Q: Determine the WKB approximation for the bound state energy of a particle of mass m in the potential: V(x) = { ax for x >=0, 2ax for x < 0 }

The WKB approximation provides a method to determine the approximate bound state energy levels of a quantum particle in a potential well. For a potential to support bound states, it must confine the particle, meaning it must rise to positive infinity on both sides of a minimum. The given potential V(x) = { ax for x >= 0, 2ax for x < 0 } does not inherently guarantee bound states if 'a' is a simple positive constant, as V(x) would tend to -∞ for x < 0. To make this a valid problem for bound sta

Q: A charged particle of mass m and charge q, is confined to a one-dimensional box of side L with 0 ≤ x ≤ L. At t > 0, an electric field E = Eoe-ati acts on the particle where a is a constant. If the particle is in the ground state when t<0, calculate the probability that it will be in the first excite

To calculate the probability that the particle transitions from the ground state to the first excited state, we employ time-dependent perturbation theory, as the electric field is a time-dependent perturbation. The unperturbed system is a particle of mass m in a one-dimensional box of side L, with energy eigenstates ψ_n(x) and eigenvalues E_n. For a particle in a 1D box (0 ≤ x ≤ L), the unperturbed energy eigenvalues are given by E_n = n²π²ħ² / (2mL²), and the corresponding normalized wavefunct

Question 1

Write the space translation operator in quantum mechanics Î(a) for a finite translation a along the x direction. Calculate the commutator X, [x,f(a)]. You may use the Baker-Campbell-Hausdorff formula: e eÂ Be¯Â = B+ [A,B]+[Â,[A,B]+..... (5) b) Consider an operator Ô for which û⁺ Ô슾 = −Ô . Show that

Accepted Answer

Quantum mechanics provides powerful tools, such as the translation operator and parity operator, to understand symmetries and properties of physical systems. The translation operator shifts a system in space, while the parity operator reflects it through the origin. Understanding their mathematical forms and commutation relations is crucial for solving problems involving spatial transformations and symmetries in quantum systems, as demonstrated in the following solutions.

Sub-question (a) deals

Question 2

Determine the wave function and energy of the ground state and first excited state for a system of two identical bosons in 1D simple harmonic oscillator. (5) b) Define the action of the permutation operator P₁₂ for a system of two particles 1 and 2 and two states |wp) and p. Show that P₁22 = Î and d

Accepted Answer

Answer available — download the full PDF for complete details.

Question 3

Write down the eigenkets | j,mj)for j = j₁ + j2 with j₁ = 2; j2 = 1/2. b) Calculate the matrix elements for J2 for a system of two spin half particles. (5+5)

Accepted Answer

Quantum mechanics provides a framework for describing the intrinsic angular momentum (spin) of particles and how these momenta combine. This question explores the principles of angular momentum addition, a fundamental concept in multi-particle systems. It specifically examines how to determine the possible total angular momentum states (eigenkets) when combining two distinct angular momenta and then delves into calculating the matrix elements of the total angular momentum squared operator for a

Question 4

Determine the first and second order perturbation correction to the ground state energy eigenvalue of the one-dimensional infinite potential well of width L (0≤x≤L) with the perturbation: H₁(x) = Vo sin (πχ/L)

Accepted Answer

Perturbation theory is a powerful approximation method used in quantum mechanics to find the energy eigenvalues and eigenstates of a system when the Hamiltonian can be split into an unperturbed part (H₀) and a small perturbation (H₁). For the one-dimensional infinite potential well of width L (0≤x≤L), the unperturbed Hamiltonian is H₀ = -ħ²/(2m) d²/dx², and the unperturbed ground state (n=1) wavefunction and energy eigenvalue are well-known.

**Unperturbed System (Ground State)**

The unperturb

Question 5

Consider the following one-dimension simple harmonic oscillator Hamiltonian operator Ĥ = (h^2/2m)(d^2/dx^2) + (1/2)mω^2x^2. Use a trial wave function (x) = N exp(-x^2/2α) with a variational parameter α to estimate the upper bound to the ground state energy.

