## Introduction

If we extend classical Physics laws to macroscopic systems, we strive to characterize just some broad characteristics of the system’s behavior. For example, we address the movement of a rigid body in its entirety but don’t really analyze the movement of all of the body’s essential elements, but quantum mechanics is different.

This is a characteristic of classical physics as applicable to macroscopic systems. The finer intricacies of the system’s behavior are overlooked, and no attempt is made to evaluate all elements of the problem. In this respect, classical physics rules are approximate natural laws.

## What is Quantum Mechanics required?

More knowledge on atoms, nuclei, and numerous elementary particles that comprise the tiny world of macroscopic particles was gained during the start of the twentieth century. The new feature of nature discovered while studying those macroscopic particles is characterized as “Quantum Phenomena,” and the basic mathematical concept is defined as “Quantum Mechanics.”

As a result, we may claim that Classical Mechanics explains macroscopic systems whilst Quantum Mechanics covers tiny ones. The type of confidence about the future that is characteristic of classical mechanics is unachievable in quantum mechanics since the initial state of a particle itself cannot be determined with sufficient accuracy (uncertainty relation). This is because quantum mechanics exclusively investigate probability.

Classical mechanics’ certainties are misleading or false, and their conformity to experiment happens because macroscopic things are made up of a large number of individual atoms that deviation or deviation from average behavior is undetectable.

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As a result, we may conclude that classical mechanics is merely an estimate of quantum mechanics, and quantum mechanics rules are the most universal laws of nature. When applied to macroscopic systems, the results are nearly comparable to that of classical mechanics.

## de Broglie

As a result, classical mechanics approach quantum mechanics. Whenever the de Broglie wavelength is tiny in comparison to the system dimensions, the two techniques merge. In an attempt to elucidate the black body spectra, Max Planck created the notion of the quantum hypothesis” in 1900.

This quantum nature was exploited by Einstein in the theory of specific heat of solids and the photoelectric phenomenon. Compton scattering later demonstrated the dual nature of light. Meanwhile, Bohr exploited this (quantum) condition to describe hydrogen spectra. All of these improvements occurred between the turn of the century and 1924.

## Matter or Wave in Quantum Mechanics?

The use of a quantum condition as a hypothesis is sometimes referred to as ‘old quantum theory.’ De Broglie introduced the concept of matter waves in 1924, and it was confirmed experimentally by Davison Germer and G P Thomson.

Since then, de Broglie, Schrodinger, Bohr, Heisenberg, Dirac, and others have demonstrated that if matter waves are regarded as standing waves, the quantum conditions that are critical to explaining many experimental facts will automatically follow from the theory. This mathematical framework, which was established between 1924 and 1926, is known as ‘Wave mechanics’ or ‘quantum mechanics.’

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## Wave Function in Quantum Mechanics

The height of the water surface fluctuates frequently in water waves, while the pressure varies periodically in sound waves. Electric and magnetic fields change with distance and time in electromagnetic waves. In quantum mechanics, the quantity whose variations makeup matter waves is called a wave function. Usually, it is represented by the Greek letter ψ (psi).

The value of the wave function associated with a moving body at a particular point x, y, and z in space at the time ‘t’ is related to the likelihood of finding the body there at that time. A wave function may have either positive or negative amplitude and the negative amplitude of a matter wave has no meaning. Hence the wave function ψ itself is said to have no physical significance.

## Probability Density in Quantum Mechanics

In optics and in sound, the square of the amplitude represents the intensity of light and sound. Similarly, the probability of experimentally finding the body described by the wave function at the points x, y, and z at the time is proportional to the value of |ψ| there at time t. The square of the absolute value of the wave function which is always positive is called ‘probability density.

A large value of the probability density |ψ| means the strong Possibility of the presence of the body and a small value of |ψ| means the slight possibility of its presence. This interpretation was first made by Max Bohr in 1926. Momentum angular momentum and energy are the physical quantities that can be established from ψ. The real problem of quantum, mechanics is to determine ψ for a body when its freedom of motion is limited by the action of external force.

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## Schrodinger’s Equation in Quantum Mechanics

Since a particle in motion has wave properties, there should be some sort of wave equation that describes the state of the particle in motion.

The equation must have two basic properties: It must be linear so that solution of the equation can be supposed to produce interference effects as in the Davisson-Germer experiment and, the coefficients of the equation must involve only constants such as h,m,e, etc and should not involve parameters of a particular kind of motion such as momentum, energy, etc.

In 1926, Schrodinger developed a second-order equation, which satisfies the above condition considering the wave function for a nonrelativistic particle. It is called the Schrodinger equation. One can get solutions for other quantum mechanical problems by applying appropriate boundary conditions. The solution yields the wave functions and hence one can calculate all measurable features of the system.

If these theoretical values agree with the experiment, then the postulate embodied in the Schrodinger equation is valid. If the theoretical value disagrees with the experimental result, then the postulate itself is discarded and some other approach is explored. It means the Schrodinger equation cannot be derived from other first principles, but it represents the first principle by itself. In other words, we can say Schrodinger Equation is really a postulate and it cannot be derived from anything else.

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