# The Laplace Method for Energy Eigenvalue Problems in Quantum Mechanics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bound States with the Laplace Method

## 3. Examples: Bound States for the Simple Harmonic Oscillator

## 4. Continuum Solutions with the Laplace Method

## 5. Contour Integral around Circular Path for the Coulomb Problem

## 6. Continuum Solutions of the Morse Potential

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Schrödinger, E. Quantisierung als Eigenwertproblem (Erste Mitteilung). Ann. Phys.
**1926**, 384, 361–376. [Google Scholar] [CrossRef] - Schlesinger, L. Einführung in die Theorie der Differentialgleichungen mit einer unabhängigen Variablen; G. J. Göschensche Verlagshandlung: Leipzig, Germany, 1900. [Google Scholar]
- Dirac, P.A.M. Complex Variables in Quantum Mechanics. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1937**, 160, 48–59. [Google Scholar] - Dirac, P.A.M. Principles of Quantum Mechanics, 3rd ed.; Clarendon Press: Oxford, UK, 1947. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M. Quantum Mechanics: Non-Relativistic Theory, 3rd ed.; Pergamon Press: Oxford, UK, 1977. [Google Scholar]
- Messiah, A. Mécanique Quantique; Dunod: Paris, 1959; translated into English as Quantum Mechanics; John Wiley and Sons: North Holland, The Netherlands, 1966. [Google Scholar]
- Konishi, K.; Paffuti, G. Quantum Mechanics: A New Introduction; Oxford University Press: Oxford, UK, 2009. [Google Scholar]
- Capri, A.Z. Nonrelativistic Quantum Mechanics; Benjamin Cummings: Menlo Park, CA, USA, 1985. [Google Scholar]
- Galler, A.; Canfield, J.; Freericks, J.K. Schrödinger’s original quantum-mechanical solution for hydrogen. Eur. J. Phys.
**2021**, 42, 035405. [Google Scholar] [CrossRef] - Tsaur, G.-Y.; Wang, J. A universal Laplace-transform approach to solving Schrödinger equations for all known solvable models. Eur. J. Phys.
**2014**, 35, 015006. [Google Scholar] [CrossRef] - Arda, A.; Sever, R. Exact solutions of the Schrödinger equation via Laplace transform approach: Pseudoharmonic potential and Mie-type potentials. J. Math. Chem.
**2012**, 50, 971–980. [Google Scholar] [CrossRef] - Das, T. A Laplace transform approach to find the exact solution of the N-dimensional Schrödinger equation with Mie-type potentials and construction of Ladder operators. J. Math. Chem.
**2015**, 53, 618–628. [Google Scholar] [CrossRef] - Chung, W.; Kim, Y.; Kwon, J. Laplace transform method in one dimensional quantum mechanics on the semi infinite axis. J. Math. Chem.
**2022**, 60, 1080–1088. [Google Scholar] [CrossRef] - Szegö, G. Orthogonal Polynomials; American Mathematical Society: Providence, RI, USA, 1939. [Google Scholar]
- Olver, F.W.; Olde, J.; Daalhuis, A.B.; Lozier, D.W.; Schneider, B.I.; Boisvert, R.F.; Clark, C.W.; Miller, B.R.; Saunders, B.V.; Cohl, H.S.; et al. (Eds.) NIST Digital Library of Mathematical Functions; National Institute of Standards and Technology: Gaithersburg, MD, USA, 2022. Available online: http://dlmf.nist.gov/ (accessed on 1 June 2022).
- Nicholls, R.W. Continuum Wavefunctions for Morse Molecules. Chem. Phys. Lett.
**1981**, 79, 317–320. [Google Scholar] [CrossRef] - Matsumoto, A. Generalized matrix elements in discrete and continuum states for the Morse potential. J. Phys. B At. Mol. Opt. Phys.
**1988**, 21, 2863–2870. [Google Scholar] [CrossRef]

**Figure 1.**The contours we consider when ${\alpha}_{\pm}$ are not integers. The branch cuts in each case are shown by the dashed red lines. In panel (

