Particle in a spherically symmetric potential

An important problem in quantum mechanics is that of a particle in a spherically symmetric potential, i.e., a potential that depends only on the distance between the particle and a defined center point. In particular, if the particle in question is an electron and the potential is derived from Coulomb's law, then the problem can be used to describe a hydrogen-like (one-electron) atom (or ion).

In the general case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form:

${\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m_{0}}}+V(r)}$

where ${\displaystyle m_{0}}$ is the mass of the particle, ${\displaystyle {\hat {p}}}$ is the momentum operator, and the potential ${\displaystyle V(r)}$ depends only on ${\displaystyle r}$, the modulus of the radius vector r. The quantum mechanical wavefunctions and energies (eigenvalues) are found by solving the Schrödinger equation with this Hamiltonian. Due to the spherical symmetry of the system, it is natural to use spherical coordinates ${\displaystyle r}$, ${\displaystyle \theta }$ and ${\displaystyle \phi }$. When this is done, the time-independent Schrödinger equation for the system is separable, allowing the angular problems to be dealt with easily, and leaving an ordinary differential equation in ${\displaystyle r}$ to determine the energies for the particular potential ${\displaystyle V(r)}$ under discussion.

Structure of the eigenfunctions

The eigenstates of the system have the form

${\displaystyle \psi (r,\theta ,\phi )=R(r)\Theta (\theta )\Phi (\phi )\,\;}$

in which the spherical polar angles θ and φ represent the colatitude and azimuthal angle, respectively. The last two factors of ψ are often grouped together as spherical harmonics, so that the eigenfunctions take the form

${\displaystyle \psi (r,\theta ,\phi )=R(r)Y_{lm}(\theta ,\phi ).\,}$

The differential equation which characterizes the function ${\displaystyle R(r)}$ is called the radial equation.

Derivation of the radial equation

The kinetic energy operator in spherical polar coordinates is

${\displaystyle {\frac {{\hat {p}}^{2}}{2m_{0}}}=-{\frac {\hbar ^{2}}{2m_{0}}}\nabla ^{2}=-{\frac {\hbar ^{2}}{2m_{0}\,r^{2}}}\left[{\frac {\partial }{\partial r}}{\Big (}r^{2}{\frac {\partial }{\partial r}}{\Big )}-{\hat {l}}^{2}\right].}$

The spherical harmonics satisfy

${\displaystyle {\hat {l}}^{2}Y_{lm}(\theta ,\phi )\equiv \left\{-{\frac {1}{\sin ^{2}\theta }}\left[\sin \theta {\frac {\partial }{\partial \theta }}{\Big (}\sin \theta {\frac {\partial }{\partial \theta }}{\Big )}+{\frac {\partial ^{2}}{\partial \phi ^{2}}}\right]\right\}Y_{lm}(\theta ,\phi )=l(l+1)Y_{lm}(\theta ,\phi ).}$

Substituting this into the Schrödinger equation we get a one-dimensional eigenvalue equation,

${\displaystyle {\frac {1}{r^{2}}}{d \over dr}\left(r^{2}{dR \over dr}\right)-{l(l+1) \over r^{2}}R+{\frac {2m_{0}}{\hbar ^{2}}}[E-V(r)]R=0.}$

This equation can be reduced to an equivalent 1-D Schrödinger equation by substituting ${\displaystyle R(r)=\chi (r)/r}$, where ${\displaystyle \chi (r)}$ satisfies

${\displaystyle {d^{2}\chi \over dr^{2}}+{\frac {2m_{0}}{\hbar ^{2}}}[E-V_{\mathrm {eff} }(r)]\chi =0}$

which is precisely the one-dimensional Schrödinger equation with an effective potential given by

${\displaystyle V_{\mathrm {eff} }(r)=V(r)+{\hbar ^{2}l(l+1) \over 2m_{0}r^{2}},}$

where the radial coordinate r ranges from 0 to ${\displaystyle \infty }$. The correction to the potential V(r) is called the centrifugal barrier term.

If ${\displaystyle \lim _{r\rightarrow 0}r^{2}V(r)=0}$, then near the origin, ${\displaystyle R\sim r^{l}}$.

