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Physics

Where a p Orbital Gets Its Shape

From Schrödinger equation to quantum shape

Where does a pp orbital get its shape?

The familiar dumbbell is not an electron flying along a path. It is a map of where a quantum wave makes the electron more or less likely to be found.

Follow the arrows

From Schrödinger's equation to a dumbbell

We solve the stationary Schrödinger equation for an electron attracted to a nucleus. Each arrow shows the next mathematical choice and what it tells us about the shape.

  1. 01

    Start with the governing equation

    For hydrogen, treat the nucleus as fixed and use its Coulomb potential:

    −ℏ22me∇2ψ+V(r)ψ=EψV(r)=−e24πε0r\begin{aligned} -\frac{\hbar^2}{2m_e}\nabla^2\psi + V(r)\psi &= E\psi \\ V(r) &= -\frac{e^2}{4\pi\varepsilon_0 r} \end{aligned}

    This equation balances the wave's kinetic energy against its electric attraction to the nucleus. Solving it gives the allowed wavefunctions ψ\psi.

    The potential depends only on distance rr, so the problem has spherical symmetry.

  2. 02

    Separate distance from direction

    Spherical symmetry lets the solution split into a radial part and an angular part:

    ψ(r,θ,ϕ)=Rnℓ(r)Yℓm(θ,ϕ)\psi(r,\theta,\phi)=R_{n\ell}(r)Y_{\ell}^{m}(\theta,\phi)

    RR controls how the wave changes with distance. YY, a spherical harmonic, describes how it changes from one direction to another.

    The separated angular equation is an angular-momentum eigenvalue problem; its solutions are spherical harmonics.

  3. 03

    Select the p pattern

    The letter p means ℓ=1\ell=1. The angular-momentum equation gives its allowed angular pattern; for the orientation called pzp_z:

    L^2Yℓm=ℓ(ℓ+1)ℏ2YℓmY10(θ,ϕ)∝cos⁡θ\begin{aligned} \hat{L}^2Y_{\ell}^{m} &= \ell(\ell+1)\hbar^2Y_{\ell}^{m} \\ Y_1^0(\theta,\phi) &\propto \cos\theta \end{aligned}

    The cosine is positive on one side of the nucleus and negative on the other. At θ=π2\theta=\frac{\pi}{2}, it is zero: that is the plane z=0z=0.

    For hydrogen's 2p2p state, the radial solution supplies the size and falloff with distance.

  4. 04

    Combine the radial and angular waves

    The hydrogen 2p2p radial solution is R21(r)∝re−r/(2a0)R_{21}(r)\propto re^{-r/(2a_0)}. Multiplying by the angular solution gives:

    ψ2pz(r,θ)∝re−r/(2a0)cos⁡θ\psi_{2p_z}(r,\theta)\propto re^{-r/(2a_0)}\cos\theta

    The radial factor sets how far the cloud extends; the cosine factor sets its direction and the nodal plane.

    The Born rule says measurement probability density is the absolute square of the wave.

  5. 05

    Square the wave: the lobes appear

    ∣ψ2pz∣2∝r2e−r/a0cos⁡2θ|\psi_{2p_z}|^2\propto r^2e^{-r/a_0}\cos^2\theta

    At θ=π2\theta=\frac{\pi}{2}, cos⁡θ=0\cos\theta=0, so the probability density vanishes across the whole z=0z=0 plane. At a given nonzero distance, cos⁡2θ\cos^2\theta is largest along +z+z and −z-z. The high-probability regions on those two sides are the lobes of a pp orbital.

The equation, drawn

The pp orbital is the probability pattern

For pzp_z, the angular wave is proportional to cos⁡θ\cos\theta. It is positive above the nucleus, negative below, and zero all across the middle plane.

Squaring the wave removes the sign but keeps the zero. Probability is concentrated on either side of the nodal plane, creating two lobes.

A cross-section through a p orbital oriented along the z axisTwo diffuse lobes lie above and below the nucleus. They have opposite wave phase, and a dashed nodal plane passes through the nucleus.positive phasenegative phasenodal planeprobability density · cross-section
The pzp_z state shown in an x–z\text{x--z} slice; in 3D, the nodal plane extends around the nucleus.

Color = wave phase. The two sides have opposite signs; neither lobe is a different kind of charge.

Density = ∣ψ∣2|\psi|^2. Both lobes have positive probability density, with zero at the nodal plane.

Rotate the axis

Three orientations, same pattern

The three diagrams show the same two-lobed pattern aimed along different axes. For an isolated atom without an external field, these orientations have equal energy; the labels choose a coordinate direction.

pxp_x
pyp_y
pzp_z

Symbol guide: what is inside Schrödinger's equation?

Symbols and SI units in the hydrogen orbital derivation
SymbolTermMeaningSI unit
ψ\psiWavefunctionThe quantum state whose spatial shape we solve for.m−3/2\mathrm{m}^{-3/2}
ℏ\hbarReduced Planck constantSets the scale of quantum effects.J s\mathrm{J}\,\mathrm{s}
mem_eElectron massSets the kinetic-energy scale of the electron.kg\mathrm{kg}
∇2\nabla^2LaplacianMeasures how the wave bends through space.m−2\mathrm{m}^{-2}
V(r)V(r)Potential energyThe electron's electric potential energy, set by distance rr.J\mathrm{J}
eeElementary chargeSets the strength of the electron–nucleus attraction.C\mathrm{C}
ε0\varepsilon_0Vacuum permittivityRelates electric charge to the Coulomb potential.F/m\mathrm{F}/\mathrm{m}
rrRadial distanceDistance from the nucleus to the electron.m\mathrm{m}
EEEnergy eigenvalueAn allowed energy of the stationary quantum state.J\mathrm{J}
a0a_0Bohr radiusSets the radial scale in hydrogen.m\mathrm{m}

The wavefunction unit assumes a normalized three-dimensional spatial wavefunction. The Laplacian's unit describes the operator, before it acts on the wavefunction.

The drawings are schematic cross-sections of probability density, not hard surfaces. An orbital is a quantum state; the electron does not trace out the pictured dumbbell.