††thanks: ORCiD: 0000-0001-6841-9850

# The envelope theory as a pedagogical tool

Claude Semay<EMAIL_ADDRESS>Maud Balcaen
<EMAIL_ADDRESS>Service de Physique Nucléaire et
Subnucléaire, Université de Mons, UMONS Research Institute for Complex
Systems, Place du Parc 20, 7000 Mons, Belgium

###### Abstract

The envelope theory is a reliable and easy to implement method to solve time
independent Schrödinger-like equations (eigenvalues and eigenvectors). It is
mainly useful to solve many-body systems since the computational cost is
independent from the number of particles. Due to its simplicity, this method
can also be used as a pedagogical tool. This is shown here for the soft-
Coulomb potential $-k/\sqrt{x^{2}+d^{2}}$ in one dimension, characterised by a
bias distance $d$. Such interaction is used for the study of excitons,
electron-hole bound pairs where the two charges are kept separated in two
different one-dimensional regions (quantum wires).

Envelope theory, Excitons, quantum wires, teaching of quantum mechanics

##  I Introduction

Finding the eigenvalues and eigenvectors of Hamiltonians is a fundamental
problem in quantum mechanics. Various methods can be used as the variational
one, the WKB approximation or the perturbation theory, for instance flug99 ;
griff18 . All these methods rely on the construct of good approximations for
the wavefunctions. The principle is different for the envelope theory (ET) in
which the Hamiltonian under study $H$ is approximated by an auxiliary
Hamiltonian $\tilde{H}$ whose solutions are exactly known hall80 ; hall83 ;
sema13 ; chev21 . The approximate eigenvalues of $H$ are obtained by an
extremisation procedure applied to the eigenvalues of the auxiliary
Hamiltonian. A practical choice for $\tilde{H}$ is a harmonic oscillator
Hamiltonian. If the accuracy is not great, the method is reliable and
analytical upper or lower bounds can be obtained in favourable situations. The
ET is mainly useful for many-body systems since the computational cost is
independent from the number of particles, but it can also be used as a
pedagogical tool as it is very easy to implement and allows the computation of
the whole spectrum.

In this paper, the ET is applied to the study of the soft-Coulomb potential
$-k/\sqrt{x^{2}+d^{2}}$ in one dimension, characterised by a bias distance
$d$. This interaction is chosen for three reasons: (i) It is characteristic of
excitons, electron-hole bound pairs where the two charges are kept separated
in two different one-dimensional regions. These systems, named quantum wires,
appear in solid state physics schr15  and in various biological processes
mara18 ; (ii) It is bounded from below and from above by two potentials for
which known exact solutions can be used as control lower and upper bounds ;
(iii) It is studied in a recent paper by the variational method gras17  which
allows interesting comparisons.

The recipe to use the ET for a two-body system in one dimension is presented
in Sect. II. The ET is applied to the one-dimensional soft-Coulomb potential
$-k/\sqrt{x^{2}+d^{2}}$ in Sect. III, where eigenvalues and eigenvectors are
computed and compared with results coming from accurate numerical calculations
and from a variational method. Concluding remarks are given in Sect. IV.

##  II The envelope theory

In this section, matters from several papers sema13 ; sema19 ; cimi21  are
collected and summarised to describe how to compute the approximate ET
solutions for the generic two-body Hamiltonian given by

$H=T(\hat{p})+V(x),$ (1)

where $x$ is the relative position, $\hat{p}=-i\hbar\,d/dx$ the momentum
conjugated with $x$, $T$ the kinetic term and $V$ the potential. In the
following, it is necessary to make the difference between the momentum as an
operator, denoted by $\hat{p}$, and the momentum as a simple variable, denoted
by $p$. The domain of $x$ can be $\mathbb{R}_{+}$ or $\mathbb{R}$. In this
last case, $V$ is an even function of $x$ for physical and practical reasons.
$T$ is always expected to be an even function of $p$ sema16 . The approximate
energy $E$ for the $n$th level is given by an extremisation procedure. But it
can be shown that it is equivalent to solve the following system for each
value of the quantum number $n$ sema13

