Form of the Up-Ladder and Down-Ladder Operators

What must the form of M⁺ be, in order that it function properly? The up-ladder operator, M⁺, is defined according to

$$M^+ \left (\sin \left (\frac{n \pi x}{L}\right )\right ) \rightarrow \sin \left ( \frac{(n+1)\pi x}{L}\right )$$ (2)

which, using Equation 1 and $\alpha = \frac{n \pi x}{L}$ and $\beta = \frac{ \pi x}{L}$ may be expressed as

$$M^+ = \cos \left ( \frac{\pi x}{L}\right ) \sin \left ( \frac... ...frac{\pi x}{L}\right ) \cos \left ( \frac{n \pi x}{L} \right )$$ (3)

i.e.,

$$M^+\vert n> \rightarrow \cos \left ( \frac{\pi x}{L}\right ) ... ...{L}\right ) \frac{\partial \vert n>}{\partial x} = \vert n+1>$$ \rightarrow \cos \left ( \frac{\pi x}{L}\right ) ... ...{L}\right ) \frac{\partial \vert n>}{\partial x} = \vert n+1> \end{displaymath}">

The down operator is obtained in an analogous way,

$$\sin \left ( \frac{(n-1)\pi x}{L}\right ) = \cos \left ( \fr... ...\frac{\pi x}{L}\right ) \cos \left (\frac{n \pi x}{L} \right )$$

making use of the even and odd properties of cosines and sines, respectively.

$$M^-\vert n> \rightarrow \cos \left ( \frac{\pi x}{L}\right ) ... ...) \frac{\partial \vert n>}{\partial x} \rightarrow \vert n-1>$$ \rightarrow \cos \left ( \frac{\pi x}{L}\right ) ... ...) \frac{\partial \vert n>}{\partial x} \rightarrow \vert n-1> \end{displaymath}">

$$M^- = \cos \left ( \frac{\pi x}{L}\right ) - \left (\frac{L}{... ...n \left ( \frac{\pi x}{L}\right ) \frac{\partial }{\partial x}$$

The operators M⁺ and M^- can be rewritten in coördinate-momentum language as

$$M^+ = \cos \left ( \frac{\pi x}{L}\right ) - \left (\frac{L}{\imath \hbar n \pi} \right ) \sin \left ( \frac{\pi x}{L}\right ) p_x$$

$$M^- = \cos \left ( \frac{\pi x}{L}\right ) + \left (\frac{L}{\imath \hbar n \pi} \right ) \sin \left ( \frac{\pi x}{L}\right ) p_x$$ (4)