Schrodinger Equation – A Non-Mathematical Insight For Everyone

Before the 1920s it was believed that particle and wave was completely two different aspects of physics, one was completely independent of the other. This is the aspect of classical mechanics. But when wave nature of light failed to explain phenomenon like compton scattering and photoelectric effect, Planck came up with his hypothesis and to explain black body radiations, quantized energy, and Einstein successfully extended the idea to light and said that even light and all other radiations were quantized, and this particle of light was called a photon. When something is quantized, it means that it is allowed to have only certain discreet values, and so this idea gave birth to the science of quantum mechanics, where quantum means a quantized or discrete characteristic. The discrete particles of light are photons. Thus, in quantum mechanics, wave and particle nature are complimentary, and the distinction between them is blurred, unlike in classical mechanics.

DeBroglie extended this notion of dual character of light to everyday objects. He said that every body in this universe can act like a wave, and also particle, but as the size increases the wave nature becomes less noticeable (because the wavelength of the wave is inversely proportional to mass, so bigger the mass, smaller the wavelength). This gave birth o the dual property of objects. Young’s double slit experiment using electron beam proved this idea to be correct. When a beam of electrons was passed through two slits, with width comparable to the wavelength of electrons, on a screen opposite to the slits one could observe an interference patter, alternate regions of dark and bright bands, which is a characteristic of a wave. Thus electrons also behaved as wave, and had a dual nature.

Hydrogen atom wave function. It shows the chances of finding a electron around the nucleus. White: Most chance; Blue : Negligible chance

When Erwin Schrodinger in a lecture said that every particle can thus have a wave associated with it, he was challenged by Paul Dirac to show what the equation was. To achieve this Schrodinger took a break, and went on a vacation, to relieve himself from all duties and dedicate all this time to finding a wave equation. The result was the celebrated Schrodinger Equation.

The schrodinger equation when solved had solutions which are called the wave function of the particles, and once this wave function is known, one can know all the properties and information related to the particle for all times. The wave function as defined by Born, is the probability wave of the particle. This probability wave tells the chances of finding that particle at a certain position and time, higher the amplitude of the wave at a certain point, higher is the probability of finding that wave there.

“If quantum mechanics hasn’t profoundly shocked you yet, you haven’t understood it yet” -Neils Bohr

Now there is a very interesting property of this wave function obtained from the schrodinger equation. Since this wave function is actually a probability wave, it tells us the probability of different configurations of the particle. It is a superposition(or in simple a sum) of different possible events related to the particle with different chances of occurring, all put together into one wave function. Depending on this probability we can tell the chances of something happening with the particle. For example in the double slit experiment above, the electron behaved as a wave, there was a probability that the electron could pass through any of the two slits, so the wave function of the electron is a sum of two probabilities, one in which it passes through slit 1 and one in which it passes through slit 2, and since both can happen, the electron’s wave function actually passes through BOTH the slits, and thus we are given with an interference pattern on a screen (which can be formed only with two waves in case of two slits). So even if one electron was made to enter the slits at a time, even then it passed through both the slits, and made a pattern. This was a remarkable demonstration of the wave nature of electrons.

Similarly in the schrodinger’s cat paradox, a cat’s wave function is made of both the probabilities of t being alive and dead, both of which exist simultaneously.  Now an important aspect of quantum mechanics is that the wave function remains a superposition of probabilities until and unless and observation is made. In case of the double slit experiment, if you try to detect the electron to see which slit it passes through, you have in a way ‘seen’ what of the two possible events has happened. Like when you roll a dice you have an equal probability of getting any number from 1 to six, but you can finally get only one of them, similarly in the above case, though there was chances of electrons passing through both the slits, once you have observed it passing through a slit, it cannot pass through the other. Observation of an event kills the probabilities of all other events to zero. Similarly, until you see, the cat can be alive or dead, but once you have seen what it is, its wave function collapses to that particular state, and probability of other states is destroyed.

