 # 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: 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: 