Exercise 4 A particle of mass m is moving in a one-dimensional harmonic oscillator potential, V (r) = Amu-r. Calculate (

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Exercise 4 A Particle Of Mass M Is Moving In A One Dimensional Harmonic Oscillator Potential V R Amu R Calculate 1 (110.79 KiB) Viewed 13 times

Exercise 4 A particle of mass m is moving in a one-dimensional harmonic oscillator potential, V (r) = Amu-r. Calculate (a) the ground state energy and (b) the mth excited state energy to first and second order perturbation theory when a small per- turbation H, = -X8 (2) (c) Compare with the exact (rigorous) eigen-energies of this problem obtained by solving the Schrodinger equation (10) Exercise 5 Consider the isotropic harmonic oscillator with the Hamiltonian HO) = + 2 ha (Q2 + + ?) + mw(z2 + y2 + x2) 2m A small perturbation H' = Azy (x2 + y) is added to the potential. (a) Find the corrections, to second order in , to the ground state energy (b) The first excited state is 3-fold degenerate. Calculate the first order correction to E and the wavefunction. r(0) 2 Hint: Note that Inz) = yny) = h (a +a) Inx) 2mw h (ay+a) Iny) 2mw h (az +a) |nz) 2mw Inz) = with (n) = |n|ny) |nz) and E = H (m2 + v + e + ) = (n+) hw ni+nyng = + [10] Exercise 6 Consider a system whose Hamiltonian is given by H = EO 3 2X 0 0 214 0 0 0 0 3 -34 0 0 -3X 5+X where <1 (a) By decomposing this Hamiltonian into h = HO + XH' , find the eigenvalues and eigenstates of the unperturbed Hamiltonian ho (b) Diagonalize û to find the exact eigenvalues of û ; expand each eigenvalue to the second power of X. (c) Using first- and second-order non-degenerate perturbation theory, find the approximate eigen- energies of Ĥ and the eigenstates to first order. Compare these with the exact values obtained in (b) [10]