$\displaystyle E$ $\displaystyle=T(p_{0})+V(x_{0}),$ (2) $\displaystyle
x_{0}p_{0}$ $\displaystyle=Q_{n}\hbar,$ (3) $\displaystyle
p_{0}T^{\prime}(p_{0})$ $\displaystyle=x_{0}V^{\prime}(x_{0}),$ (4)

where $X^{\prime}(z)=dX/dz$, $x_{0}>0$, $p_{0}>0$ and $Q_{n}$ is defined by

$\displaystyle Q_{n}$ $\displaystyle=n+\frac{1}{2}\quad\textrm{for}\quad
x\in\mathbb{R},$ (5) $\displaystyle Q_{n}$
$\displaystyle=n_{\textrm{o}}+\frac{1}{2}\quad\textrm{for}\quad
x\in\mathbb{R}_{+},$ (6)

where $n=0,1,2,3,\ldots$ and $n_{\textrm{o}}=2n+1=1,3,5,7,\ldots$ The
restriction to odd numbers $n_{\textrm{o}}$ insures that the wavefunction
vanishes at $x=0$. The structure of $Q_{n}$ comes from the use of the harmonic
oscillator Hamiltonian as the auxiliary Hamiltonian. Consequently, the
corresponding approximate ET wavefunction is an harmonic oscillator one. For
$x\in\mathbb{R}$, it is written griff18

$\langle
x|n\rangle=\psi_{n}(x)=\left(\frac{\lambda^{2}}{\pi}\right)^{1/4}\frac{1}{\sqrt{2^{n}n!}}H_{n}(\lambda
x)\exp\left(-\lambda^{2}x^{2}/2\right),$ (7)

where $H_{n}$ is an Hermite polynomial and $\lambda=\sqrt{Q_{n}}/x_{0}$. For
$x\in\mathbb{R}_{+}$, $\psi_{n_{\textrm{o}}}(x)$ must be multiplied by
$\sqrt{2}$ to keep the correct normalisation. It can be shown that

$\langle n|x^{2}|n\rangle=x_{0}^{2}\quad\textrm{and}\quad\langle
n|\hat{p}^{2}|n\rangle=p_{0}^{2}.$ (8)

So, $p_{0}$ can be interpreted as the mean momentum of each particle and
$x_{0}$ as the mean distance between the two particles. These physical
quantities appear in three equations giving the definition of the energy (2),
the rule for the quantisation (3) and the equation of motion (4) as a
transcendental equation. If this system has a nice semi-classical
interpretation sema13 , the ET is a full quantum method giving eigenvectors
and eigenvalues. In natural units, $\hbar=1$ in (3).

The structure of $T$ is usually simpler that the one of $V$. That is the
reason why it is generally more convenient to work in the configuration space.
But the symmetry between the momentum and the relative distance in the system
(2)-(4) indicates that computations can be performed indifferently in the
configuration space (with $\hat{p}=-i\hbar\,d/dx$) or in the momentum space
(with $\hat{x}=i\hbar\,d/dp$).

In the ET, for each level $n$, the kinetic part $T(p)$ and the potential part
$V(x)$ are respectively “enveloped” by the following tangent quadratic
functions (see the example in Sect. III.3)

$\displaystyle\tilde{T}(p)$
$\displaystyle=T(p_{0})+\frac{T^{\prime}(p_{0})}{2p_{0}}\left(p^{2}-p_{0}^{2}\right),$
(9) $\displaystyle\tilde{V}(x)$
$\displaystyle=V(x_{0})+\frac{V^{\prime}(x_{0})}{2x_{0}}\left(x^{2}-x_{0}^{2}\right).$
(10)

This is the origin of the name of the method hall80 ; hall83 . The enveloping
functions are such that $\tilde{T}(\hat{p})+\tilde{V}(x)=\tilde{H}$ which is
the auxiliary Hamiltonian for the $n$th level ($p_{0}$ and $x_{0}$ depend on
$n$ by the set of equations (2)-(4)). It can be checked from (8)-(10) that
$E=\langle n|\tilde{H}|n\rangle$. If $T(p)$ ($V(x)$) is quadratic,
$\tilde{T}(p)=T(p)$ ($\tilde{V}(x)=V(x)$). By defining the two functions
$b_{T}(p^{2})=T(p)$ and $b_{V}(x^{2})=V(x)$, it can be shown that $E$ is an
upper (lower) bound of the genuine eigenvalue if $b_{T}^{\prime\prime}$ and
$b_{V}^{\prime\prime}$ are both concave (convex) functions. If the second
derivative is vanishing for one of these functions, the variational character
is solely ruled by the convexity of the other one. It is easy to check that
the situation $b_{T}^{\prime\prime}=b_{V}^{\prime\prime}=0$ corresponds to the
harmonic oscillator Hamiltonian for which the system (2)-(4) gives the exact
solution. In the other cases, the variational character of the solution cannot
be guaranteed. Let us note that $E$ has an analytical form for a large variety
of Hamiltonians for arbitrary values of the dimension sema19 ; sema15a .