**a**), we show a Hankel-like contour, in panel (

**b**) a contour that runs to infinity, and in panel (

**c**) we show the dog-bone contour that surrounds the two branch points when the branch cut is drawn between them.

**Figure 2.**Contours that lead to the correct solution for the wave function and provide the quantization condition for the energy. The contour is always a closed contour in the counter-clockwise direction encircling the point $z=-\lambda $, as shown in panel (

**a**). In some cases, shown in panel (

**b**), a branch point remains at $z=\lambda $ and the branch cut runs from there to $z=\infty $ along the real axis.

**Figure 3.**Four possible contours for solving the one-dimensional simple harmonic oscillator using the second ansatz. The contours must lie within the white “cone” regions as they go to infinity (there is no other restriction on the contours except not to cross the branch cut for finite values of z). In panel (

**a**), we show the Hankel contour ${\gamma}_{4}$, which goes around the branch cut, and ${\gamma}_{5}$, a contour that runs parallel to the real axis. In panel (

**b**), we show two contours starting from the branch point and running to infinity either below (${\gamma}_{6}$) or above (${\gamma}_{7}$) the branch cut.

**Figure 4.**Rotated ’dog-bone’ shaped contour for evaluating the contour integral for the continuum wave functions of the Coulomb problem in two and three dimensions.

**Figure 5.**Procedure for determining the phases of the integrand in Equation (55) along the vertical pieces of the contour (we deformed the contour slightly for clarity in the image). (

**a**) We pick a reference point $z={0}^{+}$ and show arrows from i and $-i$ to the reference point. (

**b**) We draw paths from the reference point ${0}^{+}$ to each point z of this piece of the deformed contour ${\gamma}_{c}$ (here running upward vertically to the right of the branch cut). The arrows drawn from the upper and lower branch points to z do not change their net direction as they move along the path from ${0}^{+}$ to z, thus the phase for $f\left(z\right)$ is the same as the phase for $f\left({0}^{+}\right)$ on the right side of the branch cut. (

**c**) For reaching the points of the contour on the left side of the branch cut, the arrow drawn from i to z needs to rotate by $2\pi $, while the net change in direction of the arrow drawn from the lower branch point is zero; note that the arrow along the path from the reference point to z is allowed to cross the branch cut, even though the path never crosses the branch cut.

**Figure 6.**Geometry for determining the relationship between the phases ${\varphi}_{1}$ and ${\varphi}_{2}$ and $\theta $. In the figures, the symbol l is used for the unknown length on each triangle, and R is chosen to equal 2, for concreteness; do not conflate l with the quantum number for total angular momentum. This case corresponds to $0\le \theta \le \pi $. Panel (

**a**) shows the geometry for the angles from the lower branch point at $-i$, while panel (

**b**) shows the results for the geometry from the upper branch point i.

**Figure 7.**Similar figure as in Figure 6, except here we have $\pi \le \theta \le 2\pi $. Again, panel (

**a**) shows the geometry from the lower branch point, while panel (

**b**) shows the geometry from the upper branch point.

**Figure 8.**Plot of the continuum Coulomb wave functions for $l=0$ and three different values of E: (

**a**) $E=0.1$; (

**b**) $E=1$, and (

**c**) $E=10$ (all energies are in Hartrees). The power series is shown in red, the contour integral with $R=1.1$ in green, and the one-dimensional integral in black. The three approximations lie on top of each other until they start to fail—the power series fails around $\xi \approx 20$, while the contour integral fails around $\xi \approx 30$. The errors typically occur due to the loss of digits of precision in the expressions being evaluated.

**Figure 9.**Plot of the continuum Morse wave function for three different values of E: (

**a**) $E=0.1$; (

**b**) $E=1$, and (

**c**) $E=10$. We use $\frac{{\hslash}^{2}{a}^{2}}{2\mu}$ as the energy unit and $\frac{1}{a}$ as the length unit. The case we consider is for ${V}_{0}={\textstyle \frac{{\hslash}^{2}{a}^{2}}{2\mu}}$. Note how the continuum wave rapidly decays for $x<0$, where the Morse potential becomes large and positive. Because the Morse potential decays to zero exponentially fast, the continuum solution rapidly looks like a simple cosine wave for large positive x with a fixed amplitude. In the region around $x=0$ we see a transition between the two behaviors.

