Q4: a) Calculate: (uz|A*|u6). b) If the Hamiltonian of harmonic oscillator defined as H = ħw + A*A and [A ‚A*] = 1, Find [A*,H]
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- If y = e-0.5 Bx is the eigen function of the simple harmonic oscillator, then the eigen value is hv 0.75 hv 0.5 hv nhy40. The first excited state of the harmonic oscillator has a wave function of the form y(x) = Axe-ax². (a) Follow theThe Brachistochrone Problem: Show that if the particle is projected withan initial kinetic energy 1/2 m v02 that the brachistochrone is still a cycloidpassing through the two points with a cusp at a height z above the initialpoint given by v02 = 2gz.
- Let F = (z^2 cos y, −xz^2 sin y, 2xz cos y − cos z).a) Show that F is irrotational.b) Find a potential function f (x, yz) such that F = ∇f , and f (0, π, π/3) = 2The Hamilton function for a point particle moving in a central potential is given by p? H + a|x|". 2m Consider the vector A = p x L+ ma where L is the angular momentum of the particle. (a) Calculate the Poisson bracket {H, Ar}, where A is the k-th component of the vector A. (b) Determine the value of the exponent n for which the vector A becomes a conserved quantity.