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

Feb 4, 2018

The General Solution is:

$\phi = A {e}^{- \frac{8 {\pi}^{2} m E}{h} ^ 2 x}$

We cannot proceed further as $v$ is undefined.

Explanation:

We have:

$\frac{\mathrm{dp} h i}{\mathrm{dx}} + k \phi = 0$

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

$\frac{\mathrm{dp} h i}{\mathrm{dx}} = - k \phi$
$\frac{1}{\phi} \setminus \frac{\mathrm{dp} h i}{\mathrm{dx}} = - k$

Now, we separate the variables to get

$\int \setminus \frac{1}{\phi} \setminus d \phi = - \int \setminus k \setminus \mathrm{dx}$

Which consists of standard integrals, so we can integrate:

$\ln | \phi | = - k x + \ln A$
$\therefore | \phi | = A {e}^{- k x}$

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

$\phi = A {e}^{- k x}$
$\setminus \setminus = A {e}^{- \frac{8 {\pi}^{2} m E}{h} ^ 2 x}$

We cannot proceed further as $v$ is undefined.