##  III The exciton Hamiltonian

The Hamiltonian $H_{\textrm{eh}}$ for the electron-hole pair is given by

$H_{\textrm{eh}}=\frac{\hat{p}^{2}}{2m}-\frac{k}{\sqrt{r^{2}+d^{2}}},$ (11)

where $r$ ($\in\mathbb{R}$) is the relative position between the electron and
the hole, $\hat{p}$ the conjugate momentum, $m$ the reduced mass of the pair
and $k=e^{2}/(4\pi\epsilon_{0}\epsilon_{r})$, with $e$ the elementary electric
charge, $\epsilon_{0}$ the dielectric permittivity in vacuo and $\epsilon_{r}$
the relative dielectric permittivity of the material.

###  III.1 Dimensionless analysis

It is always interesting to work with dimensionless variables in order to make
apparent what are the relevant physical quantities and to simplify the
formulation of the problem. By using the effective Bohr radius
$a_{\textrm{B}}=4\pi\epsilon_{0}\epsilon_{r}\hbar^{2}/(me^{2})$ and the
effective Rydberg constant
$\textrm{R}_{\textrm{y}}=me^{4}/(32\pi^{2}\epsilon_{0}^{2}\epsilon_{r}^{2}\hbar^{2})=k/(2a_{\textrm{B}})$,
a dimensionless Hamiltonian $H=H_{\textrm{eh}}/(2\textrm{R}_{\textrm{y}})$ can
be defined in terms of a dimensionless bias parameter $D=d/a_{\textrm{B}}$ and
a dimensionless position $x=r/a_{\textrm{B}}$

$H=-\frac{1}{2}\frac{d^{2}}{dx^{2}}-\frac{1}{\sqrt{x^{2}+D^{2}}},$ (12)

whose dimensionless eigenvalues will be designed by $E$. Only dimensionless
quantities will be used in the following. To recover physical quantities, $x$
must be multiplied by $a_{\textrm{B}}$ and $E$ by $2\textrm{R}_{\textrm{y}}$.
Both $a_{\textrm{B}}$ and $\textrm{R}_{\textrm{y}}$ are pertinent units for
contexts in which excitons appear. $D$ is around unity for typical
semiconductors.

###  III.2 Bounding potentials

The soft-Coulomb potential $-1/\sqrt{x^{2}+D^{2}}$ is bounded from below by
the Coulomb potential $-1/|x|$ for $x\in\mathbb{R}$ (see Fig. 1). So,
according to the comparison theorem sema11 , the eigenvalues of

$H_{\textrm{C}}=-\frac{1}{2}\frac{d^{2}}{dx^{2}}-\frac{1}{|x|}$ (13)

are lower bounds of the eigenvalues of $H$. The spectrum of $H_{\textrm{C}}$
is analytical jara09

$E_{\textrm{C}}=-\frac{1}{2(n+1)^{2}},$ (14)

with $n=0,1,2,3,\ldots$ associated with wavefunctions vanishing at $x=0$ and
with $2n$ nodes at finite $x$ values on $\mathbb{R}$. To correspond to our
numbering scheme (5)-(6), (14) must be rewritten

$E_{\textrm{C}}=-\frac{2}{(n_{\textrm{o}}+1)^{2}}.$ (15)

For large values of $n_{\textrm{o}}$, the eigenfunctions of $H$ and
$H_{\textrm{C}}$ are characterised by a large extension, that is to say a
significant part in regions where both potentials are very similar. So it is
expected that $E$ and $E_{\textrm{C}}$ will be very close in these situations.

