Scaling Rule, Energy Distribution and Energy Level Analysis for Morse Oscillator by Virtue of Hermann-Feynman Theorem

I.   INTRODUCTION

Morse oscillator [1] is an important model in molecular physics and frequently used in molecular physics and chemical physics. It has brought continuous interest of physicists [2–5] from different points of view since the pioneer work of Morse in 1929 (see also the book Quantum Chemistry [6]). For a particle in the Morse potential, the Hamiltonian is

where $D$ and $a\ $describe the depth and width of the potential well respectively. It is clear that the Morse potential is the combination of a short-range repulsion (the term $D\mathrm{e}^{-2x/a}$, the range of potential is $a/2$) and a longer-range attraction potential (the term $-2D\mathrm{e}^{-x/a}$, the range of potential is $a$).

The energy level formula of the Morse oscillator has been known for long,

where $n=0,1,\cdots$. The minimum of $V\left( x\right)$ lies at $x=0$, $V\left( 0\right) =-D$ and $V\left( +\infty \right) =0$. Thus $D$ is called the (classical) dissociation energy. Since $E_{0}= -D \left(1-\dfrac{1}{2} \sqrt{\dfrac{\varepsilon }{D}}\right) ^{2} > -D$, the true energy required for dissociation is $-E_{0}=D\left( 1-\dfrac{1}{2}\sqrt{\dfrac{\varepsilon }{D}} \right) ^{2} < D$.

In this work we shall derive a scaling rule for the Morse oscillator which can help us to analyze its energy level more conveniently. Then we explore the separate contribution from the repulsive potential and the attractive potential so that people may predict the behavior and character of the oscillator more deeply. Our discussion is arranged as follows. In Sec. II we derive a scaling theorem for the Morse oscillator. Based on this rule and by virtue of Hermann-Feynman theorem [7, 8], we respectively discuss Morse oscillator's energy level and energy distribution in Sec. III and IV. In Sec. V we derive some new recursive relations of the energy matrix elements of the potential and other similar operators.

II.   NEW SCALING THEOREM FOR MORSE OSCILLATOR

For this purpose, let us consider the more general Hamiltonian

where $D_{+},D_{-}$ are two parameters. The reason we introduce two different parameters lies in that we can use Hermann-Feynman (H-F) theorem to derive the energy contribution from $D_{+}{\rm{e}}^{-2x/a}$ and $-2D_{-}{\rm{e}}^{-x/a}$respectively (see Eqs. (16$-$18) below). The minimum of $W \left( x\right)$ lies at

The (classical) dissociation energy is $W \left( a\log \dfrac{D_{+}}{ D_{-}}\right) = -\dfrac{D_{-}{^{2}}}{D_{+}}$.

For arbitrary positive free parameters $\alpha ,\beta ,\lambda$ and $\mu$ we make a scaling transformation from $H\left( x,p_{x}\right) _{\left( D_{+},D_{-},a,m\right) }\rightarrow H\left( x,p_{x}\right) _{\left( \alpha D_{+},\beta D_{-},\lambda a,\mu m\right) },$ then we have

which implies the scaling property

The scaling in quantum optics is named squeezing [9, 10]. Note that the energy of Morse oscillator is invariant no matter the Hamiltonian is expressed as $H\left( y^{\prime },p_{y}^{\prime }\right)$ or as $H\left( x,p_{x}\right)$ if their parameters are the same, so from Eq.(7) we have the energy scaling rule for the Morse oscillator

Particularly, taking $\alpha =\lambda =1,$ $\beta = \dfrac{D_{+}}{D_{-}},\ \mu = \left( \dfrac{D_{-}}{D_{+}}\right) ^{2},$ we have

Correspondingly,

III.   ENERGY LEVEL ANALYSIS

Note that

comparing Eq.(12) with Eq.(1), and according to the energy formula (2) which corresponds to the Hamiltonian (1), we know that the energy level for $H\left( x,p_{x}\right) _{\left( D_{+},D_{+},a,m^{\prime }\right) }$ is

Then from Eq.(11) we know that the eigen-energy of $H\left( y^{\prime },p_{y}^{\prime }\right) _{\left( D_{+},D_{-},a,m\right) }$ is

and it is determined by the potential shape. Eq.(14) is the result of using the scaling rule.

IV.   THE ENERGY DISTRIBUTION

Hence by H-F theorem and using Eq.(14) we derive

Since $\left\langle n\left|-2 D_{-} {\rm{e}}^{-x/a}\right| n\right\rangle < 0,$ this equation implies that

This inequality answers why the $n$ quantum number, which obeys the inequality, $n_{0}\leqslant n < 2n_{0}+1/2,$ does not relate to physical states, although the corresponding energies seem well-behaved ($0\geqslant E_{n}>-\dfrac{D_{-}{^{2}}}{D_{+}}$, the bottom of the potential well). Hence Eq.(15) reveals that the stronger attractive potential $\left( D_{-}\right)$ or the weaker repulsive potential $\left( D_{+}\right)$, the more bound-states are allowed in the theory.

We also have

It is remarkable that

The total potential energy is

While the kinetic energy contribution is

Combining Eq.(20) and Eq.(21) one can really check:

as expected.

