Quantum Wave function

A wave function is the mathematical object that quantum mechanics uses to describe a particle. It is a complex-valued function of position and time, usually written Ψ(x,t). You cannot measure the function itself. But its squared magnitude tells you where the particle is likely to be found. On this page I'll go through what the wave function means and which equation it follows. Then I'll work through two cases, a particle in a box and the hydrogen atom. The videos at the end show these same functions as moving 3D pictures.

What does a wave function tell you ?

Let's start with the question most people ask first. If Ψ is a wave, what is actually waving ? The accepted answer is statistical, and it is due to Max Born. Ψ is a probability amplitude, and the probability density is its squared magnitude.

In one dimension, the probability of finding the particle between x = a and x = b is the area under |Ψ|² over that interval.

P(a ≤ x ≤ b) = ∫_a^b |Ψ(x,t)|² dx

The particle must be somewhere, so the same integral over the whole x axis must be 1. This is the normalization condition. A solution that does not meet it is multiplied by a constant until it does. On this page i is the imaginary unit, the same number that engineers usually write as j. The squared magnitude is |Ψ|² = Ψ^*Ψ, where Ψ^* is the complex conjugate.

The phase of Ψ is where the complex values matter. Multiplying the whole function by e^iθ changes nothing you can measure, because |e^iθ| = 1. But when two wave functions are added, their relative phase decides whether they reinforce or cancel each other. That is the interference that many of the videos show as changing colors.

Ψ is not measured directly : only |Ψ|² appears in a measurement, and it gives a probability density rather than a position.
The total probability is 1 : the integral of |Ψ|² over all space must equal 1, so a solution is scaled to meet this.
A common phase factor has no effect : e^iθΨ gives the same probabilities as Ψ. Only phase differences between added terms matter.

Which equation does the wave function follow ?

A wave function is not arbitrary. It has to solve the Schrodinger equation, which plays the role that Newton's law plays in classical mechanics. For one particle of mass m in a potential V(x), the time-dependent form is the following.

iℏ ∂Ψ/∂t = -(ℏ²/2m) ∂²Ψ/∂x² + V(x)Ψ

Here ℏ is the reduced Planck constant, h/2π. The left side gives the change in time, and the right side is the energy operator H applied to Ψ. H is called the Hamiltonian. The first term on the right is the kinetic energy, and the second is the potential energy.

When V does not depend on time, we can separate the variables. Try Ψ(x,t) = ψ(x) e^-iEt/ℏ. The time factor cancels on both sides, and what is left is the time-independent equation.

-(ℏ²/2m) d²ψ/dx² + V(x)ψ = Eψ

This is an eigenvalue problem, Hψ = Eψ, with the same structure as A v = λ v on the Eigen page. The allowed energies E are the eigenvalues, and the stationary wave functions ψ are the eigenfunctions. In a stationary state, |Ψ|² = |ψ|² does not change with time. Only the phase rotates, at the angular frequency E/ℏ.

The equation is also linear. So any sum of solutions is again a solution. A sum of two stationary states with energies E₁ and E₂ is not stationary. Its probability density moves back and forth at the angular frequency (E₂ - E₁)/ℏ, and that motion is what most animations of a wave function show.

The Schrodinger equation sets how Ψ changes in time : it is first order in time and second order in space.
Stationary states are eigenfunctions of H : the time-independent equation Hψ = Eψ gives the allowed energies as eigenvalues.
Superposition is allowed because the equation is linear : a sum of stationary states gives a probability density that moves in time.

What does the wave function of a particle in a box look like ?

The simplest case that still shows quantization is a particle trapped between two walls. Let's put V = 0 for 0 < x < L and an infinite potential outside. The particle cannot be outside the box, so ψ must be 0 at x = 0 and at x = L. These two boundary conditions are what select the allowed energies.

Inside the box the equation is ψ'' = -(2mE/ℏ²)ψ, and its solutions are sines and cosines. A cosine is not 0 at x = 0, so only the sine is left. The sine must also be 0 at x = L, so a whole number n of half wavelengths must fit into the box. This gives the eigenfunctions and the eigenvalues below.

