MPH-004 Solved Assignment January 2026 | QUANTUM MECHANICS-I | IGNOU MSCPH

Q: i) Normalize the wave function: ψ(x,0) = N sin (πx/L) + sin (2πx/L), 0≤x<L. ii) Calculate the expectation value of x for a particle in this state.

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Q1. a). Electrons are accelerated through a potential 150 V and incident on a crystal with interatomic spacing d = 0.20 nm. Calculate the de Broglie wavelength and the first-order Bragg diffraction angle.

Key Points:

De Broglie wavelength: λ = h/p, demonstrating wave-particle duality.
Electron momentum: p = √(2mₑeV) when accelerated by potential V.
De Broglie wavelength for electrons: λ = h / √(2mₑeV).
Bragg's Law: 2d sinθ = nλ for constructive interference in crystals.

Answer: To determine the de Broglie wavelength of electrons accelerated through a potential and the first-order Bragg diffraction angle, we utilize fundamental principles of quantum mechanics and crystallography as taught in MPH-004. According to the de Broglie hypothesis, all matter exhibits wave-like properties. The wavelength (λ) of a particle is inversely proportional to its momentum (p), given by the relation λ = h/p. For an electron accelerated through a potential difference V, its kinetic energy...

Q1. b). Estimate the uncertainty in the energy of a photon localized within a distance of 0.1 nm.

Key Points:

Heisenberg's Uncertainty Principle (HUP) dictates fundamental limits for conjugate variables.
Position-momentum uncertainty: Δx Δp ≈ ħ for estimation.
Photon energy is directly proportional to its momentum: E = pc.
Calculate momentum uncertainty (Δp) from given position uncertainty (Δx).

Answer: To estimate the uncertainty in the energy of a photon localized within a distance, we employ Heisenberg's Uncertainty Principle, a cornerstone of quantum mechanics as discussed in MPH-004. This principle establishes a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. Specifically, the position-momentum uncertainty principle states that Δx Δp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the unc...

Q1. c). i) Normalize the wave function: ψ(x,0) = N sin (πx/L) + sin (2πx/L), 0≤x<L. ii) Calculate the expectation value of x for a particle in this state.

i)) Normalize the wave function: ψ(x,0) = N sin (πx/L) + sin (2πx/L), 0≤x<L. (200 words)
ii)) Calculate the expectation value of x for a particle in this state. (200 words)

Key Points:

Normalization condition: ∫|ψ(x)|² dx = 1 over the specified domain.
Orthogonality of particle-in-a-box wavefunctions simplifies integral cross-terms.
Integral of sin²(nπx/L) over [0, L] is L/2, crucial for normalization.
Expectation value of position <x> is calculated as ∫ψ* x ψ dx.

Answer:

Q1. d). A quantum mechanical particle in one dimension has the wave function: ψ(x) = x^n : x > 0 ; n > 1. Use the Schrödinger equation to determine the corresponding potential V(x).

Key Points:

Time-independent Schrödinger Equation (TISE) is: - (ħ²/2m) d²ψ/dx² + V(x)ψ = Eψ.
Given ψ(x) = xⁿ, the first derivative dψ/dx is n * x^(n-1).
The second derivative d²ψ/dx² is n * (n-1) * x^(n-2).
Substitute ψ(x) and d²ψ/dx² into the TISE to solve for V(x).

Answer: To determine the potential V(x) for the given wave function ψ(x) = xⁿ (for x > 0, n > 1), we utilize the time-independent Schrödinger equation (TISE). The TISE is a fundamental equation in quantum mechanics that describes the stationary states of a quantum system. The time-independent Schrödinger equation is given by: - (ħ² / 2m) * (d²ψ / dx²) + V(x)ψ(x) = Eψ(x), where ħ is the reduced Planck constant, m is the particle's mass, V(x) is the potential energy, and E is the total energy of the part...

Q2. a). A 1.5 mA beam of electrons enters a sharply defined boundary with a velocity 3.0×10^6 ms-1, and then its velocity reduces to 2.0×10^6 ms⁻¹, due to the difference in potential. Calculate the transmitted and reflected currents.

