Quantum mechanics

The Schrödinger equation

The fundamental equation describing how a quantum wavefunction evolves in time — the foundation of all quantum mechanics.

The Schrödinger equation is the fundamental equation of quantum mechanics, playing a role analogous to Newton's second law in classical mechanics: it tells us how a quantum system's wavefunction $\psi$ evolves in time.

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

The left side describes the wavefunction's evolution in time. The right side is the Hamiltonian operator $\hat{H}$ acting on $\psi$ , made of a kinetic term (the second spatial derivative — the 'wigglier' $\psi$ is, the higher the kinetic energy) and a potential term $V(x)$ . This is the time-dependent equation; when the system is in a stationary state (definite energy $E$ ), there's a simpler time-independent version: $\hat H \psi = E\psi$ .

Definition: The wavefunction and its probabilistic interpretation

The wavefunction $\psi(x,t)$ is a complex number at every point in space, not directly observable. What's measurable is $|\psi(x,t)|^2$ — the probability density of finding the particle at position $x$ at time $t$ (the Born interpretation, 1926). Being a probability, $\psi$ must satisfy the normalization condition $\int |\psi|^2\,dx = 1$ .

This is a genuine numerical solution of the equation above (split-step Fourier method), not a cartoon animation. Try the 'Tunneling through a barrier' mode to see an effect only quantum mechanics predicts.

Tunneling: when a particle crosses where classical physics forbids it

Consider a particle of energy $E$ approaching a barrier of height $V_0 > E$ . Classically, it must bounce back — it lacks the energy to climb over the barrier. But the Schrödinger equation's solution shows the wavefunction doesn't vanish abruptly inside the barrier; it decays exponentially, so a small part still 'leaks' through to the other side — the particle appears past the barrier with nonzero probability, despite never having had enough energy to go 'over the top'.

Tunneling isn't a mathematical curiosity: it explains alpha decay in radioactive nuclei, is the operating principle of the scanning tunneling microscope (STM), and is the source of leakage current in very small transistors.

Example: Why macroscopic objects don't visibly tunnel

If tunneling is real, why do we never see a rolling ball 'tunnel through' a wall in everyday life?

Solution

Tunneling probability decays exponentially with particle mass and barrier thickness. For a ball (mass ~ $10^{23}$ times an electron's), this probability is so tiny it's practically zero — you'd need to wait vastly longer than the age of the universe to see it once. The effect is only significant for particles with masses around that of an electron or proton.

For a time-independent Hamiltonian, separated solutions have the form $\psi_n(\mathbf r,t)=\phi_n(\mathbf r)e^{-iE_nt/\hbar}$ . The probability density $|\psi_n|^2$ is stationary although the wavefunction's phase changes; superpositions of energy eigenstates generally produce time-dependent densities. The Hamiltonian determines the spectrum, and its eigenstates satisfy the problem's boundary conditions. When the potential depends on time, this separation generally fails, but the evolution equation still gives unitary state dynamics.

Quick check

The quantity $|\psi(x,t)|^2$ in quantum mechanics represents:

Quantum tunneling allows a particle to:

References

Erwin Schrödinger (1926). Quantisierung als Eigenwertproblem
David J. Griffiths (2018). Introduction to Quantum Mechanics