Question

The differential equation is `(dphi)/dx+kphi=0` where `k=(8pi^2mE)/h^2`E,m,h are constants.Find what is `(h/(4pi))` If `m*v * x ~~(h/(4pi))`?

Answer 1

The General Solution is:

`phi = Ae^(-(8pi^2mE)/h^2x)`

We cannot proceed further as `v` is undefined.

Explanation:

We have:

`(dphi)/dx + k phi = 0`

This is a First Order Separable ODE, so we can can write:

`(dphi)/dx =- k phi`
`1/phi \ (dphi)/dx =- k`

Now, we separate the variables to get

`int \ 1/phi \ d phi = - int \ k \ dx`

Which consists of standard integrals, so we can integrate:

`ln | phi | = -kx + lnA`
`:. |phi| = Ae^(-kx)`

We note that the exponential is positive over its entire domain, and also we have written `C=lnA`, as the constant of integration. We can then write the General Solution as:

`phi = Ae^(-kx)`
`\ \ = Ae^(-(8pi^2mE)/h^2x)`

We cannot proceed further as `v` is undefined.