Briefly, this is possible because the implementation of the correct quantization condition requires knowledge of QMF singularities and residuals. The residuals are easily calculated by substituting only a few terms of the Laurent expansion into the QHJ equation. As a byproduct of the study of finite state wavefunctions, we have another way of obtaining energy eigenvalues.

This result refers to the zeros of the bound state wave function in the complex plane. In the case of exactly solvable models, the moving poles of the QMF appear only on the real line and all such poles correspond to the nodes of the wave function. The number of such poles increases with energy according to the well-known theorems about the numerical nodes of the wave function.

QUANTUM

## HAMILTON-JACOBI FORMALISM

Classical Hamilton-Jacobi Theory

### QUANTUM HAMILTON-JACOBI FORMALISM 7

In the limit h —> 0, the QHJ equation changes to the classical Hamilton-Jacobi equation (2.1.2). QUANTUM HAMILTON-JACOBI FORMALISM 8 The generalized phase-space-variable conjugate of Ji is known as the angle variable ui{ and is given by the transformation equations.

## 2.1.6) The constant Vi are just the frequencies associated with the periodic motion, and

Quantum Hamilton-Jacobi Equation

Exact Quantization

*Boundary condition*

## 2.3.1)From (2.2.1) it is seen that the QMF satisfies the following equation

*QUANTUM HAMILTON-JACOBI FORMALISM 10 Requirement (2.3.1 ) has two interpretations: (1) as a form of the correspondence**Exact Quantization Condition**QUANTUM HAMILTON-JACOBI FORMALISM 11**Connection with Schrodinger Equation**QUANTUM HAMILTON- JACOBI FORMALISM 12**Singularities of QMF**QUANTUM HAMILTON-JACOBI FORMALISM 13 In the next section we show how to calculate the eigen-values by taking Morse**Energy Spectrum of Morse Oscillator**QUANTUM HAMILTON-JACOBI FORMALISM 14**QUANTUM HAMILTON-JACOBI FORMALISM 15**QUANTUM HAMILTON-JACOBI FORMALISM 16 The correct sign for d 0 is chosen by the condition of square integrability on the wave*

For ground state, first excited state, second excited state • • •, p(x, E) has zero, one, two • • • poles in the potential well. The number of poles p(x, E) in the potential well gives the degree of excitation of the system. The wave function for bound states in one dimension has nodes whose number increases with energy; the wave function for the nth excited state has n nodes in the classical region.

2.4.2) these nodes are reflected as poles in the QMF, and the remainder of the QMF at each pole is - ih. In the next section, we show how to calculate the eigenvalues by taking Morse. The remainder of x(0 at t = 0 is obtained from the integral. which in the t variable gives the remainder to dj.

## II is the contour integral for the contour 71 enclosing the pole y = 0 and Ir l{

*QUANTUM HAMILTON-JACOBI FORMALISM 17**CALCULATION OF WAVE-FUNCTION FOR ES MODELS 20 singularities of the potential. We will make an assumption that QMF has no**Harmonic Oscillator**CALCULATION OF WAVE-FUNCTION FOR ES MODELS 21**CALCULATION OF WAVE-FUNCTION FOR ES MODELS 22 The sum moving pole terms**Morse Oscillator**CALCULATION OF WAVE-FUNCTION FOR ES MODELS 24**CALCULATION OF WAVE-FUNCTION FOR ES MODELS 25**CALCULATION OF WAVE-FUNCTION FOR ES MODELS 26**Poschl-Teller Potential**CALCULATION OF WAVE-FUNCTION FOR ES MODELS 27**CALCULATION OF WAVE-FUNCTION FOR ES MODELS 28**CALCULATION OF WAVE-FUNCTION FOR ES MODELS 29**Eckart Potential**CALCULATION OF WAVE-FUNCTION FOR ES MODELS 30**CALCULATION OF WAVE-FUNCTION FOR ES MODELS 32 Using the values of 61,6', and E in (3.4.16) one gets the differential equation for**Hydrogen Atom**CALCULATION OF WAVE-FUNCTION FOR ES MODELS 34*

In this chapter, we use the QHJ formalism described in the previous chapter to find bound state wavefunctions for several exactly solvable potential problems in one dimension. According to the well-known node theorems of the wave function, the nth excited state corresponds to n zeros on the real line and there will be corresponding n (moving) poles in the QMF and the residue at each pole will be — i as discussed in Chapter 2. We will assume that the QMF has no other singularities in the finite complex plane.

