SCHRODINGER WAVE EQUATION & DERIVATION

SCHRODINGER WAVE EQUATION 

         In 1926 Erwin Schrodinger an Austrian Physicist developed wave equation to describe dual character of electron.

He determined the form of equation which are used to calculate the energy of system made of nucleus and electron.

According to schrodinger atom as positively charged nucleus surrounded by standing wave.

The equation which describe the wave along any of the three axes X,Y,Z is called schrodinger wave equation.

                             DERIVATION

Consider a partical P (e^-) moving with angular velocity around the nucleus in circular path of radius "A" in time "t" covering.

Consider tringle PBC

We known that

Sinθ=  PC ⁄ PB

Sinθ= ψ⁄A

Ψ=A Sinθ                                               -------------------(1)

As We know that

Θ= ωt                                                       ------------------(2)

Put eq (1) and (2)

Ψ= A Sin ωt                                           -------------------(3)

                      where Θ= ωt

                               t=θ⁄ω

                                  = 2π⁄ω

As we know that frequency (ʋ)

ʋ= 1⁄t

ʋ= 1⁄2π⁄ω

   =ω⁄2π

          ω= 2π ʋ                                          ----------------(4)

put eq (4) in eq (3)

ψ= A Sin 2π ʋt                                         -------------------(5)

We know that velocity

ʋ= x⁄t

t= x/ʋ                                                      ----------------------(6)

put (6) in (5)

ψ =A Sin 2πʋ.X/t                               ----------------------------(7)

frequency

ʋ= C/λ                                                     ------------------------------(8)

put (8) in (7)

ψ =A Sin 2π.C/λ.X/t

ψ =A Sin 2πx/λ                                         -----------------------------(9)

(9) is the main equation which are than differential with respect” X” and than “t” for time dependent SWE.

Differentiate eq (9) with respect to “X”

ψ =A Sin 2πx/λ

d/dx.ψ= d/dx A Sin 2πx/λ

d/dx.ψ= A Cos 2πx/λ. d/dx. 2πx/λ

dψ/dx=A Cos 2πx/λ.2π/λ

dψ/dx= 2π/λ A Cos 2πx/λ                       -----------------------(10)

Differentiate eq “x” with respect to “x”

d/dx .d/dx=d/dx 2π/λ A Cos 2πx/λ

d²λ/dx²= 2π/λ A (-Sin 2πx/λ).2π/λ

d²λ/dx²= -4π²/λ² (A Sin 2πx/λ)                 ---------------------(11)

put eq (9) in (11)

d²λ/dx²= -4π²/λ.ψ

d²λ/dx²+ 4π²/λ.ψ= 0                                   ---------------------------(12)

according to de Broglie

λ= h/mv                          “teke square on both side”

λ²= h²/mv²

put value of λ²  in eq (12)

d²ψ/dx²+4π²m²v²/h.ψ=O                                       -----------------------(13)

according to bohr atomic model energy of electron is the sum of P.E and K.E

E= K.E+P.E

E=½mv²+v

½mv²= E-V                                              “Divide both side by m”

½ mv²/m= E-V/m

V²= 2(E-V)/m

Put value of V² in eq (13)

 d²ψ/dx²+4π²m²/h². 2(E-V)/m.ψ=O

d²ψ/dx²+4π²m/h². 2(E-V).ψ=O

d²ψ/dx²+8π²m/h². (E-V)ψ=O

according to wave motion electron in three axis.

d²ψ/dx²+ d²ψ/dy² +d²ψ/dz²+8π²m/h². (E-V)ψ=O

Now according to laplecion operator

∆²= d²/dx²+ d²/dy² +d²/dz²

Put ∆² in equation

∆²ψ+8π²m/h². (E-V)ψ=O

h²/8π²m. ∆²ψ= -Eψ +Vψ

Eψ= -h²/8π²m. ∆²ψ+ Vψ

Eψ=( -h²/8π²m. ∆²+ V)ψ                                      -------------------------(14)

There are two equation of schrodinger one is time depending which are double derivative with respect to position and single derivative W.R.T. Time.

                                         Time independing equation are only double derivative of with respect to position.