Erwin Schrödinger published his equation in 1926, and it changed the field of quantum physics. Up to that time, physicists had developed limited principles governing the behavior of subatomic particles (that is, quantum particles). But physicists could describe little of this behavior mathematically. For example, they could not calculate: If an electron gun shoots an electron through two slits towards a detection screen, where will it land on the detection screen? This is the famous double slit experiment.

*Double slit experiment with electrons*

This inability to calculate outcomes of experiments with equations is similar to the situation of classical physics before Isaac Newton. Before Newton, physicists knew such principles as: if you drop two balls of different weights out a window, they fall at the same speed. But physicists couldn’t describe, for example, this experiment mathematically: If a player hits a billiard ball at Point A with a cue stick at Angle B with Force C, where will the billiard ball land?

*Perfectly Elastic Collision*

After Newton developed his Laws of Motion, physicists could answer this question. They could measure all the initial conditions of the billiard ball, take into account the force and angle of the cue stick, and calculate, using Newton’s Laws, where the ball would land. The field of physics governed by Newton’s Laws of Motion became known as mechanics.

Schrodinger played a similar role to that of Newton in evolving the field of quantum physics into the field of quantum mechanics (with a couple of big caveats to this statement). Schrodinger made it possible to measure the initial properties of quantum particles and calculate what would happen next. For example, he made it possible to calculate where particles would land if shot from an electron gun through two slits towards a detection screen.

*One of several possible presentations of Schrodinger’s Equation*

Here’s one of the caveats to comparing Schrodinger to Newton: Whereas many physicists contributed to the steps which led to Schrodinger’s Equation, Newton worked largely alone. And when Newton used the paradigm of describing physical motions with mathematical equations, he was pioneering a new way of doing physics. By the time Schrodinger wrote his equation, describing physical interactions with mathematical equations was expected.

Months prior to publication of Schrodinger’s Equation, Werner Heisenberg had published an alternative way to do the calculations that Schrodinger’s Equation made possible. However, Heisenberg’s approach involved non-intuitive math (vector mechanics) that was unfamiliar to physicists at the time. Due to the non-intuitive math of Heisenberg’s approach, even today, when students learn quantum mechanics, they start by learning Schrodinger’s Equation.

Another caveat to the comparison of Schrodinger’s Equation and Newton’s Laws of Motion: In the double slit experiment, Schrodinger’s equation can be used to calculate only the probabilities as to where the electrons will land on a detection screen. The particle could land anywhere on the screen, and maybe even off the screen; but the probabilities are higher in some places than others. With Newton’s Laws of Motion, we can predict exactly where a billiard ball will land. Newton’s Laws are deterministic not probabilistic: do this and you get that.

This difference between quantum mechanics and Newton’s mechanics is due to the difference between the quantum world and the classical world of tables and chairs. In the quantum world, a specific electron can behave randomly. However, when large numbers of electrons are involved, Schrodinger’s Equation can calculate the probabilities and accurately calculate the proportion that will land in various positions.

Schrodinger’s Equation can be used to calculate the outcomes of many different types of experiments, not just the double slit experiment. However, it is limited to calculations of the behavior of electrons and other particles when they are relatively slow-moving. Slow-moving, here, means traveling slower than a significant fraction of the speed of light.

It is also limited to calculations involving particles with relatively low energy levels. Later equations by Paul Dirac, Richard Feynman, and other physicists were needed for calculations involving fast-moving particles like photons or high-energy particle interactions like in particle accelerators. These later equations can take into account the effects of Special Relativity and other factors.

Schrödinger’s Equation along with Heisenberg’s vector mechanics are important because they moved quantum physics into a new stage: quantum mechanics. With quantum mechanics, physicists could describe quantum behavior with mathematical precision. Physicists were more able to make predictions as to outcomes of quantum experiments. Schrodinger’s Equation continues to be used to describe and predict the behavior of electrons and other particles when relatively slow-moving and low-energy.

Now, before ending, again, we have to keep in mind some importance of this equation.

- The electronic structure of atoms and molecules can be well explained using Schrodinger’s equation. The shape of orbitals and their orientations can be described and is documented well in the literature as well as in standard quantum physics textbooks.
- The solution of Schrodinger’s equation results in quantized properties of the quantum particles involved. The quantization is the consequence of periodic boundary conditions involved. Today we know that the energy quantization is playing a key role in fabricating various quantum devices such as laser diodes and light emitting diodes.
- Solving Schrodinger’s equation for a solid is a many-body problem and it involves the solution of complex differential equations. The problem can be worked by expressing the differential Schrodinger’s equation in momentum space/reciprocal space. In momentum space, the differential Schrodinger’s equation can be expressed as a set of linear algebraic equations rather than complex differential equations. Those linear equations can be solved using available computational numerical techniques accurately to predict the ground state properties of solids.
- The solution of Schrodinger’s equation in reciprocal space results in the electronic band structure of solids. The classification of solids (metals, semiconductors, and insulators) can be well explained using band theory of solids, which is just a consequence of the solution of the Schrodinger's equation in momentum space.
- Most of the semiconductor properties are well explained from the consequences of Schrodinger’s equations, such as the origin of the band gap, the behavior of dopants and their energy levels present in relation to the observed band structure etc.
- Today it is possible to obtain the suitable dopants for a given semiconductor just by solving the corresponding Schrodinger’s equation, before conducting experiments.
- The real materials are impure, they contain various impurities as well as crystalline defects. The ground state properties of these defects can be well described by solving the Schrodinger’s equation.
- Performing calculations based on Schrodinger's equation for a given system commonly referred as ab-initio calculations or first-principles calculations. Because the equation works based on various forces present between the atoms and the electrons present in the system, without any experimental inputs.

There are still many applications are there to explain the importance of Schrodinger’s equation. That's it.😊

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