The Hamiltonian of consisting of a mass' m less sing of length. a simple pendulum attached' to a I is H= B+mgl(1- Cose). mass- 子 Lagrangian, the value of
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- 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?Suppose that you have the Lagrangian L = (;2 + ʻr²) + 20 for a 2D 20 system in plane polar coordinates (r, 0). Determine the Hamiltonian.In a Hamiltonian system, what are the conditions for fixed points?
- (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.It has been previously noted that the total time derivative of a functionof qi and t can be added to the Lagrangian without changing the equationof motion. What does such an addition do to the canonical momenta andthe Hamiltonian? Show that the equations of motion in terms of the newHamiltonian reduce to the original Hamilton’s equations of motion.Write down the inertia tensor for a square plate of side ? and mass ? for a coordinate system with origin at the center of the plate, the z-axis being normal to the plate, and the x- and y- axes parallel to the edges.
- -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.Prove the following: if the Hamiltonian is independent of time, then ∆E doesn't change in time. Show work and be explicit to prove the statement.