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Revisiting the bound states of a confined delta potential

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The stationary states of a particle under the influence of a delta potential confined by impenetrable walls are investigated using the method of expansion in orthogonal functions. The eigenfunctions of the time-independent Schrödinger equation are expressed in closed form by using a pair of closed-form expressions for series available in the literature. The analysis encompasses both attractive and repulsive potentials with arbitrary couplings. Confinement significantly impacts the quantum states and introduces a scenario of double degeneracy including the ground state. Analysis extends to discuss the transition to unconfinement. This research holds particular significance for educators and students engaged in mathematical methods applied to physics and quantum mechanics within undergraduate courses, offering valuable insights into the complex relationships among profiles of potentials, boundary conditions, and the resulting quantum phenomena.
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Revisiting the bound states of a confined
delta potential
Luiz G. M. Ramos1and Antonio S. de Castro2
Departamento de F´ısica,
Universidade Estadual Paulista “J´ulio de Mesquita Filho”,
Guaratinguet´a, SP, Brazil
1E-mail: luiz.monteiro@unesp.br
2E-mail: antonio.castro@unesp.br
Abstract
The stationary states of a particle under the influence of a delta potential confined by
impenetrable walls are investigated using the method of expansion in orthogonal func-
tions. The eigenfunctions of the time-independent Schr¨odinger equation are expressed in
closed form by using a pair of closed-form expressions for series available in the litera-
ture. The analysis encompasses both attractive and repulsive potentials with arbitrary
couplings. Confinement significantly impacts the quantum states and introduces a sce-
nario of double degeneracy including the ground state. Analysis extends to discuss the
transition to unconfinement. This research holds particular significance for educators and
students engaged in mathematical methods applied to physics and quantum mechanics
within undergraduate courses, offering valuable insights into the complex relationships
among profiles of potentials, boundary conditions, and the resulting quantum phenom-
ena.
Keywords: confinement, delta potential in a box, expansion in orthogonal functions
1 Introduction
One-dimensional quantum mechanics presents exactly solvable problems that offer valu-
able insights into quantum physics. Examples include the free particle confined by impen-
etrable walls (the particle in a box) and a particle subjected to a Dirac delta potential.
These problems are frequently featured in introductory textbooks due to their educational
significance (see, e.g. [1]-[9]).
The stationary states of a free particle confined within a box are typically identified by
directly solving the time-independent Schr¨odinger equation. This solution incorporates
homogeneous Dirichlet boundary conditions at the walls of the box and results in an
infinite set of eigenenergies, each distinguished by an integer quantum number. As the
confinement is gradually lifted, the energies of the system approach a continuous spectrum,
emphasizing the significant influence of confinement on the nature of the stationary states.
The bound states of a particle interacting with a delta potential, represented by
αδ (x), where αdenotes the potential strength, are typically approached using the con-
tinuity of the eigenfunction and the jump discontinuity of its first derivative at x= 0 plus
homogeneous Dirichlet boundary conditions at infinity. In the case of an attractive delta
potential (α > 0), a single negative-energy bound state emerges. There is no bound-state
solution for a repulsive delta potential (α < 0).
The examination of a particle influenced by a delta potential within a confined space
has been a topic explored in educational literature as well. The quantization condition
for an attractive delta function potential was found by Epstein [10] employing match-
ing conditions on the eigenfunction and its first derivative on the support of the delta
function, along with homogeneous Dirichlet boundary conditions on the enclosing walls.
Lapidus [11] solved the problem presuming a functional form for the eigenfunction in ac-
cordance with the boundary conditions. The quantization condition for both attractive
and repulsive delta function potentials was derived by Atkinson e Crater [12] through the
expansion of eigenfunctions in orthogonal functions, a method they termed the Green’s
function technique. Subsequently, again presuming a functional form for the eigenfunc-
tion, the problem was approached by Cohen and Kuharetz [13], supersymmetry techniques
by Goldstein, Lebiedzik and Robinett [14], and factorization method by Pedram and Va-
habi [15].