The QMF turns out to be meromorphic, and to fix its form, one needs to know the behavior of QMF for large x in the complex z-plane. In the remaining sections of this chapter we give the details of the calculation of the bound state wave functions for harmonic oscillator, Morse oscillator, Poschl Teller and Eckart potentials and hydrogen atom. For this potential a change of variable is required and we always try to bring the QHJ equation in the new variable to the same form as the above equation.

The QMF p(x, E) has n poles corresponding to the zeros of the wave function, and the residue at each of these poles is - i. The sign of p(x,E) is determined by the square integrability condition of the wave function. 3.1.5) When the above value of p(x, E) is substituted into the wave function equation, the wave function is limited to large x if we choose the positive sign of \iux.

CALCULATION OF WAVE FUNCTION FOR ES MODELS 22 The sum moving pole terms The sum moving pole terms. 3.1.14). The correct sign for c is chosen by the condition of square integrability on the wave function which sets c = — j. The wave function for the Morse oscillator is given by Compare this with the standard Laguerre differential equation. Substituting the value of ijarid c into equation (3.2.13) we have.

3.5.6) Select the appropriate residue using the square integrability property of the wave function 6 and obtain. Substituting (3.5.5) for q in (3.5.4) and expanding the various terms of the resulting equation for large r and comparing the leading terms we obtain.

CONDITIONS FOR

## QUASI-EXACT SOLVABILITY

Introduction to QES

CONDITIONS FOR QUASI-EXACT SOLVABILITY 36 make a simplifying assumption that the point at infinity is an isolated

## 4.1.1) We find that for all the QES potential models studied by us, (4.1.1) and our

Sextic oscillator

Circular potential

CONDITIONS FOR QUASI-EXACT SOLVABILITY 37

### CONDITIONS FOR QUASI-EXACT SOLVABILITY •AH

*A Representation of QES Quantization Rule*

We now highlight some common features of the exactly solvable models studied in this thesis and reported in the article [5]. For the exactly solvable model, the QMF written in terms of suitable variables has the form y. The residuals b\, b\, • • • are calculated using the QHJ equation and require a condition such as that proposed by Leacock and Padgett, or .

### CONDITIONS FOR QUASI-EXACT SOLVABILITY 39

CONDITIONS FOR QUASI-EXACT SOLUBILITY 40 where Res stands for the residue and the middle term n corresponds to the contribution.

CONDITIONS FOR QUASI-EXACT SOLVABILITY 40 where Res stand for the residue and the middle term n, corresponds to the contri-

*Sextic Oscillator*

### CONDITIONS FOR QUASI-EXACT SOLVABILITY 41

*Sextic Oscillator with a Centrifugal Barrier*

4.4.3) p(x, E) has poles at x = 0 and since the potential is symmetric, it has moving poles on either side of the origin. We will only consider the case s > | so the coefficient of the centrifugal term, i is positive.

CONDITIONS FOR QUASI-EXACT SOLVABILITY 43

*Circular Potential*

CONDITIONS FOR QUASI-EXACT SOLVABILITY 45

CONDITIONS FOR QUASI-EXACT SOLVABILITY 46

CONDITIONS FOR QUASI-EXACT SOLVABILITY 47

*Hyperbolic Potential*

CONDITIONS FOR QUASI-EXACT SOLVABILITY 48

### CONDITIONS FOR QUASI-EXACT SOLVABILITY 4!)

If we equate the sum of all residuals of fixed poles, the moving poles, with the pole at infinity, we have the following relationship. In the cases where for certain ranges of potential parameters both residuals are acceptable, one must accept all such answers and work out the consequences. In addition to the QES potentials above, we now take three classes of QES potentials [9,10] and find the quasi-exact solubility conditions within our approximation.

The point y = 0 corresponds to a complex value of x and therefore we cannot insist on the finiteness of the wave function at x = 0. We will only note that choosing a positive sign in (4.4.19) leads us to the correct condition.

CONDITIONS FOR QUASI-EXACT SOLVABILITY 52

CONDITIONS FOR QUASI-EXACT SOLVABILITY 53

CONDITIONS FOR QUASI-EXACT SOLVABILITY 54

CONDITIONS FOR QUASI-EXACT SOLVABILITY 55

### CONDITIONS FOR QUASI-EXACT SOLVABILITY 56

If we now compare the sum of residues due to fixed poles, the moving poles, and that at infinity, to zero, we have the following relation.

CONDITIONS FOR QUASI-EXACT SOLVABILITY 58

### CONDITIONS FOR QUASI-EXACT SOLVABILITY 59

*Quartic Oscillator*

Therefore the assumption that the point at infinity is an isolated singular point of the QMF is inconsistent with QHJ for the real parameter a, ft, 7 and 6. We repeat the analysis given for the sextic oscillator, assuming that the point at infinity is a isolated singular point, a pole of some order m. Thus, the wave function of the corresponding bound state for large x We shall conclude this chapter with a brief analysis of the quartic and harmonic oscillator and give some remarks on polynomial potentials of degree other than six.