**Table 1.**Quantum-mechanical potentials with bound states that are analyzed in this work. For each, we give the form of the potential, the general form of the wave function where $\Phi $ is the unknown part determined by using the Laplace method, and the form for the independent variable $\xi $ used for each problem; in one-dimension, the independent variable is x, in two dimensions it is $\rho =\sqrt{{x}^{2}+{y}^{2}}$, and in three dimensions it is $r=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$. Notably, $0\le \xi <\infty $ in all cases but the last (where $-\infty \le \xi \le \infty $). In the table, m is the z-component angular momentum quantum number with eigenvalues $\hslash m$ and l is the total angular momentum quantum number with eigenvalue ${\hslash}^{2}l(l+1)$. Moreover, $\mu $ is the mass of the (effective) particle, $\omega $ is the angular frequency of the oscillator, e is the magnitude of the charge of an electron, ℏ is Planck’s constant, E is the energy of the corresponding energy eigenstate, a is a real constant with units of inverse length, and ${V}_{0}$ has units of energy.

Problem | Potential | Independent Variable | Wavefunc. Form |
---|---|---|---|

1D SHO, Even | $V=\frac{1}{2}\mu {\omega}^{2}{x}^{2}$ | $\xi =\frac{\mu \omega}{\hslash}{x}^{2}$ | $\Phi \left(\xi \right)$ |

1D SHO, Odd | $V=\frac{1}{2}\mu {\omega}^{2}{x}^{2}$ | $\xi =\frac{\mu \omega}{\hslash}{x}^{2}$ | $x\Phi \left(\xi \right)$ |

2D SHO | $V=\frac{1}{2}\mu {\omega}^{2}{\rho}^{2}$ | $\xi =\frac{\mu \omega}{\hslash}{\rho}^{2}$ | ${\rho}^{\left|m\right|}\Phi \left(\xi \right){e}^{im\varphi}$ |

3D SHO | $V=\frac{1}{2}\mu {\omega}^{2}{r}^{2}$ | $\xi =\frac{\mu \omega}{\hslash}{r}^{2}$ | ${r}^{l}\Phi \left(\xi \right){Y}_{l}^{m}(\theta ,\varphi )$ |

2D Coulomb | $V=-\frac{{e}^{2}}{\rho}$ | $\xi =\sqrt{\frac{-2\mu E}{{\hslash}^{2}}}\rho $ | ${\rho}^{\left|m\right|}\Phi \left(\xi \right){e}^{im\varphi}$ |

3D Coulomb | $V=-\frac{{e}^{2}}{r}$ | $\xi =\sqrt{\frac{-2\mu E}{{\hslash}^{2}}}r$ | ${r}^{l}\Phi \left(\xi \right){Y}_{l}^{m}(\theta ,\varphi )$ |

Morse Potential | $V={V}_{0}\left({e}^{-2ax}-2{e}^{-ax}\right)$ | $\xi =\frac{2\sqrt{2\mu {V}_{0}}}{a\hslash}{e}^{-ax}$ | ${\xi}^{\frac{\sqrt{-2\mu E}}{a\hslash}}\Phi \left(\xi \right)$ |

1D SHO Method 2 | $V=\frac{1}{2}\mu {\omega}^{2}{x}^{2}$ | $\xi =\sqrt{\frac{\mu \omega}{\hslash}}x$ | ${e}^{-\frac{\mu \omega}{2\hslash}{x}^{2}}\Phi \left(\xi \right)$ |

**Table 2.**More details on the solutions for the wave functions using the Laplace method. The 1D SHO can be solved in two ways with the Laplace method, and we treat them both. In the second column, we provide the form of the Schrödinger equation obtained by the substitutions detailed in Table 1. Notably, for all but the last case, the coefficient of the first derivative term is either an integer or a half-odd integer, except in the Morse potential. It is always larger than 1 for all of these cases except the first row. Moreover, ${a}_{0}$ is the reduced Bohr radius (${a}_{0}={\hslash}^{2}/\mu {e}^{2}$).

