Making Waves: The Schrodinger Equation

Erwin Schrodinger (1887-1961) was a titan of theoretical physics who died exactly 60 years ago, leaving behind an equation named after him that has become a pillar of quantum mechanics, the branch of physics that studies the behavior of particles (including light) at the sub-atomic level.

At the sub-atomic level, things are not what one would normally expect to see. Thus, a particle which is “there” doesn’t really exist until it is observed. Moreover, a particle is not always a point-like object; it can also be a wave (the famous wave-particle duality aspect of quantum mechanics). Even weirder, everything in quantum mechanics is a matter of probability, never certainty. Thus, one can only make statements like: “this equation tells us what the chances are of actually observing what was observed”.   All these weirdness is captured succinctly by the Schrodinger’s equation, named after the eminent Austrian physicist.

Erwin Schrodinger (1887 – 1961)

A Bit of History

Quantum mechanics was still in its infancy when Schrodinger was a young physicist at the University of Zurich. In 1924, Louis de Broglie postulated that not just light, but matter, exhibited particle-wave duality. Inspired by de Broglie’s work, Schrodinger started to model the motion of an electron around its nucleus as a wave rather than an orbiting particle. He soon hit a mental road block and in late 1925, he decided to take a break in the secluded Swiss mountain resort (accompanied by his mistress). Amazingly, by January 1926, he cracked the problem with an elegant equation that describes the evolution of a hydrogen-like atom. Schrodinger’s wave equation appeared in volume 28, number 6 of the Physical Review with the somewhat dull title, “An Undulatory Theory of the Mechanics of Atoms and Molecules.”

Birth of an Equation

Here is a compact form of this famous equation:

The leading “actor” in Schrodinger’s equation is \Psi ,the wave function defined over times and location.  Quantum mechanics says that when you observe a particle, you sort of “collapse” the wave function, turning it into a statement of the particle at that location. Messing up this neat interpretation, however, is the role of chance; you only know the probability that you were able to observe the particle at that moment. Seen from this viewpoint, the wave function is like a statistical distribution, that is, a distribution of probabilities, where what you observe depends on how nature spins an invisible roulette wheel as it were. 

Looking at the left side of the equation again, what we have is the rate of change of the wave function multiplied by a constant i times h-bar, where i denotes the complex number equals to the square root of -1, and h-bar is Dirac’s constant, which in turn is equal to Plank’s constant (h) divided by 2π. Ignoring the constant term, the left side of the equation is roughly the “velocity” of the wave function, that records how fast this function evolves over time.

What determines this “velocity”? The right-hand side of the equation gives the explanation. It says that two things influence the wave velocity: the kinetic and potential energy of the wave-particle.

The kinetic energy is captured by the first term on the right:

where m is the mass of the particle, and square of the inverted triangle term is the Laplacian of the wave function (roughly speaking, the acceleration or the rate at which velocity of changing)

The potential energy of the particle is measured by V (the second term on the right-hand side of the equation). Like the wave function itself, V is a function of time and space. Potential energy arises because no particle lives in isolation but is surrounded by an electromagnetic field, and V represents the external effect of the electromagnetic field that the particle is in.

We can simplify the right-hand side by factoring the wave function outside the kinetic and potential energy like this:

Doing so simplifies the interpretation of Schrodinger’s equation. The term inside the big bracket is called the Hamiltonian. Therefore, Schrodinger’s equation says that kinetic and potential energies are transformed into the Hamiltonian which acts upon the wave function to move it in time and space.

Summing Up

Ultimately, the Schrodinger equation is fundamentally important not only because it captures the mysterious probabilistic particle-wave essence of quantum mechanics (which in turn revolutionized the science of semiconductors, lasers, superconductors, quantum computers, nano technology and many exotic materials), it also beautifully complements the achievements of classical mechanics. Just as Issac Newton ushered in a new era in physics when he devised his universal law of gravity and equations of motion, three centuries later, Erwin Schrodinger made a similar contribution with an equation that is the quantum equivalent to the classical laws of motion and conservation of energy in classical physics.

Further reading:

Farmelo, Graham (ed), It Must be Beautiful: Great Equations of Modern Science, London: Granta Publications, 2002.

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