Constructing the Ladder Operator

We start by remembering the trigonometric definition of the sine of the sum of two angles, i.e.,

$$\sin ( \alpha + \beta ) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$ (1)

We seek an operator, M⁺ which takes a state |n> and ladders it up to the next state, i.e., |n+1>. Since the particle in a box problem is already solved, i.e., we know that |n> is $sin \frac{n \pi x}{L}$ we can employ this knowledge to construct our ladder operators. Symbolically, we define the up-ladder operator through the statement

$$M^+\vert n> \rightarrow \vert n+1>$$ \rightarrow \vert n+1> \end{displaymath}">

and we will require the inverse (the down-ladder operator), i.e.,

$$M^-\vert n> \rightarrow \vert n-1>$$ \rightarrow \vert n-1> \end{displaymath}">

where the down-ladder operator is lowering us from n to n-1 as usual.

One knows that there is a lower bound to the ladder, i.e., that one can not ladder down from the lowest state (n=1), i.e.,

M^-|lowest> = 0