CONDITIONS FOR QUASI-EXACT SOLVABILITY 60

CONDITIONS FOR QUASI-EXACT SOLVABILITY 61

*Summary and Observations*

CALCULATION OF

## WAVE-FUNCTIONS FOR QES MODELS

*CALCULATION OF WAVE-FUNCTIONS FOR QES MODELS 63**Sextic Oscillator**CALCULATION OF WAVE-FUNCTIONS FOR QES MODELS 64**CALCULATION OF WAVE-FUNCTIONS FOR QES MODELS 65**CALCULATION OF WAVE-FUNCTIONS FOR QES MODELS 66**CALCULATION OF WAVE-FUNCTIONS FOR QES MODELS 67 where N is the normalizing factor. The value of a n is given bv**CALCULATION OF WAVE-FUNCTIONS FOR QES MODELS 68**Hyperbolic Potential**CALCULATION OF WAVE-FUNCTIONS FOR QES MODELS 69 Therefore the above equation becomes**CALCULATION OF WAVE-FUNCTIONS FOR QES MODELS 70**CALCULATION OF WAVE-FUNCTIONS FOR QES MODELS 71**CALCULATION OF WAVE-FUNCTIONS FOR QES MODELS 72 On integrating and substituting the values of b x , b z and c x we get the expression for**CALCULATION OF WAVE-FUNCTIONS FOR QES MODELS 73**Concluding Remarks**CALCULATION OF WAVE-FUNCTIONS FOR QES MODELS 74*

Since o3 has two values, the exact value is fixed by the square integrability condition on the wave function. The general strategy for obtaining the wave functions is the same as that discussed for exactly solvable models in Chapter 3. Thus for a fixed value of n, and hence for a given set of potential parameters, the wave functions for all states that can be solved have the same number of zeros equal to n.

CALCULATION OF WAVE FUNCTIONS FOR QES MODELS 69 Therefore, the above equation becomes Therefore, the above equation becomes. Since the potential is symmetrical, there are moving poles on either side and we therefore assume that P(y) has the form shown below. 5.2.25) The wavefunction for this model is calculated as follows. CALCULATION OF WAVE FUNCTIONS FOR QES MODELS 72 When integrating and substituting the values of bx, bz and cx we get the expression for When integrating and substituting the values of bx, bz and cx we get the expression for the wave function in the variable x if.

The general feature of the zeros of the wave function for the sextic oscillator also holds for the QES hyperbolic potential. In particular, for a given potential, all exactly solvable wave functions have the same total number of (real and complex) zeros. Our study of bound wave functions in this chapter shows the following similarities and differences between the exact solvable and QES models.

For each bound state in one dimension, the kth excited state wavefunction has fc nodes on the real axis. This statement is general and applies to all models including exactly solvable and QES potentials, the study in Chapter 3 shows that the QMF for exactly solvable models has moving poles which are consistent with the nodes of the wave function. However, this property is not true for QES potentials where the QMF has poles away from the real axis, in addition to the poles on the real axis corresponding to the nodes of the wave function.

For the QES potentials, only part of the energy spectrum and the corresponding wave functions can be calculated accurately.

## CONCLUSIONS AND OUTLOOK

### CONCLUSIONS AND OUTLOOK 76

The quantization condition given by Leacock and Padgett applies to discrete systems that can be reduced to one-dimensional problems. It will be interesting to formulate an exact quantization condition for non-separable systems in higher dimensions, and examine its relation to the well-known existing semi-classical schemes and see applications for chaotic systems. Here again, any idea about the knowledge of the location of moving poles has some relation to classical trajectories.

We have tried to study the QHJ formalism for a harmonic oscillator, which is a test case of any computational scheme. One can compute the asymptotic value of the QMF for large x and use this answer as input to numerical integration of QHJ. Detailed research is ongoing and an interesting approximation scheme for a harmonic oscillator is expected based on the preliminary results.

When bound states are calculated for some potentials such as Rosen Morse hyperbolic potential, it is found that the application of boundary condition p(x, E) - ^ pci{x, E), leads one carefully to select different residues for different potential series boundaries . Beyond this and further, further analysis leads to different set of energy spectrum and wave functions for such different ranges of parameters in the potential. This corresponds to the well-known result about phases of super-symmetry in Rosen Morse potential 13].

Other such potentials, for example the trigonometric Scarf potential [14], which exhibit different phases for different ranges of potential parameters, can also be explored in ours.

CONCLUSIONS AND OUTLOOK 77

REFRENCES