Answered: For the Hamiltonian H= [H₁ [H₁X]]=… | bartleby

Related questions

Question

ol
For the Hamiltonian H=
[H₁ [H.x]] = 4²kx
m
P
2m
- kx²
Ex² demonstrate that
2

Expert Solution

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Blurred answer

Similar questions

Consider the 2D harmonic oscillator Hamiltonian: 87 - 12/24 (13² + 8² ) + 2 ²³² (1² + 8² ) Ĥ mw² 2m Unless otherwise specified, we will work in the eigenstates that satisfy: Ĥ|nz, ny) = Enz,ny |nx, ny) x, with Eng,ny = ħw(nx + Ny + 1). (a.) Some energy levels are degenerate. For example, E 2ħw can be achieve with (nx, ny) = (1, 0); (0, 1). This energy level has a degeneracy D(2ħw) = 2. What is the degeneracy of energy level E (where N is a positive integer)? = Nhw (b.) Consider the state (0)) = √ (12,0) + 2 |1, 1) + (0,2)). (c.) Calculate (Ĥ), (px), (py), and (âŷ) for the state above. = What is (t)) at a later time t > 0?
2) ( Consider an electron trapped in a one-dimensional anharmonic potential. The Hamiltonian for this system is given as: H mw?£? + 2m + B&3 . Regarding the cubic term as being small, apply non- degenerate perturbation theory by letting the perturbation be: A, = ß2³. а) Calculate to first order the ground state energy of this anharmonic oscillator. b) ( | Calculate to second order the ground state energy of this anharmonic oscillator. c) ( Find the lowest order correction to the ground state wave function.
(7) Suppose the Hamiltonian for a particle in three dimensions is given by H = +V(f). Here, the 2m operator î represents the radial direction relative to the origin of coordinates. In other words, the potential energy exhibits spherical symmetry. Show that the three operators, H, L.,Ľ commute.
-ax (ii) Show that Y, = A,e¯* is an eigenfunction of the simple harmonic 1 ocillator Hamiltonian above when a = 2h Vkm. Find the corresponding eigenvalue. Interpret the result.
The Hamiltonian matrix has been constructed using an orthonormal basis. (1 1 0V (1 0 1) A = (2 1 0 )+(0 2 2 \2 1 4 where H = Hº + V and cis a constant. 1 2 0/ b) Use time-independent perturbation theory to determine the eigenvalues with corrections up to second order.