Solutions for potentials of interest

Five special cases arise, of special importance:

1. V(r) = 0, or solving the vacuum in the basis of spherical harmonics, which serves as the basis for other cases.
2. ${\displaystyle V(r)=V_{0}}$ (finite) for ${\displaystyle r<r_{0}}$ and infinite elsewhere, or a particle in the spherical equivalent of the square well, useful to describe bound states in a nucleus or quantum dot.
3. As the previous case, but with an infinitely high jump in the potential on the surface of the sphere.
4. V(r) ~ r² for the three-dimensional isotropic harmonic oscillator.
5. V(r) ~ 1/r to describe bound states of hydrogen-like atoms.

We outline the solutions in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box. This article relies heavily on Bessel functions and Laguerre polynomials.

Vacuum case

Let us now consider V(r) = 0 (if ${\displaystyle V_{0}}$, replace everywhere E with ${\displaystyle E-V_{0}}$). Introducing the dimensionless variable

${\displaystyle \rho \ {\stackrel {\mathrm {def} }{=}}\ kr,\qquad k\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {2m_{0}E \over \hbar ^{2}}}}$

the equation becomes a Bessel equation for J defined by ${\displaystyle J(\rho )\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {\rho }}R(r)}$ (whence the notational choice of J):

${\displaystyle \rho ^{2}{d^{2}J \over d\rho ^{2}}+\rho {dJ \over d\rho }+\left[\rho ^{2}-\left(l+{\frac {1}{2}}\right)^{2}\right]J=0}$

which regular solutions for positive energies are given by so-called Bessel functions of the first kind' ${\displaystyle J_{l+1/2}(\rho )}$ so that the solutions written for R are the so-called Spherical Bessel function ${\displaystyle R(r)=j_{l}(kr)\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {\pi /(2kr)}}J_{l+1/2}(kr)}$.

The solutions of Schrödinger equation in polar coordinates for a particle of mass ${\displaystyle m_{0}}$ in vacuum are labelled by three quantum numbers: discrete indices l and m, and k varying continuously in ${\displaystyle [0,\infty )}$:

${\displaystyle \psi (\mathbf {r} )=j_{l}(kr)Y_{lm}(\theta ,\phi )}$

where ${\displaystyle k\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {2m_{0}E}}/\hbar }$, ${\displaystyle j_{l}}$ are the spherical Bessel functions and ${\displaystyle Y_{lm}}$ are the spherical harmonics.

These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves ${\displaystyle \exp(i\mathbf {k} \cdot \mathbf {r} )}$.

Sphere with finite "square" potential

Let us now consider the potential ${\displaystyle V(r)=V_{0}}$ for ${\displaystyle r<r_{0}}$ and ${\displaystyle V(r)=0}$ elsewhere. That is, inside a sphere of radius ${\displaystyle r_{0}}$ the potential is equal to V₀ and it is zero outside the sphere. A potential with such a finite discontinuity is called a square potential.[1]

We first consider bound states, i.e., states which display the particle mostly inside the box (confined states). Those have an energy E less than the potential outside the sphere, i.e., they have negative energy, and we shall see that there are a discrete number of such states, which we shall compare to positive energy with a continuous spectrum, describing scattering on the sphere (of unbound states). Also worth noticing is that unlike Coulomb potential, featuring an infinite number of discrete bound states, the spherical square well has only a finite (if any) number because of its finite range (if it has finite depth).

The resolution essentially follows that of the vacuum with normalization of the total wavefunction added, solving two Schrödinger equations — inside and outside the sphere — of the previous kind, i.e., with constant potential. Also the following constraints hold:

1. The wavefunction must be regular at the origin.
2. The wavefunction and its derivative must be continuous at the potential discontinuity.
3. The wavefunction must converge at infinity.

The first constraint comes from the fact that Neumann N and Hankel H functions are singular at the origin. The physical argument that ψ must be defined everywhere selected Bessel function of the first kind J over the other possibilities in the vacuum case. For the same reason, the solution will be of this kind inside the sphere:

${\displaystyle R(r)=Aj_{l}\left({\sqrt {2m_{0}(E-V_{0}) \over \hbar ^{2}}}r\right),\qquad r<r_{0}}$

with A a constant to be determined later. Note that for bound states, ${\displaystyle V_{0}<E<0}$.

