SCHRODINGER WAVE EQUATION (Time Independent)

     Let us suppose an electron moving with velocity of light, it undergoes wave motion.

So, if we want to describe  the motion of the electron, we have to use  quantum mechanics and quantum mechanics principle.

     Here, electron moving like a wave there must be a vibration, frequency and amplitude there must be phase.

 Let ψ = ψ₀ sin(ωt-kx). . . .(1)

 it is simple harmonic wave equation.

 where, ψ= wave function,

             ω= angular velocity,

              t = time,

              k= wave vector and

              x= displacement

 Diff. eq(1) with respect to x 

we get,

        dψ/dx = ψ₀ cos(ωt-kx)(-k)

Diff. again with respect to x

we get,

         d²ψ/dx²= -ψ₀ sin(ωt-kx)(-k)(-k)

                       = -k²ψ₀ sin(ωt-kx)

                       = -k²ψ  [ψ = ψ₀ sin(ωt-kx). . . .(1)]                                                                                     d²ψ/dx² +k²ψ =0 . . . .(2)

we know k=2π/λ

putting the value of k in eq(2)

we get,

         d²ψ/dx² + 4π²/(λ²ψ) =0 . . . .(3)

   according to de-broglie hypothesis

                λ=h/(mv)

similarly, λ²= h²/(m²v²)

putting the value of λ² in eq(3)

we get,

         d²ψ/dx² + 4π²m²v²ψ/h²=0 . . . .(4)

         E=K+V

         E=1/2 (mv²) + v

         (E-V) = 1/2 (mv²)

        2(E-V)= mv²

multiply m both side,

we get,

 2m(E-V) = m²v²

putting the value of m²v² in eq(4)

  d²ψ/dx² + 4π²/h² 2m(E-V)ψ=0

according to bohr

  ħ=h/2π or ħ²=h²/4π²

 d²ψ/dx² +2m(E-V)ψ/ħ²=0

This equation represent time independent Schrodinger wave in one dimension.

In three dimension it may be written in the following way

d²ψ/dx² + d²ψ/dy² + d²ψ/dz² + 2m(E-V)ψ/ħ² =0

∆²ψ+ 2m(E-V)ψ/ħ²=0

where ∆=id/dx + jd/dy + kd/dz.

QUANTUM NUMBER

Quantum numbers are mathematical expression of any particular electron in an atom.

     In atoms electrons are well arranged in different orbits, sub-orbits and orbitals. In an orbital two electrons have their specific identity to address in its shell position and direction of spin are to be express. All they are denoted using specific mathematical number called the quantum numbers.

Quantum number of four types:-

i) principle quantum number(n)

     The idea of principle quantum number came from Bohr's suggestion of different energy levels, it indicates the shell position of an electron i;e, expression for  the energy of electron for the different shells

Values are

K shell(1st shell) - n=1

L shell(2nd shell) - n=2

M shell(3rd shell) -n=3

N shell(4th shell) - n=4

ii) Angular momentum quantum number(l)

     The idea of angular momentum quantum number came from  Sommerfeld's suggestion, which explain how electron having same energy may have different angular momentum. Its express the sub-shell position of the electron and gives an idea of the angular momentum value of the electron.

For the different sub-shell its value are:-

S sub-shell - 0

P sub-shell - 1

D sub-shell - 2

F sub-shell - 3

iii)Magnetic quantum number(m)

  The idea of magnetic quantum number causes from Zeeman and

stark's observation of different spectral line is applied electromagnetic field.

Magnetic quantum express the orbital position of the electron.

For the different orbitals its values are

For the orbitals of the

iv) Spin quantum number(s)

     The idea of spin quantum number came from the works of  Wolfgang Pauli.

     It express the direction of spin of the electron in the orbital. For clockwise spin the value is +1/2 and for anticlockwise spin the value is -1/2.

HEISENBERG'S UNCERTAINTY PRINCIPLE

      It is impossible to determine simultaneously both the position and momentum of an object with the same certainty i;e, both position and momentum of an object can not be determine at any certain time.

         If, uncertainty in position=𐤃x,
             uncertainty in momentum=𐤃p,
then according to Heisenberg's finding  𐤃x×𐤃p ≥ h/4π
                                                               or 𐤃x×m×𐤃v ≥ h/4π
                                                               or 𐤃x×𐤃v  ≥ h/4π m
Where 𐤃v= uncertainty in velocity,
              h= plank's constant,
    and   m= mass of the electron,
 Thus, position and velocity of an object cannot be simultaneously known with certainty. The uncertainty principle applied to the location and momentum along the same axis.
    Explanation of principal:-- 
    To locate the position of electron in atom, we have to use light. If the wavelength of the incident light be 'λ'. Therefore the wave character of electron is following interference
 1) The position of electron shall change by +λ, i;e, the position of electron be uncertain within range - λ to + λ.
 2) To reduce the uncertainty in position and determine the in the position more certainty (or more accuracy) is required.
 3) If light of shorter wavelength be used then the momentum of the photon being too high(p&1/ λ).
 4) The momentum of electron shall get disturb by higher order when the photon colloids with electron and vice-versa.
      So, both cannot be determine at any certain time.
  Significance of uncertainty principle.
   i) It rules out the existence of definite path or trajectories.
    ii) the effect of Heisenberg uncertainty principle is significant only for motion of microscopic objects.