This study aims to offer a thorough comprehension of the bound states of a particle
under the influence of a delta potential embedded in a box. To take advantage of the
symmetry of parity, the support of the delta potential is strategically placed at the center
of a box that is, in turn, centered at the origin. Adopting the method of eigenfunction
expansion, we extend the analysis to encompass attractive and repulsive potentials and
include a clear presentation of corresponding closed form for the normalized eigenfunctions
of the time-independent Schr¨odinger equation. Additionally, by identifying the proper
coupling constant, we explore the double degeneracy observed in the strong coupling
limit and delineate the transition of the stationary states to the unconfined environment.
The elegant method of expansion in orthogonal functions in a finite region of space,
and its close connection with Fourier series, can be of interest of teachers and students
of mathematical methods applied to physics and quantum mechanics of undergraduate
courses.
1
2 The confined delta potential
The time-independent Schr¨odinger equation is expressed as follows:
2
2m
d2
dx2+V(x)ψE(x) = E(x), ψE(x)= 0.(1)
Here, the pair (E, ψE) represents the sought-after solution to the eigenvalue problem.
ERand ψE(x) is required to be single-valued, finite, and continuous everywhere. Ad-
ditionally, the normalization condition is imposed to ensure the probability interpretation
of the eigenfunction: Z+
−∞
dx |ψE(x)|2= 1.(2)
As a second-order differential equation, the time-independent Schr¨odinger equation has
two linearly independent solutions corresponding to the same energy eigenvalue. While
twofold degeneracy is generally precluded in describing one-dimensional bound states, as
ensured by the nondegeneracy theorem, it becomes a possibility for a strongly singular
potentials (see, e.g. [16]). For a system confined by symmetric impenetrable walls around
the origin
V(x) =
v(x),|x| L/2
,|x|> L/2,
(3)
the time-independent Schr¨odinger equation takes the form
d2ψk(x)
dx2+k22m
2v(x)ψk(x) = 0,|x|< L/2,(4)
where
k=r2mE
2.(5)
Furthermore,
ψk(x)=0,|x|L/2,(6)
and Z+L/2
L/2
dx |ψk(x)|2= 1.(7)
In the case where v(x) = 0 (free particle in the box) the solutions of (4) become
solutions of an eigenvalue problem:
d2
dx2ϕκ(x) = κ2ϕκ(x), ϕκ(x)= 0,
(8)
ϕκ(±L/2) = 0,
in such a way that d2/dx2is a self-adjoint operator on the interval [L/2,+L/2]. With
n= 1,2,3, . . ., the characteristic pairs (κ,ϕκ) are segregated by parity as follows:
κ2n1= (2n1) π/L,
(9)
ϕ2n1(x) = cos κ2n1x,
2
and
κ2n= (2n)π/L,
(10)
ϕ2n(x) = sin κ2nx.
It is noteworthy that Z+L/2
L/2
dx |ϕn(x)|2=L
2.(11)
The set ϕn(x) is both complete and orthogonal, satisfying:
Z+L/2
L/2
dx ϕ
n(x)ϕen(x) = 0 for n=en. (12)
Even if v(x)= 0, it is feasible to represent ψk(x) through the series
ψk(x) =
X
n=1
a(k)
nϕn(x).(13)
Manifestly ψk(x) is now expanded into Fourier series with period L. When v(x) = v(x),
the Fourier series takes the form of either a cosine series for even-parity eigenfunctions or
a sine series for odd-parity eigenfunctions. In order to determine the Fourier coefficients
a(k)
nto make ψk(x) solution of (4), a systematic approach is adopted. This involves
substituting expression (13) into (4), multiplying the result by ϕ
n(x), integrating term by
term over the interval [L/2,+L/2], and applying the orthogonality properties expressed
by (12).
