Physical Chemistry 1) Recall that first-order perturbation theory can be used to compute the energies of...
Let Ho be the Hamiltonian of the non-relativistic hydrogen atom neglecting spin. Consider H1 = e|E\r cos 0 with e|E|af < 1. This Hamiltonian describes a weak constant electric field in the z-direction interacting with the atomic dipole. We want to understand the effect such a field has on the first excited energy level, E2, of hydrogen. Remember that this energy level is degenerate with corresponding eigenstates |2lm) Use first-order perturbation theory to find the aproximate energies of Ho+ H1...
Consider a quantum mechanical system with 4 states and an unperturbed Hamiltonian given by 1 0 0 0 Ho E0 0 2 0 a small perturbation is added to this Hamiltonian 0 0 1 0 where e is much smaller than E a) [10pts] What are the energy eigenvalues of the unperturbed system of the following states? 1 o 2o 0 and which energy levels are degenerate? b) [10pts Find a good basis for degenerate perturbation theory instead of c)...
h2 d2 1 2m dx22 m ω2 + γχ4, use perturbation theory to estimat 1. For the HamiltonianH - the ground state energy (A) What would be a good choice for the reference, or unperturbed, Hamiltonian? (B) The ground state wavefunction for harmonic oscillator is ψ(x) e 2 wit mc /h. W energy rite down the expression the first order perturbation contribution to the (C) Evaluate the integral from part (B). The relevant integral should be in the Useful Integral...
Please solve the problem as soon as possible. Problem 1: Consider a two level system with Hamiltonian: Using the first order time-dependent perturbation theory, obtain the probability coefficients cn (t) if the perturbation is applied at t >0 and the system is originally in the ground state. Hint: When solving the problem, first you may need to find the energies and wave functions of the unperturbed Hamiltonian A0. Problem 1: Consider a two level system with Hamiltonian: Using the first...
sorry i have only this information. Part B (Advanced: grade +0,+I,+2) Use degenerate perturbation theory to compute the second order energy shifts if 2 Solve for the eigenvalues of the total Hamiltonian by exact diagonalization of the matrix. Compare the exact solution to the approximated one given by the perturbation theory, e.g. using a sketch of the energies as a function ofA, for different values of the inter-atomic detuning Δ-ah- a2. Identify the regime when the perturbation theory gives a...
Exercise 4: Fine structure of hydrogenic atoms a) Consider a Hamiltonian H-Ho + λΗ. with Mr a small perturbation. Show that in (non-degenerate) perturbation theory the first order correction to the unperturbed, discrete energy level E(Holis given by and the second order by b) Apply this to evaluate the first order corrections to the energy levels (the so-called fine structure) of a hydrogenic atom, that arise due to relativistic corrections. Confirm that the answer for the total first order correction...
In the class lecture notes, the result of the first-order perturbation calculation for the energy of lithium was given as E(1) = 2J, +J Lists – K,625 - This energy is added to the E© value obtained by treating each electron as if it were in an independent hydrogen-like orbital in order to estimate the total energy of the lithium atom. 1s 2s The calculation to determine the E9) energy correction involves solving many integrals because the wavefunction for lithium...
Total: 24 pts] In our derivation of the first-order energy corrections for the spin-orbit coupling we have claimed that the unperturbed eigenstates of the hydrogen atom, {Inl misms >}, are not "good” wavefunctions to determine the first-order energy correction for the spin-orbit coupling e2 ) 1 H'so = (8160) m2c273 S.L. 8760 ) m a) (14 pts] To confirm our claim, calculate the commutators (S · L, L?) and (S · L, Lx]. What do you conclude? What do you...
1- 5. Two particles each of mass m are fixed at the end of a rigid rod of length 2a. This rod lies in the xy plane and is free to rotate in that plane about an axis passing through the midpoint of the rod and perpendicular to it (that is, parallel to the z-axis). Neglect the inertial properties of the rod in the rest of this question z-axis 1. Derive the classical expression for the kinetic energy of the...
1- 5. Two particles each of mass m are fixed at the end of a rigid rod of length 2a. This rod lies in the xy plane and is free to rotate in that plane about an axis passing through the midpoint of the rod and perpendicular to it (that is, parallel to the z-axis). Neglect the inertial properties of the rod in the rest of this question z-axis 1. Derive the classical expression for the kinetic energy of the...