Bound states bring the novelty as compared to the vacuum case that E is now negative (in the vacuum it was to be positive). This, along with third constraint, selects Hankel function of the first kind as the only converging solution at infinity (the singularity at the origin of these functions does not matter since we are now outside the sphere):

${\displaystyle R(r)=Bh_{l}^{(1)}\left(i{\sqrt {-2m_{0}E \over \hbar ^{2}}}r\right),\qquad r>r_{0}}$

Second constraint on continuity of ψ at ${\displaystyle r=r_{0}}$ along with normalization allows the determination of constants A and B. Continuity of the derivative (or logarithmic derivative for convenience) requires quantization of energy.

Sphere with infinite "square" potential

In case where the potential well is infinitely deep, so that we can take ${\displaystyle V_{0}=0}$ inside the sphere and ${\displaystyle \infty }$ outside, the problem becomes that of matching the wavefunction inside the sphere (the spherical Bessel functions) with identically zero wavefunction outside the sphere. Allowed energies are those for which the radial wavefunction vanishes at the boundary. Thus, we use the zeros of the spherical Bessel functions to find the energy spectrum and wavefunctions. Calling ${\displaystyle u_{l,k}}$ the k^th zero of ${\displaystyle j_{l}}$, we have:

${\displaystyle E_{kl}={u_{l,k}^{2}\hbar ^{2} \over 2m_{0}r_{0}^{2}}}$

So that one is reduced to the computations of these zeros ${\displaystyle u_{l,k}}$, typically by using a table or calculator, as these zeros are not solvable for the general case.

In the special case ${\displaystyle l=0}$ (spherical symmetric orbitals), the spherical Bessel function is ${\displaystyle j_{0}(x)={\frac {\sin x}{x}}}$, which zeros can be easily given as ${\displaystyle u_{0,k}=k\pi }$. Their energy eigenvalues are thus:

${\displaystyle E_{k0}={(k\pi )^{2}\hbar ^{2} \over 2m_{0}r_{0}^{2}}={k^{2}h^{2} \over 8m_{0}r_{0}^{2}}}$

3D isotropic harmonic oscillator

The potential of a 3D isotropic harmonic oscillator is

${\displaystyle V(r)={\frac {1}{2}}m_{0}\omega ^{2}r^{2}.}$

In this article it is shown that an N-dimensional isotropic harmonic oscillator has the energies

${\displaystyle E_{n}=\hbar \omega {\Bigl (}n+{\frac {N}{2}}{\Bigr )}\quad {\text{with}}\quad n=0,1,\ldots ,\infty ,}$

i.e., n is a non-negative integral number; ω is the (same) fundamental frequency of the N modes of the oscillator. In this case N = 3, so that the radial Schrödinger equation becomes,

${\displaystyle \left[-{\hbar ^{2} \over 2m_{0}}{d^{2} \over dr^{2}}+{\hbar ^{2}l(l+1) \over 2m_{0}r^{2}}+{\frac {1}{2}}m_{0}\omega ^{2}r^{2}-\hbar \omega {\bigl (}n+{\tfrac {3}{2}}{\bigr )}\right]u(r)=0.}$

Introducing

${\displaystyle \gamma \equiv {\frac {m_{0}\omega }{\hbar }}}$

and recalling that ${\displaystyle u(r)=rR(r)\,}$, we will show that the radial Schrödinger equation has the normalized solution,

${\displaystyle R_{n,l}(r)=N_{nl}\,r^{l}\,e^{-{\frac {1}{2}}\gamma r^{2}}\;L_{{\frac {1}{2}}(n-l)}^{(l+{\frac {1}{2}})}(\gamma r^{2}),}$

where the function ${\displaystyle L_{k}^{(\alpha )}(\gamma r^{2})}$ is a generalized Laguerre polynomial in γr² of order k (i.e., the highest power of the polynomial is proportional to γ^kr^2k).

