Physical Chemistry-I Mid-Semester Exam

Course Overview

• Course Title: Physical Chemistry-I
• Course Code: CHCHC03
• Degree: M.Sc.
• Semester: 1st
• Total Pages: 1
• Exam Duration: 1 hour and 30 minutes

Examination Instructions

• All questions must be attempted.
• Missing data can be assumed as necessary.

Questions Breakdown

• Question 1(a): Involves operators ( A ) and ( B ) with a commutation relation ( [A, B] = 1 ). Task is to find ( [A, B^2] ) (2 marks).

• Question 1(b):

  • Statements assessed for hermitian nature:
    • ( (A + A^*) ) is hermitian.
    • ( i(A + A^*) ) is hermitian.
  • Additional inquiry: Is ( A = -\frac{d}{dx} + x ) hermitian? (2 marks + 1 mark for justification).

• Question 2(a): Evaluate the value of the commutator ( [, [, A]] ) (2 marks).

• Question 2(b):

  • Calculate the expectation values of position ( x ) and momentum ( p_x ) for a particle in a one-dimensional box of width ( a ).
  • Given normalized wavefunction:
    [ Ψ_n(x) = \left(\frac{2}{a}\right)^{1/2} \sin\left(\frac{n\pi x}{a}\right) ] (3 marks).

• Question 3(a): Computes the probability of finding an electron in a 1-D box of specified length (exact length not provided).

Key Concepts

• Operators & Commutation Relations: Fundamental in quantum mechanics for understanding how physical quantities relate to each other.
• Hermitian Operators: Critical for defining observable quantities; have real eigenvalues.
• Expectation Values: Describes average outcomes in quantum states, crucial for predicting measurement results in quantum systems.
• Wavefunction Normalization: Ensures probabilities derived from the wavefunction sum to 1, enabling physical interpretation.

Additional Notes

• Ensure to understand and apply the principles of quantum mechanics related to operators, eigenvalues, and wavefunctions.
• Utilize mathematical proofs and justifications effectively when answering theoretical questions.