_{1}

In this paper, a quantum mechanical Green’s function for the quartic oscillator is presented. This result is built upon two previous papers: first [1], detailing the linearization of the quartic oscillator (qo) to the harmonic oscillator (ho); second [2], the integration of the classical action function for the quartic oscillator. Here an equivalent form for the quartic oscillator action function in terms of harmonic oscillator variables is derived in order to facilitate the derivation of the quartic oscillator Green’s Function, namely in fixing its amplitude.

Following Schiff [

implements the superposition principle satisfied by the wave function because it satisfies a linear partial differential equation, the Schrödinger equation. (Note

Here we show that the Dirac-Feynman [

where

Part II summarizes the results needed from [

Part III begins from first principles and expresses the Action function in terms of harmonic oscillator (ho) variables

In Part IV, we then address the missing piece for the Green’s function, namely, the Amplitude

Part V outlines the extension of these results to the hierarchy of all even power potentials.

The linearization map [

Note both systems are assumed to have the same mass m.

Specifically, the invertible linearization map to the quartic oscillator with mass m and space coordinate y is stated in two parts. First,

or

where y is the space coordinate of the quartic oscillator and we have used the representation

and similarly for sgn(y). This implements the physical requirement that

the potential energies at the two different times, coupled with matching of the signs of the space coordinates. One cycle of the qo corresponds to one cycle of the ho, of course the periods are different.

Second,

and

which results by requiring

Given the matching of the potential energies, the matching of the velocities and the masses of the oscillators for all values of

Further we need

Note: Our convention

Finally, and key to the interchangeability of the qo and ho variables needed here is the standard change of variables in differentiation given by the following: First, it follows from (2.2) that

Second, it follows from (2.3) that

Note from (2.7) and (2.8) that

As stated in the Introduction, the object of this paragraph is to express the defining expression for the qo action in terms of the ho variables and integrate it.

We start with an expression of the qo action in qo variables and transform it to ho variables

Employing

where

Now

where

Therefore

and

Continuing, we have for the first term in the final expression (3.2a)

Therefore

Now paralleling the development in [

with

Therefore,

Hence,

Finally,

where

and

Now we verify that this is indeed

This checks with (4.3) in [

Next,

This checks with (4.3) in [

Similarly

Using the results in Part II, (3.10) can be directly shown to be equal to the result (4.2) in [

where

and

The significance of

(It is important to correct some exponent typos in [

reported. The minus sign on lhs of (2.8) should be a plus. The exponent in (2.9a) and (3.1) in [^{2}

terms should read −1/4. The terms involving

exponent of 1/2. The corresponding equations and pages should be corrected in the arXiv article. Sorry for any inconvenience, but again no errors in the final results!)

Here, (assuming the Dirac-Feynman form of the Green’s function) an amplitude

expression

where

Thus we have on the lhs of (4.1)

where

And we have the rhs of (3.1)

where

The 2nd term in the lhs of (4.1) is equal to the sum of the 2nd and 3rd terms in the rhs of (4.1) for our conservative system.

This leaves only the 1st term of the lhs of (4.1) and the 1st term in the rhs of (4.1). Equating their coefficients and cancelling common factors we obtain

Proceeding with the evaluation of (4.4), we have for the lhs

and for the rhs

Therefore equating (4.5) to (4.6) and canceling the common factors including one given by (2.9), we have

or

or

where a = constant.

Before completing our discussion of the amplitude, we start with the observation that there exists a

Now set

Now set

Here using

Equivalently from (3.18)

We turn to how do we use this structure of the Green’s function to bring it to the form (1.1).

There are two quadratures necessary to fix the connection between the coordinates.

To obtain (1.1) we set

In this section we present a brief outline of the extension of these results to the hierarchy of attractive potentials given by even powers of the space coordinate [

Fundamental to this outline the mapping of the harmonic oscillator extremals onto the extremals of a each member of an hierarchy of attractive oscillators with coordinates

which is the generalization of (2.1). The generalization of (2.2) is given by:

and

and

These mappings take the space-time extremals of the linear oscillator with coordinates

With these mapping in hand, all of the analyses presented in Parts II - IV can then be extended to the members of the hierarchy including the analysis of the corresponding

A quantum mechanical Green’s function

The linearization map originally given in [

The author wishes to thank Professor Howard Lee for insightful discussions and his constant encouragement. The idea to emphasize the quartic oscillator was his.

Finally, the author wishes to acknowledge those who participated in a seminar organized by Robert Varley and David Edwards in AY 2006-2007 to study Feynman Path Integrals, and especially two students, Emily Pritchett and Justin Manning. The seminar provided the original motivation for exploring the extent of the connection between the linear oscillator and the Feynman’s Path Integral Method. As an offshoot of this seminar, special thanks go to my Department of Mathematics colleague Robert Varley, who spent enumerable hours over a four year period of time following the seminar discussing this work with me. His comments, questions and posing of challenging related problems helped to clarify for me many aspects of this work.

Robert L. Anderson, (2016) Green’s Function for the Quartic Oscillator. Applied Mathematics,07,1571-1579. doi: 10.4236/am.2016.714135