Problem | Laplace Form of the Schrödinger Equation | ${\mathit{\alpha}}_{\pm}=\frac{\mathit{\beta}\mathit{\lambda}\pm \mathit{\delta}}{2\mathit{\lambda}}$ |
---|---|---|

1D SHO, Even | $\xi {\Phi}^{\u2033}+\frac{1}{2}{\Phi}^{\prime}+\left(\frac{E}{2\hslash \omega}-\frac{1}{4}\xi \right)\Phi =0$ | $\frac{1}{4}\pm \frac{E}{2\hslash \omega}$ |

1D SHO, Odd | $\xi {\Phi}^{\u2033}+\frac{3}{2}{\Phi}^{\prime}+\left(\frac{E}{2\hslash \omega}-\frac{1}{4}\xi \right)\Phi =0$ | $\frac{3}{4}\pm \frac{E}{2\hslash \omega}$ |

2D SHO | $\xi {\Phi}^{\u2033}+\left(\right|m|+1){\Phi}^{\prime}+\left(\frac{E}{2\hslash \omega}-\frac{1}{4}\xi \right)\Phi =0$ | $\frac{2\left|m\right|+1}{2}\pm \frac{E}{2\hslash \omega}$ |

3D SHO | $\xi {\Phi}^{\u2033}+(l+\frac{3}{2}){\Phi}^{\prime}+\left(\frac{E}{2\hslash \omega}-\frac{1}{4}\xi \right)\Phi =0$ | $\frac{2l+3}{4}\pm \frac{E}{2\hslash \omega}$ |

2D Coulomb | $\xi {\Phi}^{\u2033}+\left(2\right|m|+1){\Phi}^{\prime}+\left(\frac{2\hslash}{{a}_{0}\sqrt{-2\mu E}}-\xi \right)\Phi =0$ | $\frac{2\left|m\right|+1}{2}\pm \frac{\hslash}{{a}_{0}\sqrt{-2\mu E}}$ |

3D Coulomb | $\xi {\Phi}^{\u2033}+2(l+1){\Phi}^{\prime}+\left(\frac{2\hslash}{{a}_{0}\sqrt{-2\mu E}}-\xi \right)\Phi =0$ | $l+1\pm \frac{\hslash}{{a}_{0}\sqrt{-2\mu E}}$ |

Morse Potential | $\xi {\Phi}^{\u2033}+\left(2\frac{\sqrt{-2\mu E}}{a\hslash}+1\right){\Phi}^{\prime}+\left(\frac{\sqrt{2\mu {V}_{0}}}{a\hslash}-\frac{1}{4}\xi \right)\Phi =0$ | $\frac{\sqrt{-2\mu E}\pm \sqrt{2\mu {V}_{0}}}{a\hslash}+\frac{1}{2}$ |

1D SHO Method 2 | ${\Phi}^{\u2033}-2\xi {\Phi}^{\prime}+\left(\frac{2E}{\hslash \omega}-1\right)\Phi =0$ | N/A |

**Table 3.**Quantization condition for the Laplace method (by tradition, the principal quantum number n starts from 0 for all cases, except the Coulomb cases, where it starts from $\left|m\right|+1$ in two dimensions and from $l+1$ in three dimensions); N is required to be a nonnegative integer from the quantization condition arising from Laplace’s method. By tracing back through the definitions of $\Phi $ and $\xi $ in each case, one obtains the standard wave functions for each problem, up to a normalization constant, that still needs to be determined. The $\delta $ in the index of the associated Laguerre polynomial in the last column of the Morse potential satisfies $\delta =\frac{\sqrt{2\mu {V}_{0}}}{a\hslash}$. All models, except for the Morse potential, have an infinite number of bound states. The Morse potential has a finite number, where we are required to have $n<\sqrt{2\mu {V}_{0}}/a\hslash $. ${H}_{n}\left(\xi \right)$ is the nth Hermite polynomial.

