Schrodinger Equation

How the Quantum Wavefunction Is Described

There isn't just one way to write the Schrodinger Equation to describe a wave function.
There isn't just one way to write the Schrodinger Equation to describe a wave function. traffic_analyzer, Getty Images

The Schrodinger equation is the mathematical equation that describes the way that the quantum wavefunction, which is the distribution of probabilities that a quantum system will be in any given quantum state, evolves over time.

The importance (and complexity) of the Schrodinger equation is spelled out by James Kakalios in his book The Amazing Story of Quantum Mechanics:

The Schrodinger equation plays the same role in atomic physics that Newton's laws of motion play in the mechanics of everyday objects.[...]

[...] Schrodinger's equation involves the rates of change of the wave function in both space and time; consequently it can't be solved without calculus. in addition, it involves imaginary numbers (the term mathematicians use when referring to the square root of negative numbers), and thus calls upon considerable imagination to interpret.

The Schrodinger equation comes from Erwin Schrodinger in the mid-1920's, when he first derived these formulas and applied them to the behavior of an electron. He soon also showed they were equivalent to the existing mathematical formalism, a matrix approach created by Werner Heisenberg. Albert Einstein himself endorsed Schrodinger's approach, because he considered it a superior approach to Heisenberg's matrix method.

Michio Kaku also sings the praises of the revolutionary nature of the Schrodinger equation in his book Parallel Worlds:

In 1925, Austrian physicist Erwin Schrodinger proposed an equation (the celebrated Schrodinger wave equation) that accurately described the motion of the wave that accompanies the electron. This wave, represented by the Greek letter psi, gave breathtakingly precise predictions for the behavior of atoms which sparked a revolution in physics. Suddenly, amost from first principles, one could peer inside the atom itself to claculate how electrons danced in tehir orbits, making transitions and bonding atoms together in molecules.

Elements of the Schrodinger Equation

There are various forms of the Schrodinger equation. Even when majoring in physics, it takes several years of study before they are understood, but you can use them to begin to make sense of what the equation is supposed to mean:

  • h-bar: Planck's constant h divided by 2π
  • m: mass of the object
  • Ψ: the Greek letter psi (pronounced "sigh") wave function (which represents the probability of the quantum system being in a given state)
  • i: the square root of -1, called an imaginary number
  • ∇: the Laplacian differential operator
  • V: the potential energy of the particle
  • H: the Hamiltonian operator on the wave function
  • E: the energy of the state

Schrodinger's Equation and Quantum Field Theory

In his book The Elegant Universe, physicist and author Brian Greene describes how Schrodinger's initial formulation fell short of a full formulation of quantum physics, especially in research conducted in the 1930s or 1940s:

[Physicists] found that Schrodinger's quantum wave equation [...] was actually only an approximate description of microscopic physics--an approximation that works extremely well when one does not probe too deeply into the microscopic frenzy (either experimentally or theoretically), but that certainly fails if one does.

The central piece of physics that Schrodinger ignored in his formulation of quantum mechanics is special relativity. In fact, Schrodinger did try to incorporate special relativity initially, but the quantum equation to which this led him made predictions that proved to be at odds with experimental measurements of hydrogen. This inspired Schrodinger to adopt the time-honored tradition in physics of divide and conquer: Rather than trying, through one leap, to incorporate all we know about the physical universe in developing a new theory, it is often far more profitable to take many small steps that sequentially include the newest discoveries from the forefront of research. Schrodinger sought and found a mathematical framework encompassing the experimentally discovered wave-particle duality, but he did not, at that early stage of understanding, incorporate special relativity.

The effort to expand Schrodinger's approach to include special relativity resulted in quantum electrodynamics, which is part of a more comprehensive class of theories called a quantum field theory.

Edited by Anne Marie Helmenstine, Ph.D.