Introduction

Application of Schrodinger equation Contains a great importance in the world of physics. It explains many phenomena like particle in a one-dimensional box,particle on a ring,The structures of atoms, orbitals, quantum numbers: values &information, More info on Schrodinger equation,angular and radial contribution,radial distribution function, Electron spin, Many electron atoms, Pauli exclusion principle, energy bands , order of filling orbitals with electrons, electronic configuration of atoms and ions, Atomic properties and periodic trends,atomic size, ionization energy, Chemical bonds, ionic,covalent, Schrodinger equation for simple molecules, Born-Oppenheimer approximation.

Some other important application of Schrodinger equation

In valence bond theory,diatomic molecules, a polyatomic molecules,hybridization, VSEPR model,resonance,trial wavefunction, variation theorem, molecular orbital theory, bonding and antibonding orbitals,bond order, second period diatomic molecules, homonuclear, heteronuclear, permanent and induced dipole moment, magnetic properties, polyatomic molecules, HOMO, LUMO, delocalized bonding, Spectroscopy, rotational spectroscopy linear, symmetric, and spherical rigid rotors , reduced mass, moment of inertia, rotational energy levels, transitions, and selection rules , Raman rotational spectroscopy, polarizability, selection rules, vibrational spectroscopy , harmonic oscillator, vibrational modes, vibrational energy levels and selection rules, Raman vibrational spectroscopy,resonance Raman spectroscopy, electronic spectroscopy , s – s *, p – p *, n – p *, d -d*, and charge transfer transitions, fluorescence and phosphorescence and photoelectron spectroscopy, lasers etc.

 Application of Schrodinger equation:- for the free particle

Application of Schrodinger equation for the Free particle is also known by No boundary state.

And We know that general form of Schrodinger equation can be written as,

∇²ψ + 2m (E-V)ψ /h²=0 (1)

Also we know that Application of Schrodinger equation

Because for free particle V=0

Therefore above equation becomes-

∇²ψ + 2m (E)ψ /h²=0 (2)

In addition Cartesian co-ordinate system-

δ²ψ/δx² + δ²ψ/δy² + δ²ψ/δz² + 2mEdψ/h² =0 (3)

So By the method of separation of variables-

ψ(x y z ) = X(x) Y(y) Z(z)

Put this value in equation (3)

YZ δ²X/δx² + XZ δ²Y/δy² + XYδ²Z/δz² + 2mEψ/h² =0

And Now devide this equation by XYZ we get

1/X δ²X/δx² + 1/Yδ²Y/δy² + 1/Z δ²Z/δz² + 2mE/h² =0

Where E= Ex î + Ey j + Ez k

(Note :- where i,j,k are vector quantities.)

Also let us put,

1/X δ²X/δx²= kx

1/Yδ²Y/δy² = ky

1/Z δ²Z/δz² = kz

Kx+ Ky + Kz + 2mE/h² = 0

Where ,

Kx = -2m Ex/ h²

Ky = -2m Ey/h²

Kz = -2m Ez/h ²

From equation ,

1/ X δ²X/δx² = -2mEx/h²

1/ X δ²X/δx² + 2mEx/h² =0

And 1/Yδ²Y/δy² +2mEy/h² =0

Hence,

1/Z δ²Z/ δz² + 2mEz/h² =0

Now , solutions for these equations –

X = Nx sin [√2mEx (x-x₀)]/h

And where, Nx = normalisation constant

X = initial displacement

Similarly ,Y = Ny sin [ √ 2mEy(y-y₀)/h]

Z = Nz sin [√2mEz(z-z₀)/h]

Total wave function is given by-

ψ( x y z ) = XYZ

Hence, ψ( x y z ) = NxNyNz sin [ √2mEx(x -x₀) /h] sin [√2mEy(y-y₀)/h ] sin [ √2mEz(z-z₀)/ h]

Put , NxNyNz =N

ψ( x y z ) = N sin [ √2mEx(x -x₀) /h] sin [√2mEy(y-y₀)/h ] sin [ √2mEz(z-z₀)/ h]

If we include time factor then equation is given by-

ψ( x y z ) = N sin [ √2mEx(x -x₀) /h] exp (-iExt/h) sin [√2mEy(y-y₀)/h ] exp(-iEyt/h) sin [ √2mEz(z-z₀)/ h]exp (-iEzt/h)

And also,

ψ( x y z ) = Nsin {√2mEx(x -x₀) /h} sin {√2mEy(y-y₀)/h } sin { √2mEz(z-z₀)/ h} [exp (-iEt/h

 

Born -Oppenheimer approximation-

Application of Schrodinger equation

The atomic and line spectra arise from the transition of an electron between the atomic energy levels .while band or molecular spectra arise from three types of energy changes i,.e molecular rotational, molecular vibrational, and molecular vibration and molecular electronic transition.

According to born -Oppenheimer approximation,

Hence The total energy of a molecule,

E=Etr + Erot+Ev+ Electronic

Where  Etr represents translation energy, Erot is rotational energy, Evib is vibrational energy and Ee is electronic energy. Furthermore, translation energy does not remain quanytised and its value is small enough that we can neglect this value. Therefore, the complete internal energy of a molecule is

E = Er+ Ev+Ee

Also, Ee>Evib>Erot>Etr

∴E = Erot+Evib+ Eel

  • Erot arises when molecule rotates about an area perpendicular to the internuclear axis and passing through C.G.
  • Evib arises due to and fro motion of nuclei of molecule.such that C.G doesn’t change.
  • Eel arise due to the transition of the electron from Groud  state to excited state due to absorption of a photon of suitable frequency

It means that a molecule can exist only in certain discrete electronic, rotational, spin and nuclear states. And the separation between electronic energy level is usually much larger than those between the vibrational energy level and in turn are much larger than the spin energy level. Consequently, because of much difference in mass between the electron and the nucleus, it is permissible as a very good approximation to regard the respective motions as mechanically reparable. And hence this simplification is usually referred to as Born-Oppenheimer approximation.

Conclusion

In this particular article Application of Schrodinger equation, we have discussed initially Application of Schrodinger equation in various fields. Along with that we have widely discussed Application of Schrodinger equation, for the free particle. And also the concept of Born-Oppenheimer approximation in simplest manner.

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