Problem | Quantization Condition | Energy Quantization, ${\mathit{E}}_{\mathit{n}}$ | Form of $\mathbf{\Phi}\left(\mathit{\xi}\right)$ |
---|---|---|---|

1D SHO, Even | $N=n=\frac{E}{2\hslash \omega}-\frac{1}{4}$ | $\hslash \omega \left(2n+\frac{1}{2}\right)$ | ${e}^{-\xi /2}{L}_{n}^{\left(-\frac{1}{2}\right)}\left(\xi \right)$ |

1D SHO, Odd | $N=n=\frac{E}{2\hslash \omega}-\frac{3}{4}$ | $\hslash \omega \left(2n+1+\frac{1}{2}\right)$ | ${\xi}^{1/2}{e}^{-\xi /2}{L}_{n}^{\left(\frac{1}{2}\right)}\left(\xi \right)$ |

2D SHO | $N=n=\frac{E}{2\hslash \omega}-\frac{2\left|m\right|+1}{2}$ | $\hslash \omega \left(2n+\left|m\right|+1\right)$ | ${e}^{-\xi /2}{L}_{n}^{\left(\right|m\left|\right)}\left(\xi \right)$ |

3D SHO | $N=n=\frac{E}{2\hslash \omega}-\frac{1}{2}\frac{2l+3}{2}$ | $\hslash \omega \left(2n+l+\frac{3}{2}\right)$ | ${e}^{-\xi /2}{L}_{n}^{\left(l+\frac{1}{2}\right)}\left(\xi \right)$ |

2D Coulomb | $N=n-\left|m\right|-1=\frac{\hslash}{{a}_{0}\sqrt{-2\mu E}}-\left(\left|m\right|+\frac{1}{2}\right)$ | $-\frac{{\hslash}^{2}}{2\mu {a}_{0}^{2}{\left(n-\frac{1}{2}\right)}^{2}}$ | ${e}^{-\xi}{L}_{n-\left|m\right|-1}^{\left(2\right|m\left|\right)}\left(\xi \right)$ |

3D Coulomb | $N=n-l-1=\frac{\hslash}{{a}_{0}\sqrt{-2\mu E}}-\left(l+1\right)$ | $-\frac{{\hslash}^{2}}{2\mu {a}_{0}^{2}{n}^{2}}$ | ${e}^{-\xi}{L}_{n-l-1}^{(2l+1)}\left(\xi \right)$ |

Morse Potential | $N=n=\frac{\sqrt{2\mu {V}_{0}}-\sqrt{-2\mu E}}{a\hslash}$ | $-\frac{{a}^{2}{\hslash}^{2}}{2\mu}{\left(n-\frac{\sqrt{2\mu {V}_{0}}}{a\hslash}\right)}^{2}$ | ${e}^{-\xi /2}{L}_{n}^{(2n-2\delta -1)}\left(\xi \right)$ |

1D SHO Method 2 | $N=n=\frac{E}{\hslash \omega}-\frac{1}{2}$ | $\hslash \omega \left(n+\frac{1}{2}\right)$ | ${H}_{n}\left(\xi \right)$ |

**Table 4.**Summary for how to convert the Schrödinger equation into the Laplace equation for the five problems with continuum solutions. For each, we give the form of the potential, the general form of the wave function where $\Phi $ is the part of the solution that is found by using the Laplace method, and the independent variable $\xi $ used for each problem. Note that $E>0$ for these continuum problems. We have m denoting the quantum number for the z-component of angular momentum and l denoting the quantum number for the total angular momentum and ${Y}_{l}^{m}(\theta ,\varphi )$ the spherical harmonic.

