# Wave Function

## What is wave function?

In quantum physics, a wave function is a mathematical description of a quantum state of a particle as a function of momentum, time, position and spin. The symbol used for a wave function is a Greek letter called psi, 𝚿.

By using a wave function, the probability of finding an electron within the matter wave can be explained. This can be obtained by including an imaginary number which is squared to get a real number solution resulting in the position of an electron. The concept of wave function was introduced in the year 1925 with the help of the Schrodinger equation.

## What is Schrodinger equation?

Schrodinger equation is defined as the linear partial differential equation describing the wave function, 𝚿. The equation is named after Erwin Schrodinger. Using the postulates of quantum mechanics, Schrodinger could work on the wave function.

Following is the equation of Schrodinger equation:

• Time dependent Schrodinger equation: $ih\frac{\partial }{\partial t}\Psi (r,t)=[\frac{h^{2}}{2m}\bigtriangledown ^{2}+V(r,t)]\Psi (r,t)$
• Time independent Schrodinger equation: $[\frac{-h^{2}}{2m}\bigtriangledown ^{2}+V(r)]\Psi (r)=E\Psi (r)$

Where,

m: mass of the particle

⛛: laplacian

i: imaginary unit

h=h/2𝝿 : reduced Planck constant

E: constant equal to the energy level of the system

## Properties of wave function

• All measurable information about the particle is available.
• 𝚿 should be continuous and single-valued.
• Using the Schrodinger equation, energy calculations becomes easy.
• Probability distribution in three dimensions is established using the wave function.
• The probability of finding a particle if it exists is 1.

## Postulates of quantum mechanics

• With the help of time-dependent Schrodinger equation, the time evolution of wave function is given.
• For a particle in a conservative field of force system, using wave function it becomes easy to understand the system.
• Linear set of independent functions is formed from the set of eigenfunctions of operator Q.
• Operator Q associated with a physically measurable property q is Hermitian.
• By performing the expectation value integral with respect to the wave function associated with the system, the expectation value of the property q can be determined.
• For every physical observable q, there is an operator Q operating on wave function associated with a definite value of that observable such that it yields wave function of that many times.

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