For a delta potential with support at the center of the box
v(x) = αδ (x),(14)
the corresponding differential equation becomes:
d2ψk(x)
dx2+k2ψk(x) = 4g
Lδ(x)ψk(x),(15)
where the dimensionless parameter gis defined by
g=mLα
22.(16)
The series representation for ψk(x) for this case has two distinctive characters. Odd
eigenfunctions remain unaffected by the delta function because δ(x)ψk(x) = δ(x)ψk(0).
In general, an even-parity eigenfunction does not vanish at the origin but its first derivative
satisfies the relation
(e)
k(x)
dx x=0+
=2g
Lψ(e)
k(0) .(17)
Here, x= 0+indicates the limit as xapproaches 0 from x > 0, and (17) implies that
ψ(e)
k(x) has a cusp at the origin when (e)
k(0) = 0. In terms of
ζ=kL
π,(18)
3
the series (13) gives rise to
ψ(e)
ζ(x) =
X
n=1
a(ζ)
2n1ϕ2n1(x),
(19)
ψ(o)
ζ(x) = a2nϕ2n(x).
The coefficients a(ζ)
2n1are determined by:
a(ζ)
2n1=8(e)
ζ(0) 2
(2n1)2ζ2, ζ = 2n1,(20)
resulting in the expression for ψ(e)
ζ(x):
ψ(e)
ζ(x) = 8(e)
ζ(0)
π2
X
n=1
ϕ2n1(x)
(2n1)2ζ2,(21)
where ψ(e)
ζ(0) behaves as a normalization constant. Setting x= 0 in (21) reveals that
X
n=1
1
(2n1)2ζ2=π2
8g.(22)
Moreover, as previously mentioned in [12], the sum on the left-hand side of (22) admits a
closed form (refer to 1.421(1) in Ref. [17]):
X
n=1
1
(2n1)2ζ2=π
4ζtan ζπ
2.(23)
Combining (22) and (23), yields the quantization condition
1
gζπ
2= tan ζπ
2.(24)
Fortunately, the eigenfunction (21) can also be expressed in a closed form by leveraging
expressions from the literature (see 1.445(6) and 1.445(8) in Ref. [17]):
X
n=1
cos nz
n2ζ2=1
2ζ2π
2
cos ζ[(2µ+ 1) πz]
ζsin ζπ ,(25)
for (2µ)πz(2µ+ 2) π, and
X
n=1
(1)ncos nz
n2ζ2=1
2ζ2π
2
cos ζ[(2µ)πz]
ζsin ζπ ,(26)
for (2µ1) πz(2µ+ 1) π. Subtracting (26) from (25) results in
X
n=1
cos (2n1) z
(2n1)2ζ2=π
4
sin ζ[z(2µ+ 1/2) π]
ζcos ζπ/2,(27)
4
for (2µ)πz(2µ+ 1) π. Especially for µ= 0, substituting zby πx/L gives
X
n=1
cos (2n1) πx/L
(2n1)2ζ2=π
4
sin ζπ (xL/2) /L
ζcos ζπ/2,(28)
for 0 xL. A symmetric extension of this result enables us to write:
ψ(e)
ζ(x) = 2(e)
ζ(0)
ζπ cos ζπ/2sin ζπ (|x| L/2)
L.(29)
The normalization condition (7) determinates ψ(e)
ζ(0):
ψ(e)
ζ(0) = eζζπ cos ζπ/2
2gr2
L1sin ζπ
ζπ 1/2
,(30)
where θζis an arbitrary phase. Therefore, the complete set of stationary-state solutions
are expressed as
Eζ=2π2ζ2
2mL2,(31)
ψ(e)
ζ(x) = eζq2
L1sin ζπ
ζπ 1/2sin ζ π(|x|−L/2)
L,
ψ(o)
ζ(x) = eζq2
Lsin ζπx
L, ζ = 2n,
(32)
where ζfor the even-parity eigenfunction is solution of the quantization condition (24).
