# What uses do products of power series have?

##### 1 Answer

One example that I find useful is the use and manipulation of the products of power series to derive **Schroedinger Equation** in Physical Chemistry, by substituting

What is accepted by Physical Chemists is that you can write out the general solution to the Equation as:

#y(x) = c_1 e^(alphax) + c_2 e^(-alphax)#

where

#psi(x) = sum_(i=1)^N c_i phi_i(x)#

where each#phi# could, for example, represent an atomic orbital, and#psi(x)# would in that case be the molecular orbital.

A common example of solving the time-dependent Schroedinger equation is (example 2-4 in *Physical Chemistry: A Molecular Approach*):

#(d^2x(t))/(dt^2) + omega^2x(t) = 0#

subject to the **boundary conditions**

To solve this one, **one would have to use identity written at the top**, with

#c_1e^(alphax) + c_2e^(-alphax)#

#= c_1(cosx + alphasinx) + c_2(cosx - alphasinx)#

#= c_1cosx + c_1alphasinx + c_2cosx - c_2alphasinx#

#= (c_1 + c_2)cosx + (c_1alpha - c_2alpha)sinx#

and it is generally written out by absorbing the arbitrary constants

#= c_3cosx + c_4sinx#

Then, substituting

#c_3cos(omegat) + c_4sin(omegat)#

for the solution to the so-called common example.

Looking at the boundary condition

#= c_3(-sin(omegat))*omega + c_4cos(omegat)*omega#

#= cancel(c_3(-sin(omega(0)))*omega)^(0) + c_4cos(omega(0))*omega#

#= c_4omega#

But we know that at

#=> c_4omega = 0# , thus satisfying the condition#(dx(0))/(dt) = 0# .

Using the

#x(0) = c_3cos(omega(0)) + cancel(c_4sin(omega(0)))^(0)#

#= c_3#

with

#color(blue)(x(t) = Acos(wt))#

which is the familiar physics equation for a transverse wave, as depicted in the image above! :)