Quantum Physics

Time Independent Schrodinger Wave Equation

As discussed in the article of time dependent Schrodinger wave equation:

V=A exp[-i/Ћ(Et-px]

= A exp(-i/Ћ Et) exp(i/Ћ)

Ψ=ψ exp(-iEt/Ћ)                                                             (1)

Where Ћ = h/2π

So, ψ is a product of a time dependent function exp(-i/Ћ Et) and a position dependent function

Ψ= A exp(-i/Ћ px)

Differentiating equation (1) w.r.t.x, We have

dψ/dx = exp(-i/ЋEt) dψ/dx

and            d2ψ/dx2= exp(-i/Ћ Et) d2ψ/dx2 (2)

Also on differentiating ψ w.r.t. t, we have

dψ/dt=ψexp (-iEt/Ћ) (I E/Ћ)

dψ/dt=-(iE/Ћ)ψ exp(-I Et/Ћ)                                           (3)

Put equations [1-3] in time dependent Schrodinger wave equation (discussed earlier),

iЋ[-iE/Ћψ exp(-iEt/Ћ)]= -Ћ2/2m[exp(i/ЋEt) d2ψ/dx2] +V ψexp(iEt/Ћ)

exp(iEt/Ћ) = -Ћ2/2m exp(i/Ћ Et) d2ψ/dx2 + V ψexp(iEt/Ћ)

Dividing throughout by expression (i/Ћ Et) we have

= (-Ћ2/2π) d2ψ/dx2+V ψ

Or            (E-V)ψ=-Ћ2/2m  dψ/dx2

or             d2Ψ/dx2 + (2m/Ћ2)(E-V)ψ (4)

Which is time independent form of Schrodinger wave equation in one dimension.

In three-dimensional form:

d2 Ψ/dx2+ d2ψ/dy2+ d2Ψ’/ d2x2+2m/Ћ2(E-V)ψ=0

In this equation, ψ’ equation, ψ’(x) is also called the wave function. The potential V(x) does not contain the time explicity and E, the total energy of the particle is a constant.

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