V.   RECURSIVE RELATIONS OF POTENTIAL’S MATRIX ELEMENTS

In this section we shall derive some recursive relations of potential’s matrix elements in energy representation. For this purpose we introduce two new operators

with free parameters $s,t.$ Noting that when $s=-1/a$, $-2/a$, $X\left( s\right) \propto -2D_{-}{\rm{e}}^{-x/a}$, $D_{+}{\rm{e}}^{-2x/a},$ respectively. Then we calculate

when $s=-t$, $\left[ X\left( s\right) ,P\left( -s\right) \right] =i\hbar$. This equation indicates that $X\left( s\right)$ is conjugate to $P\left( -s\right)$, in this sense, we may look them as a pair of generalized coordinate-like operators.

We also have the commutation relation

Further

so

Taking the matrix elements of Eq.(28) in energy-eigenstates basis, we obtain

If we define

then Eq.(29) is simplified as

this is a new recursive relation for $X_{k}.$ Eq.(31) allows us to calculate matrix elements $X_{k}=\left\langle m\left|{\rm{e}}^{kx/a}\right| n\right\rangle$ recursively for integers $k$.

Based on Eq.(31) and using induction method, in the following we can prove that when $k\geqslant 0$, $\left\langle m\left|{\rm{e}}^{kx/a}\right| n\right\rangle=0$ if $\left\vert m-n\right\vert >k$.

First we assume that $n_{0}=\dfrac{D_{-}}{D_{+}}\sqrt{\dfrac{D_{+}}{ \varepsilon }}-\dfrac{1}{2}$ is not an integer, so $k^{2}-\left( m+n-2n_{0}\right) ^{2}\neq 0$ for any integers $k,m,n$.

(1) For $k=0$ case, it is obvious that $\langle m|1| n\rangle=0$ if $\left\vert m-n\right\vert >0$.

(2) Supposing $\left\langle m\left|{\rm{e}}^{kx/a}\right| n\right\rangle=0$ for any nonnegative integer $k\leqslant k_{0}$ and $\left\vert m-n\right\vert >k$, then for $k=k_{0}+1$ and $\left\vert m-n\right\vert >k,$ Eq.(31) reads

Since $\left\vert m-n\right\vert >k$, so $k^{2}-\left( m-n\right) ^{2}\neq 0$. We also have $k^{2}-\left( m+n-2n_{0}\right) ^{2}\neq 0$ by assumption, so $X_k=\left\langle m\left|{\rm{e}}^{kx/a}\right| n\right\rangle=0$.

The above (1) and (2) complete the induction and prove that $\left\langle m\left|{\rm{e}}^{kx/a}\right| n\right\rangle=0$ when $\left\vert m-n\right\vert >k\geqslant 0$ and $n_{0}$ is not an integer.

Since $\left\langle m\left|{\rm{e}}^{kx/a}\right| n\right\rangle$ is a continuous function of the parameters $D_{+},D_{-},a,m$, particularly, a continuous function of $n_{0}$, so $\left\langle m\left|{\rm{e}}^{kx/a}\right| n\right\rangle=0$ when $\left\vert m-n\right\vert >k\geqslant 0$ even if $n_{0}$ is an integer number.

The matrix elements of $P\left( s\right)$ can be derived by Eq.(25) straightforwardly,

So we immediately get the conclusion that $\langle n|P(s)| n\rangle \equiv 0$ for any real parameter $s$, and $\langle m|P(k / a)| n\rangle =0$ if $\left\vert m-n\right\vert >k\geqslant 0$.

Examples,

1. In $m=n$ case, $A_{k}=k^{2}\left[ k^{2}-4\left( n-n_{0}\right) ^{2}\right] ,$

$k=1$:

Although ${\rm{e}}^{x/a}$ is also positive-definite operator, Eq.(34) does not imply that we should have $n<n_{0}-\dfrac{1}{2}$ to guarantee $\left\langle n\left|{\rm{e}}^{x/a}\right| n\right\rangle > 0$, because $\left\langle n\left|{\rm{e}}^{x/a}\right| n\right\rangle$ is in fact infinite when $n_{0}-1/2\leqslant n < n_{0}$.

2. For $k=0$ case, Eq.(26) becomes

therefore we have

Thus Eq.(19) is simply the special case of Eq.(36) when $m=n$.

Finally, we provide an example similar to Morse-like potentials arising in a class of diatomic molecules, it is the Lennard-Jones potential [11] with the form

where $r$ is the distance between two atoms, $\varepsilon$ and $\sigma$ are two parameters of the potential curve, the minimum of $U\left( r\right)$ lies at $r=2^{1/6}\sigma .$ Comparing with the Morse potential $V\left(x\right)=D\left(\rm{e^{-2\mathit{x}/\mathit{a}}}-2\rm{e^{-\mathit{x}/\mathit{a}}}\right)$ in Eq.(1) one can see $\displaystyle\left(\sigma/r\right)^{12}=\left[\left(\sigma/r\right)^6\right]^2$ is analogous to $\rm{e^{-2\mathit{x}/\mathit{a}}}= \left(\rm{e^{-\mathit{x}/\mathit{a}}}\right)^2,$ thus one may discuss scaling rule and energy distribution for Lennard-Jones potential following the above procedures of this paper, this may enrich molecular simulation theory.

VI.   CONCLUSION

In summary, for the first time we have derived the scaling rule for Morse oscillator such that the Hermann-Feynman theorem can be applied to investigating the contribution of the attractive and the repulsive potentials to the total energy. Further, we have derived the recursive relations of the matrix elements of the potentials of Morse Hamiltonian, all these discussions enrich the Morse oscillator theory and offer a practical way for calculating the corresponding matrix elements. For the generalized Hermann-Feynman theorem suitable for ensemble average and its applications we refer to Refs. [12, 13].