Problem | Potential | Independent Variable | Wavefunc. Form |
---|---|---|---|

2D Free Particle | $V=0$ | $\xi =\sqrt{\frac{2\mu E}{{\hslash}^{2}}}\rho $ | ${\rho}^{\left|m\right|}\Phi \left(\xi \right){e}^{im\varphi}$ |

3D Free Particle | $V=0$ | $\xi =\sqrt{\frac{2\mu E}{{\hslash}^{2}}}r$ | ${r}^{l}\Phi \left(\xi \right){Y}_{l}^{m}(\theta ,\varphi )$ |

2D Coulomb | $V=-\frac{{e}^{2}}{\rho}$ | $\xi =\sqrt{\frac{2\mu E}{{\hslash}^{2}}}\rho $ | ${\rho}^{\left|m\right|}\Phi \left(\xi \right){e}^{im\varphi}$ |

3D Coulomb | $V=-\frac{{e}^{2}}{r}$ | $\xi =\sqrt{\frac{2\mu E}{{\hslash}^{2}}}r$ | ${r}^{l}\Phi \left(\xi \right){Y}_{l}^{m}(\theta ,\varphi )$ |

Morse Potential | $V={V}_{0}\left({e}^{-2ax}-2{e}^{-ax}\right)$ | $\xi =\frac{2\sqrt{2\mu {V}_{0}}}{a\hslash}{e}^{-ax}$ | ${\xi}^{i\frac{\sqrt{2\mu E}}{a\hslash}}\Phi \left(\xi \right)$ |

**Table 5.**Final differential equation and exponents ${\alpha}_{\pm}$ for continuum cases to be solved by the Laplace method. The second column summarizes the final form of the Schrödinger equation obtained by the substitutions detailed in Table 4. Note that because $\delta =0$ for the free-particle cases, there is only one exponent for those problems. In all problems except for the Morse potential, we have $\overline{\lambda}=1$.

Problem | Laplace Form of the Schrödinger Equation | ${\mathit{\alpha}}_{\pm}$ |
---|---|---|

2D Free Particle | $\xi {\Phi}^{\u2033}+\left(2\right|m|+1){\Phi}^{\prime}+\xi \Phi =0$ | $\left|m\right|+\frac{1}{2}$ |

3D Free Particle | $\xi {\Phi}^{\u2033}+2(l+1){\Phi}^{\prime}+\xi \Phi =0$ | $l+1$ |

2D Coulomb | $\xi {\Phi}^{\u2033}+\left(2\right|m|+1){\Phi}^{\prime}+\left(\frac{2\hslash}{{a}_{0}\sqrt{2\mu E}}+\xi \right)\Phi =0$ | $\left|m\right|+\frac{1}{2}\mp \frac{i\hslash}{{a}_{0}\sqrt{2\mu E}}$ |

3D Coulomb | $\xi {\Phi}^{\u2033}+2(l+1){\Phi}^{\prime}+\left(\frac{2\hslash}{{a}_{0}\sqrt{2\mu E}}+\xi \right)\Phi =0$ | $l+1\mp \frac{i\hslash}{{a}_{0}\sqrt{2\mu E}}$ |

Morse Potential | $\xi {\Phi}^{\u2033}+\left(2i\frac{\sqrt{2\mu E}}{a\hslash}+1\right){\Phi}^{\prime}+\left(\frac{\sqrt{2\mu {V}_{0}}}{a\hslash}-\frac{1}{4}\xi \right)\Phi =0$ | $\frac{i\sqrt{2\mu E}\pm \sqrt{2\mu {V}_{0}}}{a\hslash}+\frac{1}{2}$ |

**Table 6.**Summary of the results of the Laplace method for continuum cases in terms of the variable $\xi $. The solution for $\Phi \left(\xi \right)$, as defined in Table 5, is expressed in terms of the confluent hypergeometric functions $M(a,b,z)$ and $U(a,b,z)$.

Problem | Confluent Hypergeometric Form of $\mathbf{\Phi}\left(\mathit{\xi}\right)$ |
---|---|

2D Free Particle | ${e}^{-i\xi}M\left(\left|m\right|+\frac{1}{2},2\left|m\right|+1,2i\xi \right)$ |

3D Free Particle | ${e}^{-i\xi}M\left(l+1,2l+2,2i\xi \right)$ |

2D Coulomb | ${e}^{-i\xi}M\left(\left|m\right|+\frac{1}{2}+\frac{i\hslash}{{a}_{0}\sqrt{2\mu E}},\phantom{\rule{3.33333pt}{0ex}}2\left|m\right|+1,\phantom{\rule{3.33333pt}{0ex}}2i\xi \right)$ |

