Particle in a spherically symmetric potential#Hydrogen-like atoms

If solved by separation of variables, the eigenstates of the system will have the form:$$\backslash psi(r,\; \backslash theta,\; \backslash phi)\; =\; R(r)\backslash Theta(\backslash theta)\backslash Phi(\backslash phi)$$

in which the spherical angles $\backslash theta$ and $\backslash phi$ represent the polar and azimuthal angle, respectively. Those two factors of $\backslash psi$ are often grouped together as spherical harmonics, so that the eigenfunctions take the form:

$$\backslash psi(r,\; \backslash theta,\; \backslash phi)\; =\; R(r)Y\_\{\backslash ell\; m\}(\backslash theta,\backslash phi).$$

The differential equation which characterises the function $R(r)$ is called the radial equation.

The kinetic energy operator in spherical polar coordinates is:$$