### One Dimensional Schrodinger's Equation for a Definite Energy (Stationary State)

Given the one-dimensional sinusoidal wave function for a particle:

and the one-dimensional Schrodinger's equation (with potential energy):

(eqn I)

We can obtain a time independent Schrodinger's equation for definite energy (stationary state).

## Separation Technique

We start by recognizing that the given particle wave function can be separated into two functions of both position and time:

where lower-case psi(x) is the time independent wave function

By taking the derivative of the above (separated) function, with respect to x, we get:

(eqn II)

(note: T = T(t))

... and taking the second derivative with respect to x gives us:

(eqn III)

Lastly, taking the derivative of the separated wave function with respect to T gives us:

(eqn IV)

By substituting equations 2, 3, and 4 into equation #1 we obtain:

Dividing through by T (tau) results in the following:

Now examine the right-hand side of the equation and recognize that the time function (T) and it's derivative are equal to the following:

Therefore, if we substitute the above derivative of T into the right hand side of our equation, the imaginary part (i) becomes 1 and the T cancels leaving us with:

Also recall that:

Which turns our equation into:

...which cancels the 2 pi terms:

Also recall that:

...and finally we arrive at:

When $$a \ne 0$$, there are two solutions to $$(ax^2 + bx + c = 0)$$ and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$