Quantum Mechanics: Particle in an Infinite Square Well

Overview of Lecture

• Continued from previous tutorial on the Schrodinger equation.
• Focus on solving the problem of a particle in an infinite square well (particle in a box).

Potential Energy Definition

• Potential V(x):
  • V(x) = ∞ for x ≤ 0 and x ≥ a.
  • V(x) = 0 for 0 < x < a.
• Infinite potential outside the boundaries means the particle cannot exist there.
• Particle confined within the walls of the well.

Schrodinger Equation Application

• Objective: Find wavefunctions (ψ) and corresponding eigenenergies (E) for the particle.
• Boundary conditions dictate:
  • ψ(x) = 0 for x ≤ 0 and x ≥ a.
  • Valid solutions for ψ must be determined within the walls (0 < x < a).

Time-Independent Schrodinger Equation

• Rearranged form:
  [ \frac{d^2ψ}{dx^2} + k^2ψ = 0 ]
  where ( k = \sqrt{\frac{2mE}{\hbar}} )
  • Leads to a second order differential equation needing solutions.
• General solution:
  [ ψ(x) = A \sin(kx) + B \cos(kx) ]

Boundary Conditions and Simplification

• At x = 0:
  • Requires B = 0 (since cos(0) = 1).
  • Simplifies to:
    [ ψ(x) = A \sin(kx) ]
• At x = a:
  • Requires ( \sin(ka) = 0 )
    • Thus, ( ka = n\pi ) (n is an integer).
    • Therefore, ( k = \frac{n\pi}{a} ).

Quantization of Energy Levels

• Each integer n corresponds to a different wavefunction and energy level.
• Normalization condition to find A:
  [ \int_{0}^{a} |ψ(x)|^2 dx = 1 ]
  • Leads to:
    [ A = \sqrt{\frac{2}{a}} ]
  • Final eigenfunctions:
    [ ψ_n(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n\pi}{a} x \right) ]_

Eigenenergies

• Express energy as:
  [ E_n = \frac{\hbar^2 k_n^2}{2m} = \frac{\hbar^2 n^2 \pi^2}{2ma^2} ]

Summary of Results

• Successfully derived wavefunctions and energies for a particle in an infinite square well.
• Each quantum state is quantized, dependent on integer n.
• Implications of these results warrant further discussion in future tutorials.