Quantum Mechanics — DU B.Sc. (Hons) Physics Previous Year Question Papers

2022 Semester : VIII- Semester (GE Paper) (a) Define a Hermitian operator and state why physical observables must be represented by them: (b) Checkif the operators and i, commute with each other by evaluating Do, Pal. (c) State the physical conditions that a wave function u(x) must satisfy to be considered acceptable. (d) Why is it impossible for a particle trapped in a one-dimensional box to have a zero ground-state energy? (e) Express the angulár momentum operator component Ly in spherical polar coordinates. (f) A wave function is given by i) = Asin(E) for Ex≤ L Determine the normalization constant A. Prove the continuity equation e + J= 0 and interpret its physical meaning regarding - 02.(a) (a) Formulate the probability current density for atre-dimensional yave function... probability conservation. (10 Marks) (b) Show that if two operators  and 8 commute, they share a common set of simultaneous eigenfunctions. (8 Marks) 03. (a) Consider a particle of mass in trapped inside a symmetric one-dimensional infinite potential well defined by: Solve the time-independent Schrödinger equation to derive the normalized energy eigenfunctions and show that the states alternate in parity. (10 Marks) b) Find the energy eigenvalues for the ground state and the first excited state of article in this symmetric: well if the total width of the well (2a) is equalto 2A- (8 Marks 04. (a) Using the power series method (Frobenius method, solve the time-independent Schrödinger equation for a 1-Ö linear harmonic oscillator to find its energy eigenvalues. (b) Given the raising operator a" and lowering operator a prove the following operational relationships; Q5: (a) For a particle in the ground state (n = 0) of a 1-p' tinear harmonic oscillator; evaluate the expectation values (x) and (Px). Use these to determine the uncertainties Ax and Ax (10 Marks) (b) A quantum harmonic oscillator has an angular frequency of a → 3 x 10"irad/s. Calculate its zero-point energy in electron-volts (eV). (8 Marks) Q6. (a) Separate the three-dimensional time-independent Schrödinger equation in spherical polar coordinates (r_D. Ф) for a central potential V(T). Explicitly isolate and solve the azimuthal (p) dependent differential equation. (10 Marks) (b) A particle is represented by an initial Gaussian wave packet w0) = ()exp(- 2) Describe qualitatively how and why this wave (8 Marks) packet spreads as time progresses (i > 0).