Schrödinger equation
In December of 1925, Erwin Schrödinger secluded himself in a mountain cabin with a mistress. He had been working on a relativistic wave equation that produced results disagreeing with Arnold Sommerfeld's formula for hydrogen spectral energies. Discouraged by these failures, he set aside the complex calculations and turned his attention to nonrelativistic approximations instead. This decision led him to publish a paper in 1926 that accurately reproduced the energy levels of the Bohr model. The equation he found was based on Louis de Broglie's hypothesis that all matter has an associated matter wave. Schrödinger used William Rowan Hamilton's analogy between mechanics and optics as his primary guide. He sought a proper three-dimensional wave equation for the electron after Peter Debye made an offhand comment about particles behaving as waves. His work formed the basis for the Nobel Prize in Physics awarded to him in 1933.
The Schrödinger equation is a linear differential equation meaning any linear combination of solutions remains a valid solution. If two state vectors are solutions, then so is their sum multiplied by any complex numbers. This property allows superpositions of quantum states to exist within the framework. A general solution can be found by taking a weighted sum over a basis of states. Physicists often choose the basis of energy eigenstates which are solutions of the time-independent version. The time-evolution operator preserves the inner product between vectors in the Hilbert space. This preservation is known as unitarity and ensures probability conservation during time evolution. If the initial state is normalized, the state at a later time will also remain normalized. The generator of this family of operators is a self-adjoint operator called the Hamiltonian. It acts as the engine driving the continuous change of the system over time.
A particle confined inside a one-dimensional potential energy box demonstrates how restraints lead to quantized energy levels. The box has zero potential energy inside but infinite potential energy outside its boundaries. The wave function must equal zero at the walls of the well because it cannot penetrate the infinite barrier. This constraint forces the wavelength to fit an integer multiple of half-wavelengths across the width. Consequently, only specific discrete values for energy are allowed for the particle. Another example involves the harmonic oscillator where a ball attached to a spring oscillates back and forth. Solutions to the Schrödinger equation for this situation involve Hermite polynomials of order n. The ground state energy is non-zero and is called the zero-point energy. The wave function for this lowest state takes the shape of a Gaussian curve. These examples illustrate that energies of bound eigenstates are discretized rather than continuous.
The original Schrödinger equation applies essentially in the nonrelativistic domain where Galilean transformations hold true. Processes that change particle number are natural in relativity so a single-particle equation has limited use there. Attempts to combine quantum physics with special relativity began by building equations from the relativistic energy-momentum relation. The Klein-Gordon equation was the first such result even before the nonrelativistic one-particle version existed. It applies to massive spinless particles but initially led to problems with negative probability density. Paul Dirac obtained his equation by seeking a differential form first-order in both time and space. He took the square root of the left-hand side of the Klein-Gordon equation using 4x4 matrices. This required factorizing it into a product of two operators known as Dirac gamma matrices. The resulting wave function became a four-component spinor field describing both particle and antiparticle states.
Max Born successfully interpreted the modulus squared of the wave function as equal to probability density just days after Schrödinger published his initial paper. In the Copenhagen interpretation, a system's wave function is viewed as a collection of statistical information about that system. While the evolution process represented by the equation is continuous and deterministic, measurements cause discontinuous stochastic changes. Other interpretations like relational quantum mechanics or QBism give the equation similar status regarding information availability. Schrödinger himself suggested in 1952 that different terms of a superposition are not alternatives but all really happen simultaneously. This view has been interpreted as an early version of Everett's many-worlds interpretation formulated independently in 1956. Bohmian mechanics reformulates quantum mechanics to make it deterministic at the price of adding a force due to a quantum potential. It attributes to each physical system not only a wave function but also a real position evolving under a nonlocal guiding equation.
Common questions
When did Erwin Schrödinger publish the paper containing his wave equation?
Erwin Schrödinger published the paper in 1926. He had secluded himself in a mountain cabin with a mistress in December of 1925 to work on the equation.
What hypothesis about matter influenced the development of the Schrödinger equation?
The Schrödinger equation is based on Louis de Broglie's hypothesis that all matter has an associated matter wave. Peter Debye also made an offhand comment suggesting particles behave as waves which prompted Schrödinger to seek a three-dimensional wave equation for the electron.
How does the Schrödinger equation explain quantized energy levels in a potential box?
A particle confined inside a one-dimensional potential energy box demonstrates how restraints lead to quantized energy levels. The wave function must equal zero at the walls because it cannot penetrate the infinite barrier, forcing the wavelength to fit an integer multiple of half-wavelengths across the width.
Why was the original Schrödinger equation limited to nonrelativistic domains?
The original Schrödinger equation applies essentially in the nonrelativistic domain where Galilean transformations hold true. Processes that change particle number are natural in relativity so a single-particle equation has limited use there compared to equations like the Klein-Gordon or Dirac equation.
Who interpreted the modulus squared of the wave function as probability density and when did this occur?
Max Born successfully interpreted the modulus squared of the wave function as equal to probability density just days after Schrödinger published his initial paper. This interpretation established the statistical nature of quantum mechanics within the Copenhagen interpretation framework.