An easy way to solve approximately a Schrödinger equation around an
equilibrium point is to use a harmonic approximation of the potential.
Expanding the soft-Coulomb potential in power of $x$ around 0, $H$ reduces to
the harmonic oscillator Hamiltonian

$H_{\textrm{HO}}=-\frac{1}{2}\frac{d^{2}}{dx^{2}}+\frac{x^{2}}{2D^{3}}-\frac{1}{D}$
(16)

at the lowest order. The corresponding eigenvalues are given by

$E_{\textrm{HO}}=\left(n+\frac{1}{2}\right)\frac{1}{D^{3/2}}-\frac{1}{D}.$
(17)

It is easy to show that the quadratic potential in $H_{\textrm{HO}}$ is always
larger that the one in $H$ (see Fig. 1). So, according to the comparison
theorem, $E_{\textrm{HO}}$ are upper bounds of $E$. Moreover, both potentials
are very similar for small values of $x$. It is then expected that $E$ and
$E_{\textrm{HO}}$ will be very close for values of $n$ close to 0,
corresponding to eigenfunctions characterised by a small extension.

Figure 1: The soft-Coulomb potential for $D=2$ (solid black) with the Coulomb
potential (dashed gray) and the quadratic potential with $D=2$ (dashed gray).

###  III.3 ET approximations

The ET solutions for $H$ are computed by solving first (3)-(4) for a
nonrelativistic kinematics and the soft-Coulomb potential. The equation for
$x_{0}$ is then

$x_{0}^{4}=Q_{n}^{2}\left(x_{0}^{2}+D^{2}\right)^{3/2}.$ (18)

Analytical solution for $x_{0}$ can be written in terms of the solutions of a
quartic equation, but the expression is so complicated that it is not usable
in practice. Graphically, it is easy to check that (18) has only one positive
solution for $x_{0}^{2}$. It is then very easy to compute it by a numerical
procedure or a software like Mathematica®. The ET approximations
$E_{\textrm{ET}}$ for the energies are upper bounds since
$b_{T}^{\prime\prime}=0$ and $b_{V}(z)=-(z+D^{2})^{-1/2}$ is a concave
function. Finally, the ET upper bounds are given by

$E_{\textrm{ET}}=\frac{Q_{n}^{2}}{2x_{0}^{2}}-\frac{1}{\sqrt{x_{0}^{2}+D^{2}}}\quad\textrm{with}\quad
x_{0}^{4}=Q_{n}^{2}\left(x_{0}^{2}+D^{2}\right)^{3/2}.$ (19)

As presented in Sect. III.2, the solutions for $H$ must be close to the
solutions for $H_{\textrm{C}}$ and $H_{\textrm{HO}}$ under certain conditions.
Let us check this from (19). Provided $D$ is not too large, it is expected
from (18) that $x_{0}^{4}\approx Q_{n}^{2}x_{0}^{3}$, that is to say
$x_{0}\approx Q_{n}^{2}$, when $n\gg 1$. In this case, the parameter $D$
disappears in the dominant term and (19) reduces to

$E_{\textrm{ET}}\approx-\frac{2}{(2n+1)^{2}}.$ (20)

This last formula can be compared with $E_{\textrm{C}}$ for odd values of $n$:
(20) is clearly an upper bound of (15).

If $n$ is a small integer and $D$ is not too small, an approximate solution
for (18) is $x_{0}^{4}\approx Q_{n}^{2}D^{3}$, that is to say
$x_{0}^{2}\approx Q_{n}D^{3/2}$. In this case, at the lowest order in $n$,
(19) reduces to

$E_{\textrm{ET}}\approx\left(n+\frac{1}{2}\right)\frac{1}{D^{3/2}}-\frac{1}{D},$
(21)

which is identical to $E_{\textrm{HO}}$. This is not surprising since the ET
gives the exact solutions for a harmonic oscillator Hamiltonian.

Figure 2: Lowest energies for $H$ with $D=2$: accurate values obtained with
the FGH method (black triangle) and upper bounds from the ET (gray diamond).
The exact energies for $H_{\textrm{C}}$ (gray circle) and $H_{\textrm{HO}}$
with $D=2$ (gray square) are also indicated.

Accurate numerical eigenvalues and eigenvectors of $H$ are computed with the
Fourier Grid Hamiltonian (FGH) method mars89 ; sema00  which is very powerful
and very easy to use for one-dimensional time independent Schrödinger-like
equations. In Fig. 2, theses energies are compared with $E_{\textrm{ET}}$ and
the other bounds $E_{\textrm{HO}}$ and $E_{\textrm{C}}$, for $D=2$ and the
lowest values of $n$. The ET approximations are good for all values of $n$.
The lower bound $E_{\textrm{C}}$ which is only defined for odd numbers $n$
improves with increasing $n$, as expected. On the contrary, the upper bound
$E_{\textrm{HO}}$ degrades rapidly with increasing $n$, as expected also. It
is worth drawing attention that the upper bounds $E_{\textrm{HO}}$ are
computed with a unique quadratic potential in (16) (see Fig. 1). The upper
bounds $E_{\textrm{ET}}$ are computed with a different envelope potential
$\tilde{V}(x)$ for each value of $n$ (see Fig. 3). Wavefunctions for the
ground state and the first excited state, computed with the FGH method and the
ET, are compared in Fig. 4 for three values of $D$. The agreement is good but
the ET approximations decrease faster with $x$. This is due to the harmonic
oscillator nature of the approximate wavefunctions which decrease as Gaussian
functions while the exact solutions decrease as exponential functions.