Accepted Answer

The variational method offers a way to estimate the ground state energy (E₀) of a quantum system when an exact solution is difficult or impossible to obtain. The fundamental variational principle states that for any normalized trial wave function ψ_t, the expectation value of the Hamiltonian Ĥ, given by E_t = ⟨ψ_t|Ĥ|ψ_t⟩, will always be greater than or equal to the true ground state energy E₀. Our primary goal is to minimize E_t with respect to any variational parameters present in the trial wav

Question 6

Determine the WKB approximation for the bound state energy of a particle of mass m in the potential: V(x) = { ax for x >=0, 2ax for x < 0 }

Accepted Answer

The WKB approximation provides a method to determine the approximate bound state energy levels of a quantum particle in a potential well. For a potential to support bound states, it must confine the particle, meaning it must rise to positive infinity on both sides of a minimum. The given potential V(x) = { ax for x >= 0, 2ax for x < 0 } does not inherently guarantee bound states if 'a' is a simple positive constant, as V(x) would tend to -∞ for x < 0.

To make this a valid problem for bound sta

Question 7

Consider the two state problem in which the unperturbed Hamiltonian Ĥo has just two eigenkets, |1) and |2) with: Ĥo|1) = E₁|1);Ĥ0|2) = E2|2), and E2 > E₁. The system is subjected to a time-dependent perturbation: (t) = Vo cos(ωt)|1)⟨2|+2)⟨1|. Calculate the probability for the system to be in the sta

Accepted Answer

To calculate the probability of the system being in state |2) at time t, given it starts in state |1) at t=0, we employ time-dependent perturbation theory. The unperturbed Hamiltonian Ĥ₀ has two eigenkets |1) and |2) with energies E₁ and E₂, respectively, where E₂ > E₁. The system is subjected to a time-dependent perturbation V̂(t) = V₀ cos(ωt) (|1)⟨2| + |2)⟨1|).

We begin by expressing the total Hamiltonian as Ĥ = Ĥ₀ + V̂(t). In the interaction picture, the state vector |Ψ_I(t)⟩ is written as a

Question 8

A charged particle of mass m and charge q, is confined to a one-dimensional box of side L with 0 ≤ x ≤ L. At t > 0, an electric field E = Eoe-ati acts on the particle where a is a constant. If the particle is in the ground state when t<0, calculate the probability that it will be in the first excite

Accepted Answer

To calculate the probability that the particle transitions from the ground state to the first excited state, we employ time-dependent perturbation theory, as the electric field is a time-dependent perturbation. The unperturbed system is a particle of mass m in a one-dimensional box of side L, with energy eigenstates ψ_n(x) and eigenvalues E_n.

For a particle in a 1D box (0 ≤ x ≤ L), the unperturbed energy eigenvalues are given by E_n = n²π²ħ² / (2mL²), and the corresponding normalized wavefunct

MPH-008 Solved Assignment — January 2026 / July 2026

QUANTUM MECHANICS-II | IGNOU Master of Science (Physics) Programme (MSCPH)

MPH-008—January 2026 / July 2026

QUANTUM MECHANICS-II

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MPH-008 Assignment Questions — January 2026

Q3. Write down the eigenkets | j,mj)for j = j₁ + j2 with j₁ = 2; j2 = 1/2. b) Calculate the matrix elements for J2 for a system of two spin half particles. (5+5)

Q4. Determine the first and second order perturbation correction to the ground state energy eigenvalue of the one-dimensional infinite potential well of width L (0≤x≤L) with the perturbation: H₁(x) = Vo sin (πχ/L)

Q5. Consider the following one-dimension simple harmonic oscillator Hamiltonian operator Ĥ = (h^2/2m)(d^2/dx^2) + (1/2)mω^2x^2. Use a trial wave function (x) = N exp(-x^2/2α) with a variational parameter α to estimate the upper bound to the ground state energy.

Q6. Determine the WKB approximation for the bound state energy of a particle of mass m in the potential: V(x) = { ax for x >=0, 2ax for x < 0 }

Q9. Using the Born Approximation, calculate the differential cross-section for a beam of particles of mass m scattered by a potential: V(r) = Vo exp(-r/a). You may use: ∫[0 to ∞] e-αx sin(βx)dx = β/(α^2 + β^2) for α > 0