At this point, we mention that beyond the solutions with ζ > 0 (Eζ>0) there is now
the possibility of an even-parity eigenfunction with ζ= 0 (Eζ= 0) or ζ=i|ζ|(Eζ<0).
Therefore, it is convenient to distinguish these three distinct classes of solutions:
ζ=|ζ|
In this case, |ζ|is solution of the transcendental equation
1
g|ζ|π
2= tan |ζ|π
2,(33)
and
Eζ=2π2|ζ|2
2mL2>0,(34)
ψ(e)
ζ(x) = eζr2
L1sin |ζ|π
|ζ|π1/2
sin |ζ|π(|x| L/2)
L.(35)
A qualitative overview about the behaviour of the solutions of the transcendental
equation for |ζ|can be obtained by plotting their left-hand and right-hand sides on
the same grid. The abscissa of of each intersection point corresponds to a solution.
A graphical analyses shows that
5
|ζ|2n, g << 1
|ζ|2n1, g 0
|ζ|2n1, g 0+
|ζ|2n2, g >> +1 (except for n= 1)
0<|ζ|<1,0< g < 1.
(36)
The solution with 0 <|ζ|<1 holds a distinct significance: |ζ| 1 for g0, and
|ζ| 0 for g1. This unique case aside, |ζ| 2nfor |g|>> 1, with the values close
to those ones corresponding to odd-parity eigenfunctions. This doublet structure in
the spectrum serves as a clear indication of the double degeneracy associated with
the limit |g| . To enhance precision and clarity, the exact numerical solution
of equation (51) for the six lowest values of |ζ|is presented in Fig.1.
ζ=i|ζ|
In this case, |ζ|satisfies the equation
1
g|ζ|π
2= tanh |ζ|π
2,(37)
and
Eζ=2π2|ζ|2
2mL2<0,(38)
ψ(e)
ζ(x) = eζr2
Lsinh |ζ|π
|ζ|π11/2
sinh |ζ|π(|x| L/2)
L.(39)
Evidently g > 0. Graphical analyses reveals that there is no solution for g < 1, and
a unique solution exists for g > 1 (|ζ| 0 for g1 and |ζ| 2g for g >> 1).
ζ= 0
The solution to the quantization condition for g= 1 can be derived as a limiting case
from the two preceding classes by employing the approximations tan |z| tanh |z|
|z|for |z|<< 1. This yields:
Eζ= 0.(40)
Moreover, utilizing the approximations sin |z| |z| |z|3/3! and sinh |z| |z|+
|z|3/3! for |z|<< 1, we obtain:
ψ(e)
ζ(x) = eζ23
L3/2(|x| L/2) .(41)
Let us consolidate the findings thus far. Generally, the stationary states are charac-
terized by non-integer quantum numbers (|ζ|). The quantization condition (33) yields
|ζ|= 2n1 for g= 0, resulting in eigenvalues and eigenfunctions consistent with those
obtained in the absence of the delta function potential. The spectrum consists of an in-
finite set of non-negative and discrete eigenvalues, corresponding to both even and odd
6
Figure 1: The lowest values of ζ=|ζ|(Eζ>0) as a function of the coupling constant g. Dashed
lines for |ζ|= 2n(n= 1,2,3, . . .), corresponding to odd-parity eigenfunctions. Continuous lines
for numerical solutions of the transcendental equation 1
g|ζ|π
2= tan |ζ|π
2, corresponding to
even-parity eigenfunctions
eigenfunctions, irrespective of the sign of the delta potential. An additional solitary
negative-energy eigenvalue, corresponding to an even-parity eigenfunction, emerges for a
sufficiently strong attractive potential (α > 22/(mL)) or an attractive delta potential
embedded in a box of sufficient size (L > 22/(m|α|)). Including the ground state for
a very strong repulsive coupling, all positive-energy eigenvalues tend to form doublets
under the influence of very strong couplings. A notable exception is the ground state for
an attractive potential. Indeed, one has
|ζ|= 2n, ψ(o)
ζ(x) = ψ2n(x),g
|ζ|2n, ψ(e)
ζ(x)(1)nψ2n(|x|), g << 1,
|ζ|2n, ψ(e)
ζ(x)(1)nψ2n(|x|), g >> +1.
