The wave function Ψ represents the ______.

Concept:

As stated by de- Broglie, we know that matter has dual nature, a wave, and a particle.

• The electrons in an atom also possess this dual nature. The wave nature is significant in them due to their small size and negligible mass.
• An orbital describes the wave-like nature of the electron.
• The orbitals are mathematical functions that are used to calculate the probability of finding an electron.
• The probability of finding an electron is maximum in these orbitals.
• The square of the wave functions gives us the probability density of the electrons.
• The wave functions have two parts :
  • Radial - giving us the spread or size of the orbitals
  • Angular- giving us the orientation of the orbitals in space.
• Plotting the wave functions in a three-dimensional coordinate system, we get the shape, size, and orientation of the orbitals.

Explanation:

Wave nature of electrons:

• SchrÖdinger tried to develop the mechanics of electrons based on wave character. This is called wave mechanics.
• Any wave motion is the period variation of some property of the system, which is thus a property of time.
• The displacement of a wave at any point may be expressed as a function of the space coordinates and time 't':

ψ = f(θ,ϕ,t)

• This displacement of the wave about its mean position is known as the amplitude denoted by 'ψ'.
• For simple harmonic motion, such as that exhibited by electrons, the amplitude is given by:

ψ  = sin( kx - ωt ) where, k = wave number, ω  = angular frequency, and x = linear displacement of the particles.

• In wave mechanics, the probability of finding a particle at some point is proportional to the square of its wave function psi at that point ψ².

Hence, the wave function Ψ represents the amplitude function.

Additional Information

• In microscopic systems, if the total energy of the system remains unchanged with time, we may express ψ  as a product of two functions, one of the space coordinates and the other as a function of time.

ψ = f(θ,ϕ,r) f(t)