The Schrödinger Equation and the Wavefunction
Chem 3240 · Lecture 3.1
Why a new equation?
- Classical mechanics fails at atomic scales
- Experiments demand: energy quantization and wave-particle duality
- Schrödinger’s equation is a fundamental law: postulated, not derived
- It has never failed when applied correctly
From classical waves to quantum waves
\[\Psi(x,t) = Ae^{i(kx-\omega t)}\]
From classical waves to quantum waves
\[\Psi(x,t) = Ae^{i(kx-\omega t)}\]
- Insert de Broglie: \(p = \hbar k\)
- Insert Planck: \(E = \hbar \omega\)
\[\Psi(x,t)=Ae^{\frac{i}{\hbar}(px-E t)}\]
What equation generates such waves? Differentiate and see.
The time-dependent Schrödinger equation
\[
-\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V(x) \Psi = i \hbar \frac{\partial \Psi}{\partial t}
\]
- Only a single time derivative, unlike the classical wave equation
- The \(i\) makes solutions oscillate in the complex plane
- Kinetic + potential on the left, total energy flow on the right
Separation of variables
\[\Psi(x,t) = \psi(x)\cdot e^{-iEt/\hbar}\]
- Time part: a rotating phase clock at rate \(E/\hbar\)
- Spatial part: the time-independent Schrödinger equation
\[
-\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V(x) \psi = E \psi
\]
The hard part is finding \(\psi(x)\); the potential \(V(x)\) defines the system
What does the wavefunction mean?
- \(\psi\) is complex: no direct physical meaning
- The Born rule extracts the measurable content:
\[p(x) = \psi^{*}(x)\,\psi(x) = |\psi(x)|^2\]
- \(|\psi(x)|^2\,dx\) = probability of finding the particle in \([x, x+dx]\)
Wavefunctions must be normalized
\[\int^{+\infty}_{-\infty} |\psi(x)|^2\, dx = 1\]
- The particle is somewhere: total probability is 1
- Normalization fixes the constant in \(\psi = N\psi'\)
- Probability in a region: \(P(a<x<b)=\displaystyle\int_a^b |\psi|^2 dx\)
Operators: the language of QM
- Every observable gets an operator that acts on \(\psi\)
| position |
\(\hat{x} = x\) |
| momentum |
\(\hat{p} = -i\hbar\,\partial/\partial x\) |
| kinetic energy |
\(\hat{K} = \hat{p}^2/2m\) |
| total energy |
\(\hat{H} = \hat{K} + \hat{V}\) |
The Hamiltonian
\[
\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x)
\]
- Quantum analog of the classical total energy \(H = \frac{p^2}{2m} + V(x)\)
- The Schrödinger equation in one line:
Expectation values
- Predictions are probability-weighted averages:
\[\langle A \rangle = \int \psi^{*}(x)\, \hat{A}\, \psi(x)\, dx\]
- \(\langle x \rangle\): mean position, \(\langle p \rangle\): mean momentum
- Variance \(\sigma^2 = \langle A^2\rangle - \langle A\rangle^2\) measures the spread of measurements
Eigenvalues and eigenfunctions
\[\boxed{\hat{H} \psi_n = E_n \psi_n}\]
- \(\psi_n\): eigenfunctions, the stationary states
- \(E_n\): eigenvalues, the allowed energies
- Boundary conditions make the spectrum discrete
- Linearity: superpositions \(\psi = \sum_n c_n \psi_n\) also solve the equation
Takeaway
The Schrödinger equation \(\hat{H}\psi = E\psi\) is the quantum equation of motion. Its solutions are wavefunctions whose square \(|\psi|^2\) gives probabilities, whose operators give expectation values \(\langle A \rangle = \int \psi^*\hat{A}\psi\,dx\), and whose eigenvalues are the quantized energies.