# Approximation Methods | Week 1

**Course Name: Approximation Methods**

**Course Link: Approximation Methods**

#### These are answers of Approximation Methods Coursera Week 1 Quiz

**Question 1Harmonic oscillator under electric fieldUse this information for Questions 1-2:Consider a 1D simple harmonic oscillator under a constant electric field E. The Hamiltonian is given as,H^=p^**

^{2}/2m+1/2mω

^{2}x

^{2}+qExThe applied electric field is small enough to justify the use of perturbation theory. That is, treat the Hamiltonian without electric field as the unperturbed Hamiltonian and the electric field term as the perturbation.Obtain the first order correction to the ground state energy.Express your answers in terms of mass m, oscillator resonance frequency omega, reduced Planck’s constant hbar, constant pi, charge q and applied electric field E. Your answer does not have to include all these parameters.

Answer: 0

**Question 2Harmonic oscillator under electric fieldWhat is the second order correction to the ground state energy?To simplify the calculation, use only the first non-zero term and ignore all higher order terms in the infinite sum required to evaluate the second order correction.Express your answers in terms of mass m, oscillator resonance frequency omega, reduced Planck’s constant hbar, constant pi, charge q and applied electric field E. Your answer does not have to include all these parameters.**

Answer: -(q^2*E^2)/(2*m*omega^2)

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 32D harmonic oscillatorUse the following information for Questions 3-5:Consider a 2D harmonic oscillator whose Hamiltonian is given asH^**

_{0}=p^

^{2}

_{x}/2m+p^

^{2}

_{y}/2m+1/2mω

^{2}(x

^{2}+y

^{2})What is the energy of the lowest-lying degenerate states?Express your answers in terms of mass m, oscillator resonance frequency omega, reduced Planck’s constant ℎbar, and constant pi. Your answer does not have to include all these parameters.

Answer: 2*hbar*omega

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 42D harmonic oscillatorWe now add a perturbation V=amω**

^{2}xy to the unperturbed Hamiltonian H^

_{0}. Here, a is a dimensionless real number, which is small.Find the first-order correction to the degenerate energy level found in Question 3.Since the unperturbed energy level is doubly degenerate, it should split into two, if this perturbation lifts the degeneracy. Enter the larger of the two first-order corrections here. If the degeneracy is not lifted, enter 0.Express your answers in terms of mass m, oscillator resonance frequency omega, reduced Planck’s constant hbar, constant pi and the dimensionless parameter a. Your answer does not have to include all these parameters.

Answer: (a*hbar*omega)/2

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 52D harmonic oscillatorEnter the smaller of the two first-order corrections here. If the degeneracy is not lifted, enter 0.Express your answers in terms of mass m, oscillator resonance frequency omega, reduced Planck’s constant hbar, constant pi and the dimensionless parameter a. Your answer does not have to include all these parameters.**

Answer: -(a*hbar*omega)/2

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 6Two-state systemUse the following information for Questions 6-15:The Hamiltonian matrix for a two-state system is given by,H^=(E**

_{1}

^{(0)} Δ Δ E

_{2}

^{(0)})where E

_{1}

^{(0)} and E

_{2}

^{(0)} are the energies of the unperturbed (Δ=0) states and E

_{1}

^{(0)}<E

_{2}

^{(0)}.This problem can be solved exactly by diagonalizing the Hamiltonian matrix to obtain the eigenvalues. What is the smaller of the two energy eigenvalues?Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.

Answer: (E1+E2)/2-sqrt((E2-E1)^2+4*Delta^2)/2

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 7Two-state systemWhat is the larger of the two energy eigenvalues?Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.**

Answer: (E1+E2)/2+sqrt((E2-E1)^2+4*Delta^2)/2

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 8Two-state systemIf Δ is small compared to E**

_{2}

^{(0)}−E

_{1}

^{(0)}, we can use the Taylor expansion of the exact solutions obtained in Questions 6-7 and retain only the lowest order term in Δ. Enter the expression for the smaller of the two energies in this approximation.Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.

Answer: E1-(Delta^2/(E2-E1))

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 9Two-state systemEnter the expression for the larger of the two energies in the same approximation used in Question 8.Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.**

Answer: E2+(Delta^2/(E2-E1))

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 10Two-state systemThis time, we will solve the same problem by using the non-degenerate perturbation theory. For this, we express the Hamiltonian as below.H^=H^**

_{0}+VH^

_{0}=(E

_{1}

^{(0)} Δ Δ E

_{2}

^{(0)}) and V=(0 Δ Δ 0)Using the non-degenerate perturbation theory, calculate the first order correction for the ground state energy E

_{1}

^{(0)}.Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.

Answer: 0

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 11Two-state systemCalculate the first order correction for the excited state energy E**

_{2}

^{(0)}.Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.

Answer: 0

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 12Two-state systemCalculate the second order correction for the ground state energy E**

_{1}

^{(0)}.Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.

Answer: -(Delta^2)/(E2-E1)

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 13Two-state systemCalculate the second order correction for the excited state energy E**

_{2}

^{(0)}.Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.

Answer: (Delta^2)/(E2-E1)

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 14Two-state systemNow let us assume the system is degenerate, E**

_{1}

^{(0)}=E

_{2}

^{(0)}=E

^{(0)}. The Hamiltonian is then written asH^=H^

_{0}+Vwhere H^

_{0}=(E

_{1}

^{(0)} Δ Δ E

_{2}

^{(0)}) and V=(0 Δ Δ 0)Using the degenerate perturbation theory, calculate the first order correction for the unperturbed energy E

^{(0)}.Since the system is doubly degenerate, it should split into two, if the perturbation lifts the degeneracy. Enter the larger of the two first-order corrections here. If the degeneracy is not lifted, enter 0.Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.

Answer: Delta

**These are answers of Approximation Methods Coursera Week 1 Quiz**

**Question 15Two-state systemEnter the smaller of the two first-order corrections here. If the degeneracy is not lifted, enter 0.Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.**

Answer: -Delta

**These are answers of Approximation Methods Coursera Week 1 Quiz**

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