Variational Principle Techniques and the Properties of a Cut-off and Anharmonic Wave Function

The variational principles are very useful analytical tool for the study of the ground state energy of any dynamical system. In this work, we have evaluated the method and techniques of variational principle to derive the ground state energy for the harmonic, cut-off and anharmonic oscillators with a ground state wave function for a one-body Hamiltonian in three dimensions.


Introduction
Since most problems in Physics and Chemistry cannot be solved exactly, one resort to the use of approximation methods.The two methods used commonly in quantum mechanics are the perturbation theory and variational method.The basic idea of the perturbation theory involves the splitting of the Hamiltonian into two pieces -the unperturbed and perturbation.This theory is useful when there is a small dimensionless parameter in the said problem called the coupling constant and the system becomes solvable when this parameter is set to zero.Thus the perturbation theory gives correction as an infinite series of terms in powers of the perturbation parameter which becomes smaller for a well-behaved system.
On the other hand, the variational method is useful for the study of the ground state system even though it is not useful for the study of the excited states of a system.The basic idea of the variational method lies in the choice of a trial wave function for the problem at hand; which consists of some adjustable parameters usually called the variational parameters.These variational parameters are usually adjusted until the energy of the trial wave function is minimized.Once this is done, the resulting trial wave function and its corresponding eigen values are variationally approximate to the exact wave function and energy.
The variational principle and simple properties of the ground state wave function with simple potential had been investigated 1 .They show that the ground state wave function can be taken to be real and non-negative and that it can not be degenerate where their parity and angular momentum of this ground state wave function has also been analysed 1 .The properties of the ground state wave function with simple and complex potential had been examined by different authors 2 -4, 11-14 .The variational perturbation theory in terms of path integral approach had been analyzed [5][6][7] with different potentials.
In this paper, we follow the approach 1 and extend their application to a cut-off harmonic and the anharmonic oscillator.

Fundamentals of variational principle
The ground state wave function has the lowest-energy eigen value of a given system.
and any other states have higher energy eigen values 0 ψ ψ where 0 E E n 〉 .For the Hamiltonian which cannot be solve for eigen states and eigen values exactly, we use a trial wave function or an ansatz as The ansatz is exact if 1 0 〈 C and C n = 0 for all 0 ≠ n and the normalization requires that The approximate energy or the expectation value with this ansatz is This equation holds if the ansaltz is normalized but if the wave function is not normalized, (5) becomes Since the functions j ψ are the exact eigen functions of H ˆ,we can write On subtracting 0 ∈ from both sides of (9) we obtain that [ ] 0 0 ≥∈ φ E (10)   This implies that the trial wave functions is always greater than or equal to the exact ground state energy 0 and looking for as low expectation value as possible by minimizing (11) leads to The success of variational method depends on a good trial wave function with the good set of parameters.This is the fundamental of the variational method.

Reality and position definiteness of the wave function
The stationary Schrödinger equation for a spinless particle in three dimensions moving under the influence of a potential V(r) is given as 1 Where H is the Hamiltonian of the system and E is the corresponding eigen values.Due to the spontaneous symmetry breaking occurs which may lead to degeneracy in the groundstate wave function the velocity dependent potential is not considered in this approach as pointed out by 8 .
The trial wave function for the system is written as Where assumption is made by the fact that the Hamiltonian in ( 13) has at least one bound state and the quantity ( ) Also using ( 14) and (15) lead to which shows that the potential energy is unaffected by the complex parameter ( ) r χ of the wavefunction.
The kinetic energy of the system is calculated via the use of the Jackson Feenberg identity 9 taking into consideration the anti-hermetian nature of the gradient operator 1 , thus Now taking the gradient of ( 14) results and by taking its complex conjugate, equation(18) becomes ( ) By virtue of the two relations (18) and ( 19), one can easily show that the following relationship holds, With equation ( 17), ( 20) and ( 21), we obtain the expectation value of the kinetic energy operator as The total Hamiltonian of the system can now be written as When compares with the expectation value of the potential energy operator, we observed that the expectation value of the kinetic energy depends on both the real and complex function ( ) As noted before, this wave function does not hold for the velocity-dependent potential.Other consequences like the non degeneracy, the angular momentum and the parity of the wave function have been explicitly exposed 1 .

Applications to the harmonic oscillators
In this section, we will use the technique of variational methods to calculate the ground eigen value of the Harmonic oscillators in three dimensions and compare the minimized energy with the exact result and then extend the applications to the cut-off and the anharmonic oscillators.

Harmonic oscillator
Here, we consider a system with the Hamiltonian where ω is the oscillator frequency.Normalizing (27) and comparing with (28) yields While the total energy for the trial wave function in terms of the variational parameter α is ( )

The cut-off harmonic oscillator
We now consider a system with the Hamiltonian Where ( ) We follow the approach 3 and use as trial function where ω′ is the variational parameter using the ansatz (32), we estimate the ground state energy of the system as and on substituting (36) into (34) yields ( ) Taylor expanding to first order the arguments in the square root of (37) leads to ( )

The anharmonic oscillator
The Hamiltonian of this system is defined as we recover the exact result (27) for both the cut-off and anharmonic oscillators because the trial wave function has the correct functional dependence on r, when the exact functional form is unknown as observed 1 , one can still use the information provided by the variational principle, when the ground state should be real and non-negative and the information should be incorporated into the trial wave function and expressed in the form

Conclusion
We have demonstrated the method and techniques of variational principle to derive the ground state energy for the harmonic, cut-off and anharmonic oscillators with a ground state wave function for a one-body Hamiltonian in three dimensions.Our result shows that the variational method is appropriate in determining the exact solutions of any physical problems.Unlike the result obtained by 1 , our choice of the trial function leads to a generalized ground state eigen value which reduces to exact result when some parameter is adjusted.The generalized ground state eigen value reduces to exact result when some parameter is adjusted.The generalization of our ansatz, equation (33) and the result equation (39) there of, to study the zero point energy of a system had been investigated.Finally, the extension of the result, equation (39) to evaluate the partition function of the Bose-Einstein system is in progress.

∈
. With this observation one look for a ground state wave function by introducing variational parameters to the ansatz ( ) real-valued function.Using(14), we calculate the expectation value of the potential energy as in (22).As noted by1 , since 0 ≥ T E one can always decrease the energy associated with the ansatz (14) by setting this implies making the phase factor a constant.As a consequence, the setting of ( )r χ ∇to zero in the wave function does not affect the result.Therefore, one is justified by choosing as state wave function, which indicates that the wave function is real and does not change sign.
minimum energy and then compare the result with the exact result given by length scale of the Harmonic oscillator as ω λ m h = and on substituting α into (30) yield the same value as the exact result given by (28).
the ground state energy with the trial function (41) results3 since the ground eigen function must be non-degenerate, we assure that to have a zero angular momentum.