Key Points:

Electron beam interaction with a boundary is modeled as a quantum potential step.
Incident current is split into reflected and transmitted currents.
Reflection coefficient R = [(v₁ - v₂) / (v₁ + v₂)]² for a potential step.
Transmission coefficient T = 1 - R, representing the probability of transmission.

Answer: The interaction of an electron beam with a sharply defined boundary, where its velocity changes, is analogous to a quantum mechanical potential step. In such a scenario, the incident beam is partly reflected and partly transmitted. The extent of reflection and transmission is determined by the reflection coefficient (R) and transmission coefficient (T), respectively. These coefficients can be calculated using the initial and final velocities of the electrons. From quantum mechanical analysis of...

Q2. b). Show that for any stationary state of a symmetric potential well: (x^2) ≠0.

Key Points:

⟨x²⟩ is the expectation value, defined as ∫ |ψ(x)|² x² dx.
The probability density |ψ(x)|² is always non-negative.
The term x² is always non-negative, zero only at x=0.
A stationary state ψ(x) cannot be zero everywhere except x=0.

Answer: For any stationary state of a quantum system, its spatial wavefunction ψ(x) is an eigenfunction of the Hamiltonian operator. The expectation value of x², denoted as ⟨x²⟩, is calculated as ∫ |ψ(x)|² x² dx. The probability density |ψ(x)|² is always real and non-negative (i.e., |ψ(x)|² ≥ 0) for any physically realizable state. Similarly, the quantity x² is also always real and non-negative (x² ≥ 0), with x² = 0 only at x = 0. For the integral ∫ |ψ(x)|² x² dx to be zero, the integrand |ψ(x)|² x²...

Q2. c). Calculate the expectation value of the potential energy for the first excited state of a simple harmonic oscillator.

Key Points:

SHO potential energy: V(x) = (1/2)mω²x².
Virial Theorem for SHO: Expectation values of kinetic and potential energy are equal, <T> = <V>.
Total energy E_n = <T> + <V> implies E_n = 2<V> for SHO.
SHO energy eigenvalues: E_n = (n + 1/2)ħω.

Answer: The potential energy for a simple harmonic oscillator (SHO) is defined as V(x) = (1/2)mω²x², where m is the mass and ω is the angular frequency. The expectation value of an observable A is fundamentally calculated as <A> = ∫ Ψ*(x) Â Ψ(x) dx, where Ψ(x) is the wavefunction for the given state. For the simple harmonic oscillator, a crucial relationship derived from the Virial Theorem simplifies this calculation significantly. The Virial Theorem states that for a power-law potential V(x) = Cx^s, ...

Q2. d). i) Write down the eigenfunctions for the 300 and 310 states of the hydrogen atom. ii) Write the eigenvalues of L^2 and Lz for the states 300 and 310. iii) Calculate the most probable value of r for the hydrogen atom in the state 210.

i)) Write down the eigenfunctions for the 300 and 310 states of the hydrogen atom. (80 words)
ii)) Write the eigenvalues of L^2 and Lz for the states 300 and 310. (80 words)
iii)) Calculate the most probable value of r for the hydrogen atom in the state 210. (240 words)

Key Points:

Hydrogen atom eigenfunctions are Ψ_{nlm}(r, θ, φ) = R_{nl}(r) Y_{lm}(θ, φ).
300 state implies n=3, l=0, m=0, while 310 state implies n=3, l=1, m=0.
Eigenvalue of L² is l(l+1)ħ² and eigenvalue of Lz is mħ.
L² eigenvalue for 300 state is 0; for 310 state is 2ħ².

Answer: The hydrogen atom is a fundamental system in quantum mechanics, serving as a cornerstone for understanding atomic structure. Its stationary states are described by eigenfunctions, Ψ_{nlm}(r, θ, φ), which are products of radial functions, R_{nl}(r), and spherical harmonics, Y_{lm}(θ, φ). These quantum numbers (n, l, m) specify the energy, orbital angular momentum, and its z-component, respectively.

