Schrödinger wave equation: wave function, probability and quantised energy

See the wave function ψ and the probability ψ² of an electron in a box and in a hydrogen atom, watch electron snapshots build up the probability pattern, and see why the energy is quantised. Free for every student.

Also searched as: schrodinger wave equation · schrodinger wave equation · wave function and probability · particle in a box · psi and psi squared · quantum mechanical model of atom

About this experiment

Aim

To see what the Schrödinger equation gives: a wave function ψ for the electron, a probability ψ² of finding it, and energies that are quantised, for an electron in a box and in a hydrogen atom.

Theory

Bohr's orbits could not explain atoms with more than one electron. The quantum mechanical model treats the electron as a wave (de Broglie: λ = h/mv) and uses the Schrödinger equation, Ĥψ = Eψ. Its solutions are the wave functions ψ. ψ itself has no direct meaning, but ψ² at a point is proportional to the probability of finding the electron there. Solutions exist only for certain energies, so energy is quantised.

The simplest case is an electron trapped in a one-dimensional box of width L with walls it cannot cross. The wave must fit in the box like a standing wave on a string: ψ = √(2/L) sin(nπx/L), n = 1, 2, 3 ... Then λ = 2L/n and the energy is E_n = n²h²/(8mL²). The lowest energy (n = 1) is not zero, and a narrower box needs more energy.

The number of nodes (places where ψ = 0 and the electron is never found) is n − 1 in the box. At a node the sign of ψ changes. For the hydrogen atom the solutions are the orbitals: ψ = R(r) × angular part, the energy is E = −13.6/n² eV, and the radial nodes number n − l − 1.

Formulas used

• Ĥψ = Eψ (Schrödinger equation)
• λ = h / (mv) = h / p (de Broglie)
• box: ψₙ(x) = √(2/L) sin(nπx/L), Eₙ = n²h² / (8 m L²)
• electron in a box: Eₙ = 0.376 n² / L² eV (L in nm)
• hydrogen: Eₙ = −13.6 / n² eV, radial nodes = n − l − 1
• probability of finding the electron between x and x + dx = ψ² dx

Procedure: what to do in the virtual lab

1. Choose the box with n = 1. Note that ψ has no node and ψ² has one hump in the middle.
2. Increase n step by step. Count the nodes (zeros of ψ) and compare with n − 1.
3. Make the box narrower (smaller L) and see that the energy goes up like 1/L².
4. Switch to the hydrogen atom and look at 1s, 2s and 2p. Where is ψ² highest?

Observations to make

• The energy grows as n², so the levels get further apart at higher n.
• The electron is never found at a node, and the probability pattern has n humps in the box.
• In hydrogen the 2s function has one radial node and the 2p function has none.

What you can change in this experiment

• System: Electron in a 1D box (the simplest case) / Electron in a hydrogen atom (radial part)
• Quantum number n
• Azimuthal quantum number l (hydrogen only, l < n)
• Width of the box L (nm, box only)
• Show electron snapshots (build up ψ²) (on / off)

Viva questions on Schrödinger wave equation

Questions an examiner may ask in the Class 11 chemistry practical. Run the experiment in the lab to find the answers yourself, then check them in the Learn tab.

1. What is the physical meaning of ψ and ψ²?
2. What is a node? How many nodes does the n-th state of an electron in a box have?
3. Why is the energy of the electron quantised?
4. What does the de Broglie relation say?

Frequently asked questions

Syllabus: Class XI Chemistry, Chapter 2 — Structure of Atom, §2.6 (quantum mechanical model of atom, Schrödinger equation)

What you do in the lab

Change the controls and watch the chemistry behave in real time in 3D. Live readouts and graphs show the numbers behind it, and the Learn tab has the theory, procedure, observations and viva questions. Works on a phone or a laptop, in English and Hindi.

Topics: Schrödinger equation · wave function · probability density · particle in a box · quantum mechanical model · quantisation of energy · de Broglie · structure of atom

More from this unit

• Build an atom: protons, neutrons and electrons
• Electron clouds: shapes of s, p and d orbitals in 3D
• Electron, proton and neutron: how they were discovered (beams in electric and magnetic fields)

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