To obtain the root functions, we rewrite Eq. (8) as where ξ$_{n}$ ( x ) ≡ 〈 x | ξ$_{n}$ 〉 . It can be easily checked that the following is a solution to Eq. (22) Because in the first step of the factorization procedure the root function ξ$_{1}$ ( x ) is equal to the first eigenfunction ψ$_{1}$ ( x ) (see Eq. (9) and Fig. 2) and we have replaced the index 1 by n , all the root functions are equal to the eigenfunctions of the Hamiltonian, that is, ξ$_{n}$ ( x ) = ψ$_{n}$ ( x ). This result is similar to Eq. (3) for x < pa . Moreover, because k$_{n}$b = k$_{n}$a − π , we have sin[ k$_{n}$ ( x − b )] = sin[ k$_{n}$ ( x − a ) + π ] = − sin[ k$_{n}$ ( x − a )]. So, these eigenfunctions are equivalent to the solutions in Eq. (3). Problem 1. Use the recursive relations (6) to show that | E$_{n}$ 〉 in Eq. (9) is the eigenfunction of the Hamiltonian with E$_{n}$ as its eigenvalue. Problem 2. Show that the eigenvalues E$_{1}$, E$_{2}$, E$_{3}$, . . . form a monotonic increasing se- quence, that is, E$_{1}$ ≤ E$_{2}$ ≤ E$_{3}$ ≤ . . . . Problem 3. Apply the factorization method to a particle in a box with no delta function potential. In the first step show that k$_{1}$ = π/a . Continue the factorization procedure to find all the eigenenergies and eigenfunctions. Hint: consider a$_{n}$ = 1 √$_{2}$$_{m}$ [ P + ic$_{n}$ cot( d$_{n}$x )], where c$_{n}$ and d$_{n}$ are real constants. ∗ Electronic address: pouria.pedram@gmail.com 1 D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, Upper Saddle River, NJ, 2004). 2 B. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors (Springer, New York, 1984), Chap. 1. 7 IV. SUGGESTED PROBLEMS ( ¯ h i d dx + i ¯ hk$_{n}$ cot( k$_{n}$x ) ) ξ$_{n}$ ( x ) = 0 , ( x < pa ) , (22a) ( ¯ h i d dx + i ¯ hk$_{n}$ cot[ k$_{n}$ ( x − b )] ) ξ$_{n}$ ( x ) = 0 , ( x > pa ) , (22b) ξ$_{n}$ ( x ) =    $^{A}$$^{sin(}$$^{k$_{n}$}$$^{x}$$^{)}$$^{,}$ ( x < pa ) , − B sin[ k$_{n}$ ( x − b )] , ( x > pa ) . (23)