Q3. a). Consider the following state vectors: |ψ) = [1, -1, i]ᵀ and |Φ) = [2, 1-i, i]ᵀ. Calculate (i) the norm of |ψ) and (ii) the inner product (Φ|ψ).

i)) the norm of |ψ) (120 words)
ii)) the inner product (Φ|ψ) (120 words)

Key Points:

Norm of a state vector ||ψ|| is calculated as √(<ψ|ψ>).
Inner product <Φ|ψ> is obtained by <Φ|ψ = Φ†ψ.
The bra vector <ψ| is the Hermitian conjugate (conjugate transpose) of the ket vector |ψ).
Complex conjugate of a complex number (a + bi) is (a - bi).

Answer:

Q3. b). If an operator Ô is Hermitian and an operator Û is unitary, show that the operator ÛÔÛ⁻¹ is Hermitian.

Key Points:

An operator X is Hermitian if its adjoint X† equals X.
An operator Û is unitary if its adjoint Û† equals its inverse Û⁻¹.
The adjoint of a product of operators (ABC)† is C†B†A†.
Using the unitary property, the term (Û⁻¹)† simplifies to Û.

Answer: To demonstrate that the operator ÛÔÛ⁻¹ is Hermitian, we must show that its adjoint is equal to itself, i.e., (ÛÔÛ⁻¹)† = ÛÔÛ⁻¹. This proof fundamentally relies on the given definitions: Ô is Hermitian, meaning Ô† = Ô, and Û is unitary, which implies Û† = Û⁻¹. We begin by taking the adjoint of the combined operator ÛÔÛ⁻¹. Using the property for the adjoint of a product of operators, which states (ABC)† = C†B†A†, we can write: (ÛÔÛ⁻¹)† = (Û⁻¹)† Ô† Û† Next, we apply the properties of unitary and H...

Q3. c). The spectral representation of an operator Ω in a two-dimensional orthonormal basis {|1), |2)} is: Ω = 3|1)‹1| + √2 |1)‹2| + √2 |2)‹1| + 2|2)‹2|. Determine the matrix elements of Ω.

Key Points:

Operator matrix elements are defined as Ω_ij = ‹i|Ω|j›.
The given operator is in its spectral (outer product) representation.
Orthonormality condition ‹i|j› = δ_ij is crucial for calculations.
Each matrix element is found by sandwiching the operator between basis vectors.

Answer: The matrix elements of an operator Ω in an orthonormal basis {|i)} are fundamentally defined as Ω_ij = ‹i|Ω|j›. For the given two-dimensional orthonormal basis {|1), |2)}, we need to determine the four matrix elements: Ω₁₁, Ω₁₂, Ω₂₁, and Ω₂₂. This process is a core concept in Quantum Mechanics-I (MPH-004) for representing operators. Given the operator's spectral representation, Ω = 3|1)‹1| + √2 |1)‹2| + √2 |2)‹1| + 2|2)‹2|, we use the orthonormality condition, which states that ‹i|j› = δ_ij (Kr...

Q3. d). Assume |1) and |2) are the orthonormal basis states of a two-dimensional Hilbert space. A Hermitian operator Â in this basis is given by the spectral representation: Â = 3|1)‹1| - 1|2)‹2|. For a normalized state given by |ψ) = (1/√2)|1) + (√3/2)|2), calculate ‹Â› and the root mean square deviation in Â.

Key Points:

Expectation value ‹Â› measures the average outcome of an observable.
Formula for expectation value: ‹Â› = ‹ψ|Â|ψ›.
Root mean square (RMS) deviation ΔÂ quantifies measurement spread.
Formula for RMS deviation: ΔÂ = √‹Â²› - ‹Â›².

Answer: The expectation value of an observable Â, ‹Â›, for a normalized state |ψ) is given by the formula ‹ψ|Â|ψ›. We are given the Hermitian operator Â = 3|1)‹1| - 1|2)‹2| and the state |ψ) = (1/√2)|1) + (√3/2)|2). The basis states |1) and |2) are orthonormal, meaning ‹i|j› = δᵢⱼ. To calculate ‹Â›: ‹Â› = ‹ψ|Â|ψ› = ((1/√2)‹1| + (√3/2)‹2|) (3|1)‹1| - 1|2)‹2|) ((1/√2)|1) + (√3/2)|2)) = ((1/√2)‹1| + (√3/2)‹2|) ((3/√2)|1) - (√3/2)|2)) (using orthonormality) = (1/√2)(3/√2)‹1|1) + (√3/2)(-√3/2)‹2|2) = 3/2 ...