3D Coulomb | ${e}^{-i\xi}M\left(l+1+\frac{i\hslash}{{a}_{0}\sqrt{2\mu E}},\phantom{\rule{3.33333pt}{0ex}}2l+2,\phantom{\rule{3.33333pt}{0ex}}2i\xi \right)$ |

Morse Potential | ${(-1)}^{\beta -1}\Gamma \left({\alpha}_{-}\right){e}^{-\xi /2}U({\alpha}_{-},\beta ,\xi )$ |

**Table 7.**Summary of the results of the continuum cases we solved with the Laplace method in terms of the original independent variable. For the free particle cases, we express the more common form of the confluent hypergeometric function. Here, ${J}_{n}\left(x\right)$ is the Bessel function of the first kind, and ${j}_{n}\left(x\right)$ is the spherical Bessel function of the first kind.

Problem | Unnormalized Wave function |
---|---|

2D Free Particle | ${J}_{\left|m\right|}\left(\sqrt{\frac{2\mu E}{{\hslash}^{2}}}\rho \right){e}^{i\left|m\right|\varphi}$ |

3D Free Particle | ${j}_{l}\left(\sqrt{\frac{2\mu E}{{\hslash}^{2}}}r\right){Y}_{l}^{m}\left(\theta ,\varphi \right)$ |

2D Coulomb | $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\rho}^{\left|m\right|}{e}^{-i\sqrt{\frac{2\mu E}{{\hslash}^{2}}}\rho}{e}^{i\left|m\right|\varphi}\phantom{\rule{0ex}{0ex}}\times M\left(\left|m\right|+\frac{1}{2}+\frac{i\hslash}{{a}_{0}\sqrt{2\mu E}},2\left|m\right|+1,2i\sqrt{\frac{2\mu E}{{\hslash}^{2}}}\rho \right)$ |

3D Coulomb | $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{r}^{l}{e}^{-i\sqrt{\frac{2\mu E}{{\hslash}^{2}}}r}{Y}_{l}^{m}(\theta ,\varphi )\phantom{\rule{0ex}{0ex}}\times M\left(l+1+\frac{i\hslash}{{a}_{0}\sqrt{2\mu E}},\phantom{\rule{3.33333pt}{0ex}}2l+2,\phantom{\rule{3.33333pt}{0ex}}2i\sqrt{\frac{2\mu E}{{\hslash}^{2}}}r\right)$ |

Morse Potential | $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\left(\frac{2\sqrt{2\mu {V}_{0}}}{a\hslash}{e}^{-ax}\right)}^{i\frac{\sqrt{2\mu E}}{a\hslash}}exp\left(-\frac{\sqrt{2\mu {V}_{0}}}{a\hslash}{e}^{-ax}\right)\phantom{\rule{0ex}{0ex}}U\left(\frac{i\sqrt{2\mu E}-\sqrt{2\mu {V}_{0}}}{a\hslash}+\frac{1}{2},2i\frac{\sqrt{2\mu E}}{a\hslash}+1,\frac{2\sqrt{2\mu {V}_{0}}}{a\hslash}{e}^{-ax}\right)$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Canfield, J.; Galler, A.; Freericks, J.K.
The Laplace Method for Energy Eigenvalue Problems in Quantum Mechanics. *Quantum Rep.* **2023**, *5*, 370-397.
https://doi.org/10.3390/quantum5020024

**AMA Style**

Canfield J, Galler A, Freericks JK.
The Laplace Method for Energy Eigenvalue Problems in Quantum Mechanics. *Quantum Reports*. 2023; 5(2):370-397.
https://doi.org/10.3390/quantum5020024

**Chicago/Turabian Style**

Canfield, Jeremy, Anna Galler, and James K. Freericks.
2023. "The Laplace Method for Energy Eigenvalue Problems in Quantum Mechanics" *Quantum Reports* 5, no. 2: 370-397.
https://doi.org/10.3390/quantum5020024