Introduction At the beginning of the twentieth century, centur y, experimental evidence suggested that atomic particles were also wave-like in nature. For example, ex ample, electrons were found to give diffraction patterns when passed through a double d ouble slit in a similar way to light waves. Th erefore, it was reasonable to assume that a wave equation could explain the behavior of atomic particles. Schrodinger was the first person to write down such a wave equation. Much discussion then centered on what the equation meant. The eigenvalues of the wave equation were shown to be equal to the energy levels of the quantum mechanical system, and the best test of the equation was when it was used to solve for the energy levels of the Hydrogen H ydrogen atom, and the energy levels were found to be in accord with Rydberg's Law. It was initially much less obvious what the wave function of the equation was. After much debate, the wave function is now accepted to be a probability distribution. The Schrodinger equation is used to find the allowed energy levels of quantum mechanical systems (such as atoms, or transistors). The associated wave function gives the probability of finding the particle at a certain position.
Schrodinger’s equation In quantum In quantum mechanics, the mechanics, the Schrödinger equation is a mathematical a mathematical equation that describes the evolution over time of a physical ph ysical system in which quantum effects, such as wave as wave – particle duality, are duality, are significant. The equation is a mathematical formulation for studying quantum mechanical systems. It is considered a central result in the study of quantum system and its derivation was a significant landmark in developing the theory of quantum mechanics. The equation is a type of differential equation kno wn as a wave-equation, which serves as a mathematical model of the movement of waves.
The time-independent Schrödinger equation states: When the Hamiltonian operator acts on a certain wave function Ψ, and the result is proportional to the same wave function Ψ, then Ψ is is a stationary a stationary state, and state, and the proportionality constant, E, is the energy of the state Ψ.
In linear algebra terminology, this equation is an eigenvalue equation. The most famous manifestation is the nonrelativistic Schrödinger equation for a single particle moving in a n electric field (but not a magnetic field):
Physical Interpretation The wave function ψ just as itself has no direct physical meanin g. It is more difficult to give a physical interpretation to the amplitude of the wave. The amplitude of the wave function ψ is certainly not like displacement in water wave or the pressure wave nor the waves in stretched string. It is a very different kind of wave. But the quantity, the squared Absolute amplitude gives the probability for finding the particle at given location in space and is referred to as probability density. It is given by P( x) =y
Where y* is the complex conjugate of y and the above product results in real number.
Normalization and Normalized wave function Since the particle exists somewhere in volume V t hen the probability of finding the particle in the given volume V is equal 1.
But, normally, the value of the above integral will not be unity but contains an indefinite constant which can be determined along with sign using above considerations. The process is called normalization and the wave function which satisfies the above condition is called normalized wave function.
References https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation https://physics.stackexchange.com/questions/76301/a-simple-explanation-for-theschr%C3%B6dinger-equation-and-model-of-atom https://simple.wikipedia.org/wiki/Schr%C3%B6dinger_equation https://www.scribd.com/document/149898444/The-Schrodinger-Wave-Equation https://www.scribd.com/document/334331006/Deriving-Time-Dependent-IndependentSchrodinger-Equations-Classical-and-Hamilton-Jacobi-Equations
