin{pmatrix}1&amp;2&amp;0\\…

A: I have used calculator to find eigen values and eigen vectors to avoid lengthy calculation.

Q: If A be nxn matrix with eigen value L then find the eigen value of inverse of A .

A: The given matrix A be an n×n matrix with eigenvalue L. Calculate the eigen value of inverse of A.