Q3. e). Derive the Heisenberg equations of motion for Ŝx and Ŝy for the Hamiltonian Ĥ = ωŜz.

Key Points:

Heisenberg equation defines operator time evolution: dÔ/dt = (1/iħ) [Ô, Ĥ].
Spin operators satisfy fundamental commutation relations, e.g., [Ŝx, Ŝz] = -iħŜy.
The given Hamiltonian for the system is Ĥ = ωŜz.
The equation of motion for Ŝx is derived using its commutator with Ĥ.

Answer: The Heisenberg equation of motion, a cornerstone of the Heisenberg picture in quantum mechanics, describes the time evolution of an operator Ô. As covered in MPH-004, for an operator not explicitly dependent on time, this equation is given by dÔ/dt = (1/iħ) [Ô, Ĥ], where Ĥ is the system's Hamiltonian. We are given the Hamiltonian Ĥ = ωŜz. To derive the equations of motion for Ŝx and Ŝy, we utilize the fundamental commutation relations for spin angular momentum operators: [Ŝx, Ŝy] = iħŜz, [Ŝy, Ŝ...

Q4. a). For the simple harmonic oscillator i) Show that [Ĥ, â] = −ħωâ. ii) Calculate the matrix element ‹n|â|n+1›.

i)) Show that [Ĥ, â] = −ħωâ. (200 words)
ii)) Calculate the matrix element ‹n|â|n+1›. (200 words)

Key Points:

Hamiltonian for SHO: Ĥ = ħω(â†â + ½) using creation/annihilation operators.
Commutator definition: [A, B] = AB - BA.
Fundamental commutation relation: [â, â†] = 1, hence [â†, â] = -1.
Result of [Ĥ, â]: [Ĥ, â] = -ħωâ, indicating â lowers energy by ħω.

Answer: The quantum mechanical simple harmonic oscillator (SHO) is a fundamental system in quantum mechanics, often analyzed using the algebraic method involving creation (â†) and annihilation (â) operators. This approach simplifies the calculation of various properties, including commutation relations and matrix elements between energy eigenstates. Part (i) demonstrates a key commutation relation involving the Hamiltonian (Ĥ) and the annihilation operator (â), which highlights how â affects the system...

Q4. b). i) Write down the angular momentum states |j, mj› and calculate the matrix elements of Ĵx, Ĵy and Ĵz for j = 1/2. ii) For the angular momentum state |3,1› show that Ĵ+|3,1› = ħ√10|3,2› and Ĵ-|3,1› = 2ħ√3 |3,0›.

i)) Write down the angular momentum states |j, mj› and calculate the matrix elements of Ĵx, Ĵy and Ĵz for j = 1/2. (240 words)
ii)) For the angular momentum state |3,1› show that Ĵ+|3,1› = ħ√10|3,2› and Ĵ-|3,1› = 2ħ√3 |3,0›. (160 words)

Key Points:

Angular momentum states |j, mj› are eigenstates of Ĵ² and Ĵz.
For j=1/2, the states are |1/2, 1/2› and |1/2, -1/2›.
Matrix elements of Ĵz for j=1/2 are (ħ/2) diag(1, -1).
Ĵx and Ĵy matrix elements are derived using ladder operators Ĵ±.

Answer: This response comprehensively addresses the requested aspects of angular momentum in quantum mechanics, focusing on the representation of states and the calculation of matrix elements for specific angular momentum values, as detailed in the sub-questions below.

Q4. c). Show that in the terms of the |↑› and |↓› basis vectors defined by Ŝz|↑› = (ħ/2)|↑›; Ŝz|↓› = -(ħ/2)|↓›, we can write: Ŝx = (ħ/2) (|↑›‹↓| + |↓›‹↑|).

