Lecture on Quantum Mechanics: Solving the Schrödinger Equation

Introduction

• Main focus: Solving the Schrödinger equation, a central element of quantum mechanics.
• Solution technique: Separation of variables.
• The Schrödinger equation is a partial differential equation (PDE).

Ordinary vs Partial Differential Equations

Ordinary Differential Equations (ODE)

• Govern how specific coordinates change with time.
• Example: projectile motion with position (x, y, z) and velocity (v_x, v_y).
• Example equations:
  • dX/dt = v_x
  • dv_x/dt = -k v_x

Partial Differential Equations (PDE)

• Involve several independent variables.
• Example: Electric field (E) as a function of x, y, z.
• Gauss's law:
  • Integral form: ∮ E · dA = Q_enclosed/ε₀
  • Differential form: ∇ · E = ρ/ε₀
• In quantum mechanics, often solved using potential (V).

Separation of Variables

• A technique to solve PDEs by assuming a solution can be written as a product of functions, each depending on a single variable.
• Example with wave equation on a string:
  • U(x, t) = X(x)T(t)
  • Separate into two ordinary differential equations.

Wave Equation

• Relation of acceleration to curvature using second derivatives.
• Separation results in ordinary differential equations (ODEs) that can be solved for X and T.

Schrödinger Equation

Time-Dependent Schrödinger Equation

• iħ ∂ψ/∂t = Hψ
• Hamiltonian operator H = kinetic + potential energy.
• Use separation of variables:
  • ψ(x, t) = X(x)T(t).

Time-Independent Schrödinger Equation

• Derived from Schrödinger equation via separation of variables.
• Hψ(x) = Eψ(x) where E is energy.

Stationary States

• Solutions to the time-independent Schrödinger equation with simple time evolution.
• Probability densities and expectation values are constant in time.

Energy of Stationary States

• Energy E corresponds to the separation constant from the Schrödinger equation.
• Variance in energy is zero, meaning no uncertainty in energy.

Superposition of Stationary States

• Solutions to the Schrödinger equation can be expressed as superpositions of stationary states.
• Time evolution of a superposition involves different frequencies based on energy differences.

Example Potentials

• Infinite Square Well: Particle confined within a region where potential is zero.
• Harmonic Oscillator: Parabolic potential like a spring.
• Delta Function, Soft Box, and Constant potentials.

Solving the Schrödinger Equation

• Infinite Square Well: Solve with boundary conditions to find discrete energy levels and wavefunctions.
• Orthogonality and completeness of solutions allow expression of arbitrary functions as sums of stationary states.

Fourier's Trick

• Use orthogonal sine functions to express any function within the well.
• Calculate coefficients for this expansion using integrals.

Numerical and Simulation Examples

• Use software tools (like Sage) to handle complex integrals and visualize solutions.
• Time evolution of wavefunctions shows oscillatory behavior and spreading of probability densities.

Conclusion

• Schrödinger equation solutions depend on potential functions.
• Separation of variables and superposition principles are fundamental techniques in quantum mechanics.

Key Concepts to Remember:

• Separation of Variables: A technique to simplify and solve PDEs.
• Time-Independent Schrödinger Equation: Provides stationary states with definite energies.
• Orthogonality and Completeness: Allows superposition of states to form any wavefunction.
• Superposition: Leads to time-dependent behavior of quantum systems.
• Tools: Sage and simulations help visualize and compute solutions.