Figure 3: Four first envelopes $\tilde{V}$ (dashed gray) for $n=\\{0,1,2,3\\}$
of the soft-Coulomb potential (solid black) for $D=2$.  Figure 4: Ground
(black) and first excited (gray) states of $H$ for three values of $D$:
accurate wavefunctions computed with the FGH method (dashed), and approximate
wavefunctions computed with the ET (solid).

###  III.4 Comparison with a variational method

The ground and first excited states of $H$ have been computed in a recent
paper by a variational method using the ground and first excited states of the
harmonic oscillator Hamiltonian as trial states gras17 . The results from this
paper are compared with the ET results in Fig. 5 as a function of $D$. Let us
mention that the dimensionless numbers obtained with the formulas in gras17
must be divided by 2 to be compared with our results, because the energies are
given in unit of $\textrm{R}_{\textrm{y}}$, while they are given in unit of
$2\textrm{R}_{\textrm{y}}$ in this paper (see (12)). It is clear that the
variational upper bounds are better that the upper bounds given by the ET.
Nevertheless, the ET can give the whole spectra with the same computational
cost. This is not the case in gras17 , where different integrations must be
performed to compute ground and first excited states. Moreover, the
variational principle states that the expectation value of a Hamiltonian for
an arbitrary trial state always gives an upper bound of the ground state. But
an upper bound of an excited state can be reliably computed with a trial state
if it is orthogonal to all exact states below this excited state griff18 . So,
for a one dimensional system, an upper bound of the ground state can be
computed with an even trial wave function and an upper bound of the first
excited state with an odd trial wave function. The computation of the other
excited states is much more complicated and requires the expansion of trial
states in an (orthonormal) basis macd33 .

Figure 5: Energies of the ground (3 curves below) and first excited (3 curves
above) states of $H$ as un function of $D$: accurate values obtained with the
FGH method (solid black), upper bounds from the variational method in gras17
(dashed gray), upper bounds from the ET (solid gray).

##  IV conclusions

The envelope theory (ET) is useful to treat problems with a large number of
particles when a great accuracy is not required buis12 ; sema21 ; buis22 . It
can also be used to produce test calculations for numerical methods more
accurate but more difficult to implement horn14 ; timo17 . Nevertheless, it
can also be used in a pedagogical context to study solutions of time
independent Schrödinger-like equations, for the following reasons:

  * •

The ET is very simple to implement since the solutions are obtained by solving
a transcendental equation, and the generalisation for many-body systems in
several dimensions is trivial sema13 .

  * •

The eigenvalues and eigenvectors can be computed for the whole spectrum with
the same computational cost, and limiting cases studied.

  * •

In the most favourable situations, analytical upper or lower bounds can be
computed.

The calculations performed above for the one-dimensional soft-Coulomb
potential can be repeated for a lot of different interactions and/or for a
semirelativistic kinematics. An interesting case is the Hulthén potential on
$\mathbb{R}_{+}$, $-k/\left(\exp{(ax)}-1\right)$, which is bounded from above
by $-k\exp{(-ax)}$ and from below by $-k/(ax)$. Analytical solutions are known
for these three interactions in a one-dimensional Schrödinger equation flug99
.

Let us mention that the results of the ET in dimensions greater than one can
be improved by combining it with the dominantly orbital state method chev21 ;
sema15b . The idea is to modify the equivalent of the quantum number $Q_{n}$
appearing in these contexts. Such improvement could be possible in one
dimension also by replacing $n+1/2$ by $n+\gamma(n)$ in $Q_{n}$, in the same
spirit as the modification of the WKB method proposed in delv21 . The function
$\gamma(n)$ could be fitted on accurate numerical results to produce better
upper bounds. Similar calculations are already performed in sing85 .

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