"I am sorry that I ever had anything to do with quantum theory."- Erwin SchrӦdinger
At the beginning of the twentieth century, experimental evidence suggested that atomic particles were also wave - like in Nature.
We can do experiment with tuning forks, double-slit apparatuses to show that all kinds of wave interfere with each other. The double-slit experiment can be done with electrons and show the interference pattern, which proves that an electron beam is wave-like nature in the same way as a light beam. In this video, furthermore describe about the double slit experiment.
By comparing the electron interference patterns with the light interference patterns, we can prove that, an electron has more momentum and shorter its wave length.
Ultimately, Scientist concluded that every type of particle also has wave-like behavior.
Physicist noticed that waves in pieces of string, sound waves in the air and electromagnetic waves all seem to follow the same laws of wave motion. So they invented an equation called the "Wave equation".
We need an equation to describe the wave-like behavior of particles. However, these particle wave are a bit different from water-waves, since they involve complex numbers rather than real number like the height of the water in the wave. So the usual equation does not apply to particles.
That's why Physicist invented a different equation that seems to accurately describe the wave-like behavior of particles, which is called the Schrödinger equation.
Erwin Schrödinger(1887 - 1961): was the first person who write down such a wave equation. This wave equation let you predict what will happen to a wave in the future, if you know what it looks like now.
In 1926 Austrian Physicist Erwin Schrödinger who came up with the wave equation. For a single particle moving around in 3-dimension box - at any time $'t'$, then the time dependent Schrödinger equation can be written as:
Here, $V$ is the potential energy of the particle; Complex number$(i)$ = $\sqrt{-1}$; $M$ = mass of the particle; $\hbar = \frac{h}{2\pi}$, ⇒ $h$ = Planck's constant.
The left - hand side of the Schrödinger equation tell us how to calculate the energy of the particles. The derivative on the left - side calculates the wavelength of the particle and therefore its momentum, then uses the momentum to calculate the kinetic energy. The left-hand side also adds in the particle's potential energy.
The derivative on the right - hand side calculates the frequency of the particle. So the Schrödinger equation relates the particle's momentum with its energy, and also relates the particle's wavelength with its frequency [1]
In some situation, the potential energy does not depend on time $'t'$. In this case the problem solved by the considering the time - Independent Schrödinger equation, where wave function $(\Psi)$ depending only on space, can be written as:
$\frac{\partial^2 \Psi}{\partial x^2} + \frac{2M}{\hbar}(E-V)\Psi = 0$ .......... (ii)
Here $E$ is the total energy of the particle, $V$ is the potential energy [2].
In Summary, Schrödinger altered the classical description of the world in two ways:
1. The state is described not by positions and velocities of the particles, but by a wave function.
2. The change of this state over time is described not by Newton's or Einstein's laws, but by the Schrödinger equation.
» Wave function $(\Psi)$:
The solution of $\Psi$ to the full equation can be written in the form:
$\Psi = \psi e^{-({\frac{2\pi i E}{h}})t}$
In quantum mechanics, a wave function is a quantity that is used to describe the displacement of a matter wave produced by particles (i.e. electrons).
The wave function holds all the information you can possibly know about the particles. The particle's position can never be known with certainty. By taking the square of the absolute value of the wave function allows us to determine the probability of finding the particle at the some given location. (or, all you can do is integrate the probability density over some region and obtain the probability of finding the particle in that region).
i.e. $\int_{V}\left | \psi(x,t)^2 \right |d^3x < \infty$
This gives the probability of finding the particle somewhere inside the given region $V$, at the given time $t$. So the squared magnitude of the wave function $\Psi$ can be interpreted as a probability density, and $\Psi$ itself is a probability amplitude.
In a region in which the particle's total energy is greater than the potential $V$, the wave function really does act like a wave, but the amplitude and wave length tend to be lower in regions of higher potential. If the total energy is less than the potential, the solutions are exponentially decaying, and are not wave-like Nature.
The wave function therefore represents one of the most important differences between classical and quantum mechanics. In classical mechanics there is no wave function, because there only need six number to describe the state of a particle (i.e. three position and three momentum coordinates). But in quantum mechanics, need entire function with an infinite number of values [3].
