How Schrödinger's 1926 Paper Revolutionized Quantum Physics
Quantization is the quantum principle that subatomic particles must have specific discrete energy levels.
Quantisierung als Eigenwertproblem (“Quantization as an Eigenvalue Problem”) by Erwin Schrodinger was a research paper published in the year 1926. This paper contributed to many concepts within Quantum Physics. His goal in writing the paper was to understand the behavior of subatomic particles like electrons inside of atoms.
Quantization and Eigenvalue Problems
Quantization is the quantum principle that subatomic particles must have specific discrete energy levels. An Eigenvalue Problem involves finding Eigenvalues (special number represented by λ) and Eigenvectors (vectors, represented by x) that satisfy an equation for a square matrix. This would be abbreviated as Ax = λx where A is the square matrix. Quantization as an Eigenvalue Problem explains 3 main concepts, wave-particle duality, the theory of quanta, and the Schrodinger equation.
Wave-particle Duality
Wave-particle duality is a phenomenon where subatomic particles not only act as particles, but they also act as waves. For example, light can be in the form of particles (photons) and in the form of waves, spreading out. Particles are discrete, having a specific position and momentum at a given time. Waves on the other hand, do not have a specific position and momentum, instead they are more continuous. Louis deBroglie (a French physicist) first proposed the theory of wave-particle duality, and Schrodinger provided the mathematical explanation behind his proposals. In this research paper, Schrodinger thought about deBroglie’s idea of wave-particle duality, and proposed that particles act not only like particles, but also like waves and provided his own mathematical foundation for it.
The Theory of Quanta
The theory of quanta, explains that subatomic particles must have specific, discrete energy levels that are deterministic. These energy levels must be whole numbers. The theory of quanta was first proposed by Max Planck in the year 1900. He was attempting to understand black body radiation (the radiation that a blackbody, or a perfect absorber of all electromagnetic radiation gives off when it’s temperature is high) and proposed that energy is only emitted and absorbed in specific, discrete packets called quanta. Then, Niels Bohr (a Danish physicist) proposed the theory that the electrons in atoms can only have discrete, specific energy levels. Schrodinger, building off of the theory made by Niels Bohr, provided a mathematical model for discrete energy levels occupied by the electrons in an atom.
Schrodinger’s Wave Equation
Schrodinger’s paper had one major purpose - that was to create a new field in quantum theory, called “Wave Mechanics”. His paper proposed the wave equation, which calculated the allowed energy levels of electrons inside of an atom. The Schrodinger equation explains how the wave function of a quantum system changes over time. A wave function is the characteristic that describes the probability of a subatomic particle being in a specific place. If information about a particular atom - such as the amount of protons and electrons inside of it is input into the equation, the equation will help find which energy levels are allowed for electrons within that specific atom. There are two types of Schrodinger Equations - the Time Dependent and Time Independent.
Time Dependent Schrodinger Equation:
The Time Dependent Schrodinger Equation shows how the wave function changes over time. This equation helps understand the possibilities of where the subatomic particle might be. If it is known that the particle is at a certain place now, then the equation can calculate a possible place it might be and the probability of finding it in that place a few seconds later.
Time Independent Schrodinger Equation:
The Time Independent Schrodinger Equation is a special case of the Time Dependent Schrodinger Equation, where “time” is not one of the variables. This equation is applied when the particle is in a completely steady and unchanging environment. An example of a particle in a steady environment is an electron orbiting the nucleus inside of a stable atom.
Quantization as an Eigenvalue Problem by Erwin Schrodinger has done a great deal to quantum physics. His wave equation provides a mathematical foundation to the changing of wave functions, the 3 key concepts he proposed in his paper enhanced other physicists' understandings and viewpoints, and his paper in general, revolutionized the ideas of quanta and wave particle duality.