0.5 1 1.5 2 2.5 3 x -1.5 -1 -0.5 0.5 1 1.5 Ψ GLYPH<LParen1> x GLYPH<RParen1> Ψ$_{1}$ Ψ$_{2}$ Ψ$_{3}$ FIG. 1: First three stationary states ψ$_{n}$ ( x ) for 2 mλ/ ¯ h 2 = 8, a = 3, A = 1, and p = 1 / 2. where k$_{n}$ = √$_{2}$$_{mE}$$_{n}$$_{/}$$_{¯}$ h . The continuity condition of the wave function at x = pa gives A/B = sin[( p − 1) k$_{n}$a ] / sin( pk$_{n}$a ). Because the delta function is infinite at x = pa , the first derivative of the wave function is not continuous and the relation between the left and right derivatives of the wave function can be obtained by integrating Eq. (2) over the small interval ( x$_{0}$ − ϵ, x$_{0}$ + ϵ ) dψ$_{n}$ ( x ) dx | pa + ϵ − dψ$_{n}$ ( x ) dx | pa − ϵ = 2 m ¯ h 2 ∫ pa + ϵ pa − ϵ V ( x ) ψ$_{n}$ ( x ) dx (4a) = 2 mλ ¯ h 2 ψ$_{n}$ ( pa ) . (4b) We substitute the eigenfunctions in Eq. (3) in Eq. (4b) and obtain the quantization condition 5,6 k$_{n}$ sin( k$_{n}$a ) = 2 mλ ¯ h 2 sin( pk$_{n}$a ) sin[( p − 1) k$_{n}$a ] . (5) The solutions to Eq. (5) give the energy spectrum of the Hamiltonian, E$_{n}$ = ¯ h $^{2}$k $_{n}$/ 2 m . Figure 1 shows the first three stationary states for 2 mλ/ ¯ h 2 = 8, a = 3, A = 1, and p = 1 / 2. III. THE FACTORIZATION METHOD To calculate the eigenvalues and eigenfunctions of a Hamiltonian operator H , we can use a general operational procedure called the factorization method. In this method the Hamiltonian is written as the product of two ladder operators plus a constant, H = a † a + E . These operators are used to obtain the eigenfunctions of the Hamiltonian. In contrast to the simple harmonic oscillator, one ladder operator is usually not sufficient to form all the Hamiltonian’s eigenfunctions and another ladder operator is needed for each eigenfunction. 3