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Schrödinger Equation Explained

Schrödinger equation is the heart of quantum mechanics. When physicists did not pay attention so much on quantum mechanics then nobody could ever imagined about dual properties of particles. But, when one of the greatest physicists Erwin Schrödinger showed with a wave equation in 1926, the physicists started to do research on dual behavior of particles. Schrödinger was the first person who wrote down such a wave equation which described about weird behavior of particles.

Schrödinger equation is a mathematical equation describing the probable distribution of a particle around its nucleus. That means it tells us the probable positions in orbitals of a particle revolving around its nucleus. It was named after Erwin Schrödinger who proposed this wave equation at the very first time.

Mathematical expression of Schrödinger equation:

The following equation is a mathematical expression for Schrödinger equation-

Ə2ψ/Əx2+Ə2ψ/Əy2+Ə2ψ/Əz2+(8π2m)/h2(E-V)ψ=0

Here,   Ə2/Əx2,Ə2/Əy2 and Ə2/Əz2 represent rate of change with respect to its coordinates.

             x, y andare the three-dimensional coordinates.

             ψ is the amplitude of the wave which is universally known as Wave function.

             m is the mass of the particle.

              h is Planck's constant which is 6.62607004 × 10-34 m2 kg/s.

              E is the total energy of the particle; is the potential energy (difference between total energy and potential    energy will give kinetic energy of the particle. That means E-V= kinetic energy of the particle).

The above equation can be simplified to the following equation -

2 ψ + (8π2m)/h2 (E-V) ψ = 0

Where, ∇ = Ə2/Əx2+ Ə2/Əy22/Əz2 

The final result of the wave equation gives a set of numbers actually called Quantum numbers, which describe about energy and orbital state; also the shape and probable distribution of the particle.

The Schrödinger equation can be compared with Newton's second law of motions. In classical mechanics, using Newton's second law of motions, i.e F=m.a,we can calculate the momentum and position of a physical system with respect to an external force F.

Similarly, by using Schrödinger equation in quantum mechanics, we can calculate the probable distribution of a particle in orbitals of nucleus. The main difference between Newton's second law and Schrödinger equation is that, Newton's laws can only be applied in bigger world and hence easily understandable. Whereas, Schrödinger equation can only be applied in atomic scale, i.e in quantum world which makes it quite difficult to understand.

Concept of wave function:


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Wave function is a mathematical description of different possible states of a particle in quantum systems. It is represented by Greek letter psi (ψ).

Wave function is a complex valued probability amplitude (probability amplitude is a complex number in quantum mechanics which is used to describe the behavior of a quantum system) and with the help of it, physicists can predict possible results of the quantum system.

During the quantum superposition state, a system exists with it's all possible states at different positions at the same time. When some observation is made, then the wave function is collapsed and we get a specific outcome. Heisenberg's uncertainty principle also predicts that we cannot measure both position and momentum of a particle at the same time. If we predict the position, then momentum cannot be predicted and vice versa. Thus, we are quite uncertain about the behavior of the particles.
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Schrödinger equation not only describe the behavior of subatomic particles, molecules or atoms but also the macroscopic systems (visible systems with naked eye). Thus, this equation can also be used to study the whole universe. As, the universe is also made of such tiny particles, hence by studying quantum mechanics, a physicist can predict some possible explanation of our universe.

Schrödinger cat experiment:


Image result for cat experiment

For defining the wave function and superposition state, Erwin Schrödinger proposed a different thought experiment in 1935. Physicist named it as Schrödinger cat experiment. In that experiment, he imagined a cat surrounded by a close box. Inside the box, there was a liquid poison in where a hammer was attached on it. If the cat touches the hammer, it will break the poison container and it will spread inside the box. At a result the cat will die. But if the cat does not touch the hammer, then nothing will happen and the cat will remain alive.

Now after that arrangement, can we say whether the cat is dead or alive without opening the box? No, we cannot. In this case, the cat will exist in both dead or alive state. That means, the cat is in superposition state by obtaining two possibilities; dead or alive. But when we open the box, that means we make a observation, the wave function collapses and we get a definite result, either dead or alive. Same thing happens with quantum particles. When we do not observe them they act like waves and when we observe them they act like particles.

About quantum mechanics:

As we know, theory of relativity and quantum mechanics are not best friends because they violate the principles of each other. So, combining the both theories are quite risky. They are like Tom and Jerry who always fight when they come closer to each other. However, physicists are now trying to make a Unified field theory which would be the The theory of everything .

If one can understand the whole quantum mechanics, then he will be able to understand the whole properties of the universe. Because, quantum mechanics studies the whole universe in a smaller scale. It describes the weird properties of particles. Delayed Choice Quantum Eraser experiment is an example, based on wave particle duality and quantum entanglement.

Greatest physicist Richard Feynman once said," I think I can safely say that nobody understands quantum mechanics." In some sense it is true, because nobody could ever understands completely the weird behaviors of quantum particles. We study about their behaviors, but nobody knows why they act like this.

Erwin Schrödinger was awarded with Nobel Prize in 1933 due to his great equation. Now his equation is the CPU in quantum mechanics and every quantum physicist remembers this equation.

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2 Comments

  1. The article was very informative up until the Schrödinger cat experiment, which was historically inaccurate. Schrödinger used the cat as a way of describing how absurd the Copenhagen Interpretation is and it was developed by Niels Bohr (whom also developed a now debunked model of the Atom). The Schrödinger Equation and the Copenhagen Interpretation are two completely different things.


    The equation has been scientifically tested and remains one of the most successful theories in the history of science that became the foundation of all Quantum Mechanical Theories to come. That is not in dispute.


    The interpretation did not even have a formula nor has it ever been formally tested. Several known phenomena like photons not oscillating, directly contradict the concept of Superposition of particles. Many experiments like the Bell Test do not test even test the principles of the interpretation, they only verify the randomness of particles interactions, not if they are in all states at the same time. In fact the only thing scientific about the interpretation is that is acceptable to use “Fudge Factors” in certain situations.


    It is time we get beyond the crutch of saying “It is what it is” or using the word superposition in physics and develop real theories based upon real phenomenon so that we can develop experiments test those concepts.

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  2. Thanks for the comment. Yea I know that Copenhagen interpretation and schrodinger equation are two completely different thing. But, the reference of cat experiment has been used for just describing the principle of wave function. I have not discussed it completely. Just a reference for understanding the topic clearly. Keep visiting. Have a good day..

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