(42)
The normalized probability densities ρ(e)
ζ(x) = |ψ(e)
ζ(x)|2for the ground state are depicted
in Fig.2 for several illustrative values of g, plotted as functions of x/L. It is evident
that the probability density exhibits a cusp at the support of the delta function either
pointing towards or away from the origin depending on whether gis negative or positive,
respectively, in accordance with (17). As gdiminishes, the cusp gradually approaches the
origin until it nearly touches the origin at the support of the delta function, mimicking the
probability density of the first-excited state as gapproaches negative infinity. Conversely,
as gincreases, the cusp moves away from the origin, amplifying the probability density
7
at that support of the delta function.
Figure 2: The probability densities ρ(e)
ζ(x) = |ψ(e)
ζ(x)|2for the ground state with gequal to 30
(dashed line with cusp point towards the origin), 2(dotted line with cusp pointing towards the
origin), 0(continuous line), 3/4(dashed line with cusp pointing from the origin) and 2(dotted
line with cusp pointing from the origin).
The confinement of a free particle by impenetrable walls has profound influence on the
quantum states and makes ζrun through discrete values. A well-defined parity function
vanishing at x=±L/2 can always be written as
Φ (x) =
X
ζ=1
cζϕζ(x) (43)
with
cζ=4
LZL/2
0
dx ϕ
ζ(x) Φ (x).(44)
For ζR, two adjacent values of kare spaced by k= 2π/L. Therefore, with FL(k) =
L2πcζ/(4π) one finds
Φ (x) = r2
π
X
Lk/π=1
kFL(k)ϕk(x),(45)
with
FL(k) = r2
πZL/2
0
dx ϕ
k(x) Φ (x).(46)
8
Note that the minimum of kis very small for large but finite L. Hence, the limit L
(∆k0) gives continuous eigenenergies
Eκ=2k2
2m>0,(47)
and Φ (x), using FL(k)F(k) as L , is written as a superposition of eigenfunctions
corresponding to the continuum as
Φ (x) = r2
πZ
0
dk F (k)ϕk(x) (48)
with
F(k) = r2
πZ
0
dx ϕ
k(x) Φ (x).(49)
It is instructive to note (49) is the unilateral Fourier transform of Φ (x) with inverse given
by (48) (see, e.g. Sec. 17.31 in [17]). As the limit L is approached, a continuum of
positive-energy unbound states emerges. However, a solitary even-parity negative-energy
stationary state becomes a possibility only for an attractive delta potential (α > 0). As
a matter of fact, using |ζ| 2g/π for g >> 1 in (38) and sinh |z| e|z|/2 for |z|>> 1 in
(39), we obtain the limiting forms
E=2
22,(50)
ψ(e)(x) = eαr
2exp
2|x|,(51)
corresponding to the bound state for the unconfined attractive delta potential.
3 Final remarks
We revisited the stationary states of a particle subject to a delta potential between im-
penetrable walls using the method of eigenfunction expansion. We extend the analysis
to encompass attractive and repulsive potentials with arbitrary couplings and included a
clear presentation of the corresponding closed form for the normalized eigenfunctions by
appealing to a pair of expressions from the literature that simplify summation to a closed
form. Additionally, we discussed the double degeneracy observed in the extreme strong
coupling limit and delineated the evolution of stationary states as the system transitions
from confinement to an unconfined environment.
Acknowledgement
We are grateful to the referees for their constructive input. Grants 149369/2021-2 and
09126/2019-3, Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq),
Brazil.
9
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10
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