1.1 Wave Functions and the Schrödinger Equation
📚 Fundamental Principles of Quantum Mechanics
Wave Function \(\psi(\mathbf{r}, t)\) completely describes the quantum state.
Born's Probabilistic Interpretation:
\[ P(\mathbf{r}, t) = |\psi(\mathbf{r}, t)|^2 \]
Time-Dependent Schrödinger Equation:
\[ i\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi = \left(-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) \right) \psi \]
Time-Independent Schrödinger Equation (\(\psi = \phi(x)e^{-iEt/\hbar}\)):
\[ -\frac{\hbar^2}{2m}\frac{d^2\phi}{dx^2} + V(x)\phi = E\phi \]
Summary
In this chapter, we learned the foundations of wave mechanics. The wave function provides a complete description of quantum states through its probabilistic interpretation. The Schrodinger equation serves as the fundamental equation governing quantum mechanics. We explored the particle in a box model, which demonstrates quantized energy levels, and the harmonic oscillator, characterized by equally spaced energy levels and zero-point energy. Finally, the uncertainty principle establishes fundamental limits on the simultaneous measurement of position and momentum.
In the next chapter, we will learn about three-dimensional systems and symmetry, particularly the exact solution of the hydrogen atom.