Key Points:

|↑› and |↓› are orthonormal eigenstates of Ŝz, forming a complete basis.
Any operator Ô can be represented as Ô = Σ_ij ‹i|Ô|j› |i›‹j| in a given basis.
The diagonal matrix elements of Ŝx in the Ŝz basis are ‹↑|Ŝx|↑› = 0 and ‹↓|Ŝx|↓› = 0.
The off-diagonal matrix elements are ‹↑|Ŝx|↓› = ħ/2 and ‹↓|Ŝx|↑› = ħ/2.

Answer: The |↑› and |↓› states represent the orthonormal eigenvectors of the spin angular momentum operator Ŝz, corresponding to eigenvalues +ħ/2 and -ħ/2, respectively. These states form a complete basis for the two-dimensional spin space. Any operator Ô in this space can be represented by its matrix elements in this basis. An operator Ô can generally be written as a sum over its outer products with its matrix elements: Ô = Σ_ij ‹i|Ô|j› |i›‹j|. For Ŝx in the {|↑›, |↓›} basis, we need to calculate the ...

MPH-004 Solved Assignment — January 2026 / July 2026

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

MPH-004—January 2026 / July 2026

QUANTUM MECHANICS-I

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MPH-004 Assignment Questions — January 2026

Q1. a). Electrons are accelerated through a potential 150 V and incident on a crystal with interatomic spacing d = 0.20 nm. Calculate the de Broglie wavelength and the first-order Bragg diffraction angle.

Q1. b). Estimate the uncertainty in the energy of a photon localized within a distance of 0.1 nm.

Q1. c). i) Normalize the wave function: ψ(x,0) = N sin (πx/L) + sin (2πx/L), 0≤x<L. ii) Calculate the expectation value of x for a particle in this state.

Q1. d). A quantum mechanical particle in one dimension has the wave function: ψ(x) = x^n : x > 0 ; n > 1. Use the Schrödinger equation to determine the corresponding potential V(x).

Q2. a). A 1.5 mA beam of electrons enters a sharply defined boundary with a velocity 3.0×10^6 ms-1, and then its velocity reduces to 2.0×10^6 ms⁻¹, due to the difference in potential. Calculate the transmitted and reflected currents.

Q2. b). Show that for any stationary state of a symmetric potential well: (x^2) ≠0.

Q2. c). Calculate the expectation value of the potential energy for the first excited state of a simple harmonic oscillator.

Q2. d). i) Write down the eigenfunctions for the 300 and 310 states of the hydrogen atom. ii) Write the eigenvalues of L^2 and Lz for the states 300 and 310. iii) Calculate the most probable value of r for the hydrogen atom in the state 210.

Q3. a). Consider the following state vectors: |ψ) = [1, -1, i]ᵀ and |Φ) = [2, 1-i, i]ᵀ. Calculate (i) the norm of |ψ) and (ii) the inner product (Φ|ψ).

Q3. b). If an operator Ô is Hermitian and an operator Û is unitary, show that the operator ÛÔÛ⁻¹ is Hermitian.

Q3. c). The spectral representation of an operator Ω in a two-dimensional orthonormal basis {|1), |2)} is: Ω = 3|1)‹1| + √2 |1)‹2| + √2 |2)‹1| + 2|2)‹2|. Determine the matrix elements of Ω.

Q3. e). Derive the Heisenberg equations of motion for Ŝx and Ŝy for the Hamiltonian Ĥ = ωŜz.

Q4. a). For the simple harmonic oscillator i) Show that [Ĥ, â] = −ħωâ. ii) Calculate the matrix element ‹n|â|n+1›.

Q4. b). i) Write down the angular momentum states |j, mj› and calculate the matrix elements of Ĵx, Ĵy and Ĵz for j = 1/2. ii) For the angular momentum state |3,1› show that Ĵ+|3,1› = ħ√10|3,2› and Ĵ-|3,1› = 2ħ√3 |3,0›.

Q4. c). Show that in the terms of the |↑› and |↓› basis vectors defined by Ŝz|↑› = (ħ/2)|↑›; Ŝz|↓› = -(ħ/2)|↓›, we can write: Ŝx = (ħ/2) (|↑›‹↓| + |↓›‹↑|).