The normalization constant N_nl is,

${\displaystyle N_{nl}=\left[{\frac {2^{n+l+2}\,\gamma ^{l+{\frac {3}{2}}}}{\pi ^{\frac {1}{2}}}}\right]^{\frac {1}{2}}\left[{\frac {[{\frac {1}{2}}(n-l)]!\;[{\frac {1}{2}}(n+l)]!}{(n+l+1)!}}\right]^{\frac {1}{2}}.}$

The eigenfunction R_n,l(r) belongs to energy E_n and is to be multiplied by the spherical harmonic ${\displaystyle Y_{lm}(\theta ,\phi )\,}$, where

${\displaystyle l=n,n-2,\ldots ,l_{\min }\quad {\hbox{with}}\quad l_{\min }={\begin{cases}1&\mathrm {if} \;n\;\mathrm {odd} \\0&\mathrm {if} \;n\;\mathrm {even} \end{cases}}}$

This is the same result as given in the Harmonic Oscillator article, with the minor notational difference of ${\displaystyle \gamma =2\nu \,}$.

Derivation

First we transform the radial equation by a few successive substitutions to the generalized Laguerre differential equation, which has known solutions: the generalized Laguerre functions. Then we normalize the generalized Laguerre functions to unity. This normalization is with the usual volume element r² dr.

First we scale the radial coordinate

${\displaystyle y={\sqrt {\gamma }}r\quad {\hbox{with}}\quad \gamma \equiv {\frac {m_{0}\omega }{\hbar }},}$

and then the equation becomes

${\displaystyle \left[{d^{2} \over dy^{2}}-{l(l+1) \over y^{2}}-y^{2}+2n+3\right]v(y)=0}$

with ${\displaystyle v(y)=u\left(y/{\sqrt {\gamma }}\right)}$.

Consideration of the limiting behaviour of v(y) at the origin and at infinity suggests the following substitution for v(y),

${\displaystyle v(y)=y^{l+1}e^{-y^{2}/2}f(y).}$

This substitution transforms the differential equation to

${\displaystyle \left[{d^{2} \over dy^{2}}+2\left({\frac {l+1}{y}}-y\right){\frac {d}{dy}}+2n-2l\right]f(y)=0,}$

where we divided through with ${\displaystyle y^{l+1}e^{-y^{2}/2}}$, which can be done so long as y is not zero.

Transformation to Laguerre polynomials

If the substitution ${\displaystyle x=y^{2}\,}$ is used, ${\displaystyle y={\sqrt {x}}}$, and the differential operators become

${\displaystyle {\frac {d}{dy}}={\frac {dx}{dy}}{\frac {d}{dx}}=2y{\frac {d}{dx}}=2{\sqrt {x}}{\frac {d}{dx}},{\text{ and }}}$

${\displaystyle {\frac {d^{2}}{dy^{2}}}={\frac {d}{dy}}\left(2y{\frac {d}{dx}}\right)=4x{\frac {d^{2}}{dx^{2}}}+2{\frac {d}{dx}}.}$

The expression between the square brackets multiplying f(y) becomes the differential equation characterizing the generalized Laguerre equation (see also Kummer's equation):

${\displaystyle x{\frac {d^{2}g}{dx^{2}}}+{\Big (}(l+{\frac {1}{2}})+1-x{\Big )}{\frac {dg}{dx}}+{\frac {1}{2}}(n-l)g(x)=0}$

with ${\displaystyle g(x)\equiv f({\sqrt {x}})\,\;}$.

Provided ${\displaystyle k\equiv (n-l)/2\,}$ is a non-negative integral number, the solutions of this equations are generalized (associated) Laguerre polynomials

${\displaystyle g(x)=L_{k}^{(l+{\frac {1}{2}})}(x).}$

From the conditions on k follows: (i) ${\displaystyle n\geq l\,}$ and (ii) n and l are either both odd or both even. This leads to the condition on l given above.

Recovery of the normalized radial wavefunction

Remembering that ${\displaystyle u(r)=rR(r)\,}$, we get the normalized radial solution

${\displaystyle R_{n,l}(r)=N_{nl}\,r^{l}\,e^{-{\frac {1}{2}}\gamma r^{2}}\;L_{{\frac {1}{2}}(n-l)}^{(l+{\frac {1}{2}})}(\gamma r^{2}).}$

The normalization condition for the radial wavefunction is

${\displaystyle \int _{0}^{\infty }r^{2}|R(r)|^{2}\,dr=1.}$

Substituting ${\displaystyle q=\gamma r^{2}\,\;}$, gives ${\displaystyle dq=2\gamma r\,dr\,\;}$ and the equation becomes

