Deriving the one dimensional, Time-Independent Schrodinger Equation (Using the separation technique)

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

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

Wave Function

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

(eqn I)

Wave Function

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:

separation

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)

separation

(note: T = T(t))

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

(eqn III)

separation

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

(eqn IV)

separation

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

separation

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

separation

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:

separation

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:

separation

Also recall that:

separation

Which turns our equation into:

separation

...which cancels the 2 pi terms:

separation

Also recall that:

separation

...and finally we arrive at:

separation

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