Excited-State Energy Eigenvalue Evaluation of the Quantum Mechanical Potential V(x)=½mω2x2+μx3 via Numerical Shooting Method

The paper deals with eigenvalues excited-state energy eigenvalues and wave-function of a particle under harmonics oscillator asymmetric potential using numerical shooting method. The numerical shooting method is generally regarded as one of the most efficient methods that give very accurate results because it integrates the Schrodinger equation directly, though in the numerical sense. If the value of parameter μ is small the energy eigenvalues of single particle will large and the parameter μ large the energy eigenvalues of single particle will small.


Introduction
Most problems encountered in quantum mechanics cannot be solved exactly. Exact solutions of the Schrodinger equation exist only for a few idealized systems. To solve general problems, one must resort to approximation methods. A variety of such methods have been developed, and each has its own area of applicability. There exist several means to study them, e.g. Wentzel-Kramers-Brillouin [1], perturbation theory [2], the quasilinearization method [3], the variational method, function analysis [3], the eigenvalue moment method [4], the analytical transfer matrix method [5][6][7] and numerical shooting method [8][9]. In this paper we consider approximation methods that deal with stationary states corresponding to time-independent Hamiltonian. To study problem of stationary states, we focus on one approximation method: numerical shooting method useful evaluate wave-function and energy eigenvalue of a particle around of attraction by the harmonics oscillator with 3 x µ potential. In Section 2, we representation of the evaluation of the energy eigenvalue and wave function of particle bound in the

Schrodinger Equation in Finite Difference formula for
We consider a particle of mass µ moving on the x-axis in a time-independent potential V(x).
The time-independent Schrodinger equation corresponding to this one-dimensional motion is Suppose a particle is bound state to around of attraction by the harmonics oscillator x 3 µ (see Fig.1).
, where µ are constants ( µ is real and positive). Substituting potential into eq.(1), and we obtain The one approach we use is the numerical shooting method. To simplify the arithmetic involved in the shooting solution, we define some new dimensionless variables. The position variable x is replaced with the dimensionless variable y: We can find the numerical solution eq.(2) by dividing y into many small segments, each of ∆y in length. The second derivative for the first term of eq.(2) can be approximated in finite difference form as follows We can obtain the form of the Schrodinger equation in term of finite difference by substituting eq.(3) into eq.(2), we get ( ) We construe the new variable for using in calculating the ground-state and excited-state energy

Numerical and Result
We construe the new variable for using in calculating the ground-state and excited-state energy eigenvalue(En), wave-function of the harmonics oscillator asymmetric potential.
1. ymin is the start position in the analysis range. 2. ymax is the ultimate position in the analysis range. 3. y is any position in the analysis range. 4. nn is a number of very small bars in the analysis range. 5. △y is the length of very small bars so that For the beginning of the numerical shooting method, we need to input parameters ymin(ymax), and Eq. (4) into mathematica program. Next, two initial wave functions are determinate by following two initial conditions as (i) (1)

Conclusion
Shrinkage and spreading of the wave function is controlled by parameter µ . The wave-functions has µ defects (density fluctuation). If the value of parameter µ is small the energy eigenvalues of single particle will large and the parameter µ large the energy eigenvalues of single particle will small.