The Schrödinger Equation

Chem 3240 · Lecture 3.1

Davit Potoyan

Entering the quantum world

  • Classical mechanics fails at atomic scales
  • Need a new equation explaining energy quantization and wave-particle duality
  • Schrödinger’s Equation is a fundamental law: postulated, not derived
  • Never failed when applied correctly

From a wave to a quantum wave

Start with a traveling wave: \[\Psi(x,t) = Ae^{i(kx-\omega t)}\]

Insert the de Broglie and Planck relations: \[p = \hbar k \qquad E = \hbar\omega\]

Quantum wave function: \[\Psi(x,t)=Ae^{\frac{i}{\hbar}(px-E t)}\]

What equation generates it?

Time part returns total energy: \[\frac{\partial \Psi}{\partial t} = -\frac{i}{\hbar} E\, \Psi\]

Spatial part returns kinetic energy: \[\frac{\partial^2 \Psi}{\partial x^2} = -\frac{2m(E - V)}{\hbar^2}\, \Psi\]

Combine via \(K = E - V\) to eliminate \(E\).

Time-Dependent Schrödinger Equation

\[-\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V(x)\,\Psi = i\hbar \frac{\partial \Psi}{\partial t}\]

  • Describes evolution of \(\Psi(x,t)\) in a potential \(V(x)\)
  • Only a single time derivative (unlike the classical wave equation)
  • The \(i\) makes solutions oscillate in the complex plane

Anatomy of the equation

  • Kinetic + potential = total energy flow
  • \(\Psi\) is generally complex: not directly measurable
  • 1D shown; 3D adds \(y, z\) terms

Separation of variables

Assume \(\Psi(x,t) = \psi(x)\,T(t)\).

Time part is a simple oscillation: \[\Psi(x,t) = \psi(x)\cdot e^{-iEt/\hbar}\]

  • The hard part is finding \(\psi(x)\), set by the potential \(V(x)\)

Time-Independent Schrödinger Equation

\[-\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V(x)\,\psi = E\,\psi\]

  • Gives the stationary states of the system
  • Our primary focus for the rest of the course

Operators: the language of QM

  • An operator \(\hat{A}\) transforms a function into another
  • Differentiate, multiply, integrate, …
  • The SE is linear: \[\hat{A}[c_1 f_1 + c_2 f_2] = c_1\hat{A}f_1 + c_2\hat{A}f_2\]

The Hamiltonian

Classical total energy: \[H(x,p) = \frac{p^2}{2m} + V(x)\]

Quantum Hamiltonian operator: \[\hat{H} = \hat{K} + \hat{V} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x)\]

The SE in operator form: \[\hat{H}\psi = i\hbar \frac{\partial \psi}{\partial t}\]

Correspondence principle

Every classical observable has a quantum operator:

Observable Classical Quantum
Position \(x\) \(\hat{x}=x\)
Momentum \(p\) \(\hat{p}=-i\hbar \frac{\partial}{\partial x}\)
Kinetic energy \(\frac{p^2}{2m}\) \(\hat{K}=\frac{\hat{p}^2}{2m}\)
Total energy \(H\) \(\hat{H}=\hat{K}+\hat{V}\)

Eigenvalues and eigenfunctions

The TI-SE is an eigenvalue problem: \[\boxed{\hat{H} \psi_n = E_n \psi_n}\]

  • \(\psi_n\): eigenfunctions (wave functions)
  • \(E_n\): eigenvalues (energies)
  • General state: \(\psi = \sum_n c_n \psi_n f_n(t)\)

Takeaway

Solving a quantum system means finding the eigenfunctions \(\psi_n\) and eigenvalues \(E_n\) of its Hamiltonian: \(\hat{H}\psi_n = E_n \psi_n\).