Introduction: The Master Equation
The time-dependent Schrödinger equation governs how quantum states evolve. It's the equation of motion for quantum mechanics, playing the role Newton's second law plays in classical mechanics.
The Equation
\[i\hbar\frac{\partial\psi(x,t)}{\partial t} = \hat{H}\psi(x,t)\]For a particle in potential \(V(x)\):
\[i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi\]Key Properties
- First order in time: Fully determined by initial condition \(\psi(x, 0)\)
- Linear: Superpositions evolve independently
- Deterministic: Given initial state, future state is determined
- Preserves normalization: Probability conserved
Worked Examples
Example 1: Free Particle
With \(V = 0\): \(i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}\)
Solution: \(\psi = Ae^{i(kx - \omega t)}\) with \(\omega = \hbar k^2/2m\)
Example 2: Time Evolution of Superposition
If \(\psi(x,0) = c_1\psi_1(x) + c_2\psi_2(x)\) where \(\hat{H}\psi_n = E_n\psi_n\):
\[\psi(x,t) = c_1\psi_1(x)e^{-iE_1 t/\hbar} + c_2\psi_2(x)e^{-iE_2 t/\hbar}\]The Quantum Connection
The TDSE is remarkable: from it emerges all quantum dynamics. Energy eigenstates are special because they only pick up a phase in time. General states are superpositions, and the time evolution creates quantum beats, tunneling, and all the phenomena we'll explore.