Problem 1 [6] Compute the eigenvalues of Hˆ = pˆ 2 2m + 1 2 mΩ...

Problem 1 [6] Compute the eigenvalues of Hˆ = pˆ 2 2m + 1 2 mΩ 2xˆ 2 + λxˆ using two different methods: 1. Complete the square in 1 2mΩ 2x 2+λx (that is, write the term as 1 2mΩ 2 (x− x0) 2+C with suitable constants x0 and C) and use the exact eigenvalues En = (n+ 1 2 )¯hω of a harmonic oscillator with potential V (x) = 1 2mω2x 2 . 2. Apply second-order perturbation theory in λ

Problem 2 [2] Compute the eigenvalues of the matrix Hˆ = 2 λ λ 3 − 2λ ! and Taylor expand them to second order in the real number λ

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genius_generous answered 1 year ago

Find Eigenvalues and eigenvectors 6 -2 2 2 5 0 -2 0 7

Find the eigenvalues and eigenfunctions of the given boundary value problem. Assume that all eigenvalues are...

Find the eigenvalues and eigenfunctions of the given boundary value problem. Assume that all eigenvalues are real. (Let n represent an arbitrary positive number.) y''+λy= 0， y（0）= 0， y'（π）= 0

Find the eigenvalues and eigenfunctions of the problem y" + λy = 0 , y(−π/2) =...

Find the eigenvalues and eigenfunctions of the problem y" + λy = 0 , y(−π/2) = 0 , y(π/2) = 0. Please explain in detail all steps. Thank you

(a) Let λ be a real number. Compute A − λI. (b) Find the eigenvalues of...

(a) Let λ be a real number. Compute A − λI. (b) Find the eigenvalues of A, that is, find the values of λ for which the matrix A − λI is not invertible. (Hint: There should be exactly 2. Label the larger one λ1 and the smaller λ2.) (c) Compute the matrices A − λ1I and A − λ2I. (d) Find the eigenspace associated with λ1, that is the set of all solutions v = v1 v2 to (A...

eigenvalues of the matrix A = [1 3 0, 3 ?2 ?1, 0 ?1 1] are...

eigenvalues of the matrix A = [1 3 0, 3 ?2 ?1, 0 ?1 1] are 1, ?4 and 3. express the equation of the surface x^2 ? 2y^2 + z^2 + 6xy ? 2yz = 16. How should i determine the order of the coefficient in the form X^2/A+Y^2/B+Z^2/C=1?

find all eigenvalues and eigenvectors of the given matrix A= [1 0 0 2 1 -2...

find all eigenvalues and eigenvectors of the given matrix A= [1 0 0 2 1 -2 3 2 1]

(a) Compute the first-order correction to all energy eigenvalues of a particle in a square well...

(a) Compute the first-order correction to all energy eigenvalues of a particle in a square well in the region 0 ≤ ? ≤ L with added potential: ?(?) = ?o? when 0 ≤ ? ≤ L/2 and ?o (L − ?) when L/2 ≤ ? ≤ L. (b) Using the given potential, use the variational theorem to compute an upper bound to the ground state using the trial function below ∅(?) = ?1 (2/L)1/2 sin ( ??/L ) + ?2...

2. (a) Find the values of a and b such that the eigenvalues of A =...

Consider the Hamiltonian of a particle in one-dimensional problem defined by: H = 1 2m P...

Consider the Hamiltonian of a particle in one-dimensional problem defined by: H = 1 2m P 2 + V (X) where X and P are the position and linear momentum operators, and they satisfy the commutation relation: [X, P] = i¯h The eigenvectors of H are denoted by |φn >; where n is a discrete index H|φn >= En|φn > (a) Show that < φn|P|φm >= α < φn|X|φm > and find α. Hint: Consider the commutator [X, H] (b)...

Part A: Compute the derivative of ?(?)=(4?^4 + 2?)(?+9)(?−6) Part B: Compute the derivative of ?(?)=...

Part A: Compute the derivative of ?(?)=(4?^4 + 2?)(?+9)(?−6) Part B: Compute the derivative of ?(?)= (9x^2 + 8x +8)(4x^4 + (6/x^2))/x^3 + 8 Part C: Compute the derivative of ?(?)=(15?+3)(17?+13)/(6?+8)(3?+11).

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