Similarly, until observed every particle can be at any position in space. An electron can be at two positions simultaneously. This is why actually atoms are stable, because if it were not so, the electrons in the atoms would have collided which would destroy the atom, but this property of being in two positions at once offers a stable mechanism, and has been observed indirectly. Another weird property with the wave function obtained from Schrodinger’s equation is quantum tunneling. When a particle is near a thin wall, classical mechanics would say that it is impossible to magically appear on the other side, but quantum mechanics  says that since the probability wave of the particle determines its position, if the wave has a non zero value for the particle at the other side of the wall, there is a chance that it will magically appear on the other side, as if it has tunneled through the wall. This property is extensively used in Scanning Tunneling Microscope (which one Nobel Prize to IBM), and also in tunnel diode, an electrical component.

“Quantum mechanics explain nature as absurd from point of view of commonsense. And yet it fully agrees with experiment.So I hope you can accept Nature as she is – absurd.”       -Richard Feynman

Since knowing the wave function, and thus the probability wave, tells you everything you need to know about all the possible characteristics, events, and configurations related to the particle, in principle you can derive all of Chemistry from the knowledge of this one equation. Since Chemistry is nothing but the reactions involved between atoms, molecules, and their stability, knowledge of their wave function will tell you all about their possible reactions and properties, since they are so small that the rules of quantum mechanics cannot be ignored, they come under the realm of schrodinger’s equation. So, a physicist, smart enough to know every property from the wave function by solving the Schrodinger’s Equation, can, in principle, put chemists out of business. Everything that is known and is to be known in the field of chemistry lies in that one equation.

“Quantum Mechanics has explained all of chemistry and most of physics” – Paul Dirac

There is no end to the weirdness of quantum mechanics brought out by the schrodinger equations’ wave function. If you could know the wave functions related to every single atom in your body, in principle you could use this information to teleport yourself to some other location, or know everything about yourself that MAY happen. How? Well if you were able to perform this herculean task of obtaining all the wave functions of all the atoms in your body, your resulting wave function would basically look like you, your physical features, but apart from that there would be waves, or more precisely, probability waves coming out of different part of you extending infinitely into the space around you. These waves would have very low amplitude, but since it will be zero only at infinite distance, the amplitude would have a finite value. This means no matter how small, there is a chance of you suddenly appearing at some other place, like electrons being in two places at once, or your car could suddenly reappear outside the closed garage, like quantum tunneling in electrons. The chances though, of these events happening is so small that they can happen once in a time longer than the age of the universe. As the mass increases, wave property diminishes, and quantum mechanics rules have negligible effect. The more the number of atoms or particles in a system(like your body or your car), lesser is the observable effects of  application of quantum mechanics to their resulting wave function. But nevertheless, since there is a finite probability, no matter how tiny, if anything has a chance of happening, it can happen, so it might happen that some day you might wake up on Venus (given that your probability wave is not zero there), though this would be disastrous for you to wake up on a planet with sulfuric acid rains, and crushing pressures. Or you might wake up to see your car is standing outside your garage.


Strange things are bound to happen in quantum mechanics, but no matter how much we neglect the possibilities predicted by it, they have been observed in the laboratory, and used to make advanced machines around us. May be this is because our logic, is actually a result of intuition, which is a result of our experiences in a world around us where classical mechanics rules, which prevents such weird stuff for happening. But in the atomic world, the scene is different, quantum mechanics sets the laws there, and weird things that go against are intuition are bound to be observed. Quantum effects are negligible in the scale we experience reality, and this is why we don’t believe them , because we cannot realise them(chances are as I said too low). Schrodinger equation, like Newton’s laws of motion, and laws of thermodynamics, cannot be proved or derived using existing laws, its validation and accuracy depends on experiments and observations, which exhibit its capabilities to a remarkable accuracy. This post was just a drop in the sea of quantum mechanics and schrodinger equation. To help make it familiar for everyone I restrained from mathematical background, but you can always know more by studying more about it. Happy Reading!

“Quantum mechanics makes absolutely no sense.” -Roger Penrose

-The Cosmogasmic Person

More posts you might want to read after this:

Reality Check (1/2) – Many Worlds Theory

Why I Think Travelling To The Future Is Even Theoretically Impossible!

Introducing: Time Crystals; The matter that is stable when it’s in motion. Stranger than fiction!












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