Q1: (1 0 0 The Hamiltonian is represented by the matrix H = 0 0 2 0 0 0 2 Where o is positive real number. (a) Find the eigenenergies for the Hamiltonian. (b) Calculate the eigenvectors corresponding to its eigenenergies.
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- The dynamics of a particle moving one-dimensionally in a potential V (x) is governed by the Hamiltonian Ho = p²/2m + V (x), where p = is the momentuin operator. Let E, n = of Ho. Now consider a new Hamiltonian H given parameter. Given A, m and E, find the eigenvalues of H. -ih d/dx 1, 2, 3, ... , be the eigenvalues Ho + Ap/m, where A is a %3|The Hamiltonian of a certain system is given by 0 0 H = ħw|0 _0 0 0 1 1 Two other observables A and B are represented by A = a|-i 0 0 0 0 1 [1 0 0 B = b|0 _2 0 Lo o 2 w, a, b are positive constant. a. Find the eigenvalues and normalized eigenvectors of H b. Suppose the system is initially in the state 2c ly(0) >= 2c where c is a real constant. Determine the normalized state |4(t) >. c. What are the eigenvectors of B? d. Find the expectation values of A and B in the state [Þ(t) >, and hence determine if A and B are conservative observablesGiven a Hamiltonian, find eigenvalues and eigenvector Н = 2 (₂² 4 2) 16 2/
- b): Consider two identical linear oscillators' having a spring constant k. The interaction potential is H = Ax|X2, where xi and x2 are the coordinates of the oscillators. Obtain the energy eigen values.The Hamiltonian of a system has the form 1 Ĥ π = -1 1¹2²2² + ²x² + 4x¹ = Fo+ Y4x¹ 2 dx2 Let un(x) = |n) be the eigenstates of Ho, with Ĥo|n) = (n + ¹) |n), n = 0, 1, 2, .... In this problem, we will first utilize the linear variational method to set up the secular determinant for finding the lowest energy state for the trial function, lp) = co|0) + C₂|2). (a) In setting up the secular determinant for this problem, we will need to evaluate the Hamiltonian and overlap matrix elements, Hmn = (m|F|n) and Smn = (m/n), respectively, for m, n = 0, 2. Find the overlap (S) matrix, assuming [0) and 2) are both normalized. 3 39 (b) If (0|x4|0) =, (0|x4|2) = (2|x4|0) =,and (2|x4|2) = 32, find expressions for all the Hamiltonian elements, Hoo, H22, Ho2, and H₂0 (you will be asked to identify one of them from a list of possibilities on the quiz). (c) Use your results for Hmn and Smn to construct the secular equation for this system and evaluate the determinant to obtain a polynomial equation for the…i li. 9 V:OV docs.google.com/forms/d/e Which function is preferable to find the magnitude of a complex number? * sqrt() cart2pol() MATLAB does not support complex arguments abs() All matrices are vectors but all vectors are not matrices in MATLAB * False True Compute 24 modulo 5. b = mod(24,5) * b =6 b =4 b =5 b = 3 O O
- A Hamiltonian is given in matrix form as ħ (wo W1 A₁ = 7/7 (@₂₁ 000) 2 a. What are the energy eigenvalues? What are the energy eigenvectors? b.问题6 The Hamiltonian operator Ĥ for the harmonic oscillator is given by h d? + uw? â2, where u is the reduced mass and w = 2Tv is the angular 2µ da? frequency. Is the function f(x) = x exp- iwa? an eigenfunction of that Hamiltonian? 2h O No. Yes, and the eigenvalue is hw. Yes, and the eigenvalue is hw. 3ħw Yes, and the eigenvalue is 4The Hamiltonian of a three-level system is represented by the matrix 22 Vo 2V + 1 22 H = 3V where Vo and A are constants with units of energy (A<< Vo). The corrected eigenstate of the energy level E3=3Vo (to first order in A is:
- Check if the following operators with the corresponding functions could form an eigen value equations or not (where Bis a constant value) No. function Оperator 3 2 3 sin(ßx) sin(Bx) d dx 4 sin(ßx) dxEvaluate the commutator è = [x², Pe** =?The Hamiltonian of a system has the form 1 A = -1¹ ²³² + ²x² + Y₁x² = H₁ + V₁x¹ 2 dx2 2 Let un(x) = |n) be the eigenstates of Fo, with Fo|n) = (n + ½) |n), n = 0, 1, 2, ... . In this problem, we will first utilize the linear variational method to set up the secular determinant for finding the lowest energy state for the trial function, lp) = co10) + C₂12). (a) In setting up the secular determinant for this problem, we will need to evaluate the Hamiltonian and overlap matrix elements, Hmn = (m|Â|n) and Smn = (m[n), respectively, for m, n = 0, 2. Find the overlap (S) matrix, assuming [0) and 12) are both normalized.