ψ_n(x) = √(2/L) sin(nπx/L), E_n = n²π²ℏ²/(2mL²), n = 1, 2, 3, ...

The factor √(2/L) is the normalization. The integral of sin²(nπx/L) over the box is L/2, so the factor makes the total probability exactly 1. Let's put numbers in. For an electron in a box of L = 1 nm, E₁ = 0.376 eV, E₂ = 1.504 eV and E₃ = 3.384 eV. A jump from n = 2 to n = 1 releases 1.128 eV. That is a photon with a wavelength of about 1099 nm, in the near infrared.

The plots below show the first three states for this box. The left panel draws ψ_n(x), and the right panel draws the probability density |ψ_n(x)|². The shaded strip on the right marks the first quarter of the box, 0 ≤ x ≤ L/4.

Figure 1. Particle in a box, n = 1, 2 and 3. Each higher state adds one node, and the probability density spreads more evenly across the box as n grows.

n = 1 : one half wavelength and no node inside the box. The particle is most likely near the middle. The probability in the shaded strip is 1/4 - 1/(2π) = 0.091.
n = 2 : one full wavelength, with a node at L/2. The probability density there is 0, so the particle is never found at the center. The shaded strip holds exactly 0.25.
n = 3 : one and a half wavelengths, with nodes at L/3 and 2L/3. The shaded strip holds 0.303.
The functions are orthogonal : the integral of ψ₁ψ₂ over the box is 0. The sine terms of a Fourier series have the same property.

The lowest energy is not zero. A particle at rest would need ψ = 0 everywhere, which is not a valid state. So confinement itself costs energy, and E₁ grows as 1/L² when the box gets smaller.

Boundary conditions create quantization : only whole numbers of half wavelengths fit, so the energy takes discrete values.
The energy grows as n² : the levels spread apart as n increases, 1 : 4 : 9 for the first three.
The state n has n - 1 nodes : a node is a point where the particle is never found.

How are the hydrogen orbitals in the videos described ?

The hydrogen atom is the case that most of the videos below animate. It is a three-dimensional problem, and the potential is the Coulomb attraction of the proton. The potential depends only on the distance r, so spherical coordinates r, θ and φ are the natural choice.

The solution separates into a radial part and an angular part, ψ_nlm(r,θ,φ) = R_nl(r) Y_l^m(θ,φ). Three whole numbers label each solution, one for each dimension. The first is the principal quantum number n = 1, 2, 3, ... The second is the orbital quantum number l = 0, 1, ..., n - 1. The third is the magnetic quantum number m = -l, ..., +l. So there are n² orbitals for each n, not counting spin. The letters s, p, d and f stand for l = 0, 1, 2 and 3.

The energy depends only on n, E_n = -13.6 eV / n². A jump from n = 2 to n = 1 releases 10.2 eV, which is a photon of 121.5 nm in the ultraviolet. The ground state 1s has no angular dependence at all.

ψ₁₀₀(r) = e^-r/a₀ / √(πa₀³), a₀ = 0.0529 nm

Here a₀ is the Bohr radius. The density |ψ|² is largest at the nucleus. But the probability of finding the electron at a distance r also grows with the area of the sphere, 4πr². The product P(r) = 4πr²|ψ|² peaks exactly at r = a₀. The mean distance is 1.5 a₀, and the probability of finding the electron inside r = a₀ is only 0.323.

The shapes in the videos come from the angular part and the nodes. An orbital has n - l - 1 radial nodes, which are spheres, and l angular nodes, which are planes or cones. So 2s has one spherical node, and 2p has one nodal plane through the nucleus. The videos usually draw a surface of constant |ψ|² or a cloud of density, and many of them use color for the phase of ψ.

Three quantum numbers label an orbital : n sets the energy and size, l sets the shape, and m sets the orientation.
The energy is -13.6 eV / n² : orbitals with the same n and different l have the same energy in this simple model.
Density and radial probability are different things : |ψ|² for 1s is largest at r = 0, but P(r) peaks at the Bohr radius.

YouTube

These videos animate the functions described above, including the hydrogen orbitals in 3D.