This study employs the use of the fourth order Numerov scheme to determine the eigenstates and eigenvalues of particles, electrons in particular, in single and double delta function potentials. For the single delta potential, it is found that the eigenstates could only be attained by using specific potential depths. The depth of the delta potential well has a value that varies depending on the delta strength. These depths are used for each well on the double delta function potential and the eigenvalues are determined. There are two bound states found in the computation, one with a symmetric eigenstate and another one which is antisymmetric. Double Delta Potential, Eigenstates, Eigenvalue, Numerov Method, Single Delta Potential I. INTRODUCTION HE delta function potential has an interesting property and it plays an important character in theoretical solid state physics. In the Kronig(Penney square well periodic potential, the periodic delta function is used to simplify the coefficients of the eigenstate of electrons so as to determine the accessible energy states and isolated energy bands on solids[1]. The potential has the form ) ( ) ( x x U αδ − = (1) where α is called the delta strength. Theoretically, this has one bound state e x m m x ℏ ℏ 2 / | | ) ( α α ψ − = (2) and the allowed energy[2] is 2 2 2ℏ α m E − = . (3) This research work aims to investigate on the bound state and energy of a particle in single and double delta function potentials. Edward Aris D. Fajardo is with the Mindanao State University, Philippines. (e(mail: edwrdaris@gmail.com). Hamdi Muhyuddin Barra is with the Mindanao State University, Philippines. (e(mail: hmdbarra@gmail.com). II.NUMERICAL METHOD The Numerov Method is based on a Taylor expansion of the function and its second derivative[3]. This is the numerical method used to solve the eigenstate of particle in delta function potential. In solving the Schrödinger equation ) ( ) ( ) ( 2 2 2 2 x E x x U dx d m ψ ψ ψ = + − ℏ , (4) implementing Numerov algorithm gives the eigenstate ) ( 6 1 1 )) ( 6 1 1 ( )) ( 6 5 1 ( 2 ) 1 ( 2 2 1 ) 1 ( 2 2 2 2 1 + − − + − + − + − − − = n n n n n n U E h m U E h m U E h m ℏ ℏ ℏ ψ ψ ψ (5) This method requires two initial conditions of the eigenstate to start the iteration for the equation. It must be noted that the wave function approaches zero as the position tends to infinity. Starting conditions could be chosen as ψ 0 = 0 and ψ 1 =1. These are reliable initial conditions and can be justified mathematically since multiplying an eigenstate with a constant does not affect the eigenvalue[4]. In all calculations, h has a value equal to 0.1 Å. To easily get values of the wave function, a computer must be used to easily solve the iterative equation. The simulation tool used here is ROOT, an object(oriented framework aimed at solving the data analysis challenges of high(energy physics[5]. III. SINGLE DELTA FUNCTION POTENTIAL The Dirac delta function, δ(x), is defined informally as follows[2]: , with 1 ) ( _ = ∫ +∞ ∞ dx x δ (7) It is infinitely high, infinitesimally narrow spike at the origin, whose area is 1[2]. However, in computational calculations, it is impossible to use an infinite value. So there must be a defined depth of the delta potential well. In the numerical calculations, this potential depth is the quantity that was derived using analytical eigenvalues. For an electron as the particle in consideration, delta strengths of 1.0 neVHm, 1.5 neVHm, 2.0 neVHm, 2.5 neVHm and 3.0 neVHm are used and the    = ∞ ≠ = 0 , 0 , 0 ) ( x if x if x δ Eigenvalues of Particle Bound in Single and Double Delta Function Potentials through Numerical Analysis Edward Aris D. Fajardo and Hamdi Muhyuddin D. Barra T World Academy of Science, Engineering and Technology International Journal of Physical and Mathematical Sciences Vol:5, No:12, 2011 2069 International Scholarly and Scientific Research & Innovation 5(12) 2011 ISNI:0000000091950263 Open Science Index, Physical and Mathematical Sciences Vol:5, No:12, 2011 publications.waset.org/14799/pdf