2. Approximation Methods (a) Perturbation Theory Consider a charged one-dimensional quantum harmonic oscillator (ID QHO) in a con- stant electric field along the x-axis of strength G. The charge on the oscillator is q, so the potential of the system will now be = {kx qGx. i. We will choose the standard QHO as the unperturbed Hamiltonian, so that we can use its eigenenergies and wavefunctions. Write down and for this problem, and thus identify the explicit form or '. ii. Write down an explicit expression involving the integrals of functions for E"), the first-order perturbative correction to the energy of the v = 1 level of this charged QHO-in-a-field. Hint: The relevant wavefunctions appear on the V2.0 information sheet. The for- mula for E1) appears in Multi-Particle Lecture 4, and will appear on the V3.0 information sheet when it is finalized and released. iii. State the numerical value of this correction for this state, and comment on whether or not this conclusion will apply to all states of this charged-QHO-in-a-field. Provide brief justification of your answers that makes explicit reference to the expression from the previous sub-question. (b) Variational method Consider the usual 1D particle in a box. (Particle of mass m, with a potential that is V(x) = 0 for 0 1. i. Identify at least two attributes that this function possesses such that it is valid for use as a trial function. ii. It can be shown that the variational integral using this function evaluates to: (462 + b W = + : 2b - 1 Determine (finite) values of b that are associated with extrema of W. Hint: You will need to use the quadratic formula for the last step, but it shouldn't be too messy. Leave your answer as a radical. iii. Selecting the permissable value of b (see above), substitute this back into the expres- sion for Wb, and compute the numerical value of this best estimate of the ground- state energy of the PIB to five significant figures (in units of 2/ma?.) iv. Calculate the numerical value (to five significant figures) of the ground-state energy for the 1D PIB in units of 2 v. Comment on your results. Is the variational estimate of the ground-state energy greater than (or equal to) the 'true' energy, as expected? By what percentage does it differ from the exact result? | * #t dt = ma2 ma? 2. Approximation Methods (a) Perturbation Theory Consider a charged one-dimensional quantum harmonic oscillator (ID QHO) in a con- stant electric field along the x-axis of strength G. The charge on the oscillator is q, so the potential of the system will now be = {kx qGx. i. We will choose the standard QHO as the unperturbed Hamiltonian, so that we can use its eigenenergies and wavefunctions. Write down and for this problem, and thus identify the explicit form or '. ii. Write down an explicit expression involving the integrals of functions for E"), the first-order perturbative correction to the energy of the v = 1 level of this charged QHO-in-a-field. Hint: The relevant wavefunctions appear on the V2.0 information sheet. The for- mula for E1) appears in Multi-Particle Lecture 4, and will appear on the V3.0 information sheet when it is finalized and released. iii. State the numerical value of this correction for this state, and comment on whether or not this conclusion will apply to all states of this charged-QHO-in-a-field. Provide brief justification of your answers that makes explicit reference to the expression from the previous sub-question. (b) Variational method Consider the usual 1D particle in a box. (Particle of mass m, with a potential that is V(x) = 0 for 0 1. i. Identify at least two attributes that this function possesses such that it is valid for use as a trial function. ii. It can be shown that the variational integral using this function evaluates to: (462 + b W = + : 2b - 1 Determine (finite) values of b that are associated with extrema of W. Hint: You will need to use the quadratic formula for the last step, but it shouldn't be too messy. Leave your answer as a radical. iii. Selecting the permissable value of b (see above), substitute this back into the expres- sion for Wb, and compute the numerical value of this best estimate of the ground- state energy of the PIB to five significant figures (in units of 2/ma?.) iv. Calculate the numerical value (to five significant figures) of the ground-state energy for the 1D PIB in units of 2 v. Comment on your results. Is the variational estimate of the ground-state energy greater than (or equal to) the 'true' energy, as expected? By what percentage does it differ from the exact result? | * #t dt = ma2 ma