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Physics · Glossary

What is Wave function and probability in three dimensions?

Definition 7.1 University Physics — Year 3 · Chapter 7 — The Schrödinger Equation in Three Dimensions

The state of a particle is a complex field ψ(r→,t)\psi(\vect r, t) with Born’s rule

 ⁣dP=∣ψ(r→,t)∣2  ⁣d3r,∫∣ψ∣2  ⁣d3r=1,\dd\mathcal P = |\psi(\vect r, t)|^2\,\dd^3r , \qquad \int|\psi|^2\,\dd^3r = 1 ,

and its evolution is the Schrödinger equation with the Laplacian in place of the second derivative:

iℏ ∂ψ∂t=−ℏ22m Δψ+V(r→) ψ,Δ=∂2∂x2+∂2∂y2+∂2∂z2.\iu\hbar\,\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\,\Delta\psi + V(\vect r)\,\psi , \qquad \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} .

Everything the Year 2 volume established survives verbatim: linearity and superposition, stationary states φ(r→) e−iEt/ℏ\varphi(\vect r)\,\eu^{-\iu Et/\hbar} with −ℏ22mΔφ+Vφ=Eφ-\tfrac{\hbar^2}{2m}\Delta\varphi + V\varphi = E\varphi, and the momentum operator, now the gradient p→^=−iℏ∇→\hat{\vect p} = -\iu\hbar\vect\nabla.

The probability current: a travelling wave transports its density like a fluid; a real (bound, stationary) wave function stands still — the atom’s electron cloud does not circulate unless the state is complex.
The probability current: a travelling wave transports its density like a fluid; a real (bound, stationary) wave function stands still — the atom’s electron cloud does not circulate unless the state is complex.
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