${\displaystyle {\frac {N_{nl}^{2}}{2\gamma ^{l+{3 \over 2}}}}\int _{0}^{\infty }q^{l+{1 \over 2}}e^{-q}\left[L_{{\frac {1}{2}}(n-l)}^{(l+{\frac {1}{2}})}(q)\right]^{2}\,dq=1.}$

By making use of the orthogonality properties of the generalized Laguerre polynomials, this equation simplifies to

${\displaystyle {\frac {N_{nl}^{2}}{2\gamma ^{l+{3 \over 2}}}}\cdot {\frac {\Gamma [{\frac {1}{2}}(n+l+1)+1]}{[{\frac {1}{2}}(n-l)]!}}=1.}$

Hence, the normalization constant can be expressed as

${\displaystyle N_{nl}={\sqrt {\frac {2\,\gamma ^{l+{3 \over 2}}\,({\frac {n-l}{2}})!}{\Gamma ({\frac {n+l}{2}}+{\frac {3}{2}})}}}}$

Other forms of the normalization constant can be derived by using properties of the gamma function, while noting that n and l are both of the same parity. This means that n + l is always even, so that the gamma function becomes

${\displaystyle \Gamma \left[{1 \over 2}+\left({\frac {n+l}{2}}+1\right)\right]={\frac {{\sqrt {\pi }}(n+l+1)!!}{2^{{\frac {n+l}{2}}+1}}}={\frac {{\sqrt {\pi }}(n+l+1)!}{2^{n+l+1}[{\frac {1}{2}}(n+l)]!}},}$

where we used the definition of the double factorial. Hence, the normalization constant is also given by

${\displaystyle N_{nl}=\left[{\frac {2^{n+l+2}\,\gamma ^{l+{3 \over 2}}\,[{1 \over 2}(n-l)]!\;[{1 \over 2}(n+l)]!}{\;\pi ^{1 \over 2}(n+l+1)!}}\right]^{1 \over 2}={\sqrt {2}}\left({\frac {\gamma }{\pi }}\right)^{1 \over 4}\,({2\gamma })^{\ell \over 2}\,{\sqrt {\frac {2\gamma (n-l)!!}{(n+l+1)!!}}}.}$

Hydrogen-like atoms

A hydrogenic (hydrogen-like) atom is a two-particle system consisting of a nucleus and an electron. The two particles interact through the potential given by Coulomb's law:

${\displaystyle V(r)=-{\frac {1}{4\pi \varepsilon _{0}}}{\frac {Ze^{2}}{r}}}$

where

• ε₀ is the permittivity of the vacuum,
• Z is the atomic number (eZ is the charge of the nucleus),
• e is the elementary charge (charge of the electron),
• r is the distance between the electron and the nucleus.

The mass m₀, introduced above, is the reduced mass of the system. Because the electron mass is about 1836 times smaller than the mass of the lightest nucleus (the proton), the value of m₀ is very close to the mass of the electron m_e for all hydrogenic atoms. In the remaining of the article we make the approximation m₀ = m_e. Since m_e will appear explicitly in the formulas it will be easy to correct for this approximation if necessary.

In order to simplify the Schrödinger equation, we introduce the following constants that define the atomic unit of energy and length, respectively,

${\displaystyle E_{\textrm {h}}=m_{\textrm {e}}\left({\frac {e^{2}}{4\pi \varepsilon _{0}\hbar }}\right)^{2}\quad {\hbox{and}}\quad a_{0}={{4\pi \varepsilon _{0}\hbar ^{2}} \over {m_{\textrm {e}}e^{2}}}.}$

Substitute ${\displaystyle y=Zr/a_{0}\,}$ and ${\displaystyle W=E/(Z^{2}E_{\textrm {h}})\,}$ into the radial Schrödinger equation given above. This gives an equation in which all natural constants are hidden,

${\displaystyle \left[-{\frac {1}{2}}{\frac {d^{2}}{dy^{2}}}+{\frac {1}{2}}{\frac {l(l+1)}{y^{2}}}-{\frac {1}{y}}\right]u_{l}=Wu_{l}.}$

Two classes of solutions of this equation exist: (i) W is negative, the corresponding eigenfunctions are square integrable and the values of W are quantized (discrete spectrum). (ii) W is non-negative. Every real non-negative value of W is physically allowed (continuous spectrum), the corresponding eigenfunctions are non-square integrable. In the remaining part of this article only class (i) solutions will be considered. The wavefunctions are known as bound states, in contrast to the class (ii) solutions that are known as scattering states.

For negative W the quantity ${\displaystyle \alpha \equiv 2{\sqrt {-2W}}}$ is real and positive. The scaling of y, i.e., substitution of ${\displaystyle x\equiv \alpha y}$ gives the Schrödinger equation:

${\displaystyle \left[{\frac {d^{2}}{dx^{2}}}-{\frac {l(l+1)}{x^{2}}}+{\frac {2}{\alpha x}}-{\frac {1}{4}}\right]u_{l}=0,\quad {\text{with }}x\geq 0.}$

For ${\displaystyle x\rightarrow \infty }$ the inverse powers of x are negligible and a solution for large x is ${\displaystyle \exp[-x/2]}$. The other solution, ${\displaystyle \exp[x/2]}$, is physically non-acceptable. For ${\displaystyle x\rightarrow 0}$ the inverse square power dominates and a solution for small x is x^l+1. The other solution, x^−l, is physically non-acceptable. Hence, to obtain a full range solution we substitute

${\displaystyle u_{l}(x)=x^{l+1}e^{-x/2}f_{l}(x).\,}$

The equation for f_l(x) becomes,

${\displaystyle \left[x{\frac {d^{2}}{dx^{2}}}+(2l+2-x){\frac {d}{dx}}+(n-l-1)\right]f_{l}(x)=0\quad {\hbox{with}}\quad n=(-2W)^{-{\frac {1}{2}}}={\frac {2}{\alpha }}.}$

Provided ${\displaystyle n-l-1}$ is a non-negative integer, say k, this equation has polynomial solutions written as

${\displaystyle L_{k}^{(2l+1)}(x),\qquad k=0,1,\ldots ,}$

which are generalized Laguerre polynomials of order k. We will take the convention for generalized Laguerre polynomials of Abramowitz and Stegun.[2] Note that the Laguerre polynomials given in many quantum mechanical textbooks, for instance the book of Messiah,[1] are those of Abramowitz and Stegun multiplied by a factor (2l+1+k)! The definition given in this Wikipedia article coincides with the one of Abramowitz and Stegun.

The energy becomes

${\displaystyle W=-{\frac {1}{2n^{2}}}\quad {\hbox{with}}\quad n\equiv k+l+1.}$

The principal quantum number n satisfies ${\displaystyle n\geq l+1}$, or ${\displaystyle l\leq n-1}$. Since ${\displaystyle \alpha =2/n}$, the total radial wavefunction is

${\displaystyle R_{nl}(r)=N_{nl}\left({\frac {2Zr}{na_{0}}}\right)^{l}\;e^{-{\textstyle {\frac {Zr}{na_{0}}}}}\;L_{n-l-1}^{(2l+1)}\left({\frac {2Zr}{na_{0}}}\right),}$

with normalization constant

${\displaystyle N_{nl}=\left[\left({\frac {2Z}{na_{0}}}\right)^{3}\cdot {\frac {(n-l-1)!}{2n[(n+l)!]^{3}}}\right]^{1 \over 2}}$

which belongs to the energy

${\displaystyle E=-{\frac {Z^{2}}{2n^{2}}}E_{\textrm {h}},\qquad n=1,2,\ldots .}$

In the computation of the normalization constant use was made of the integral [3]

${\displaystyle \int _{0}^{\infty }x^{2l+2}e^{-x}\left[L_{n-l-1}^{(2l+1)}(x)\right]^{2}dx={\frac {2n(n+l)!}{(n-l-1)!}}.}$

References

1. A. Messiah, Quantum Mechanics, vol. I, p. 78, North Holland Publishing Company, Amsterdam (1967). Translation from the French by G.M. Temmer
2. Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 775. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
3. H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry, Van Nostrand, 2nd edition (1956), p. 130. Note that convention of the Laguerre polynomial in this book differs from the present one. If we indicate the Laguerre in the definition of Margenau and Murphy with a bar on top, we have ${\displaystyle {\bar {L}}_{n+k}^{(k)}=(-1)^{k}(n+k)!L_{n}^{(k)}}$.

• Gerald Teschl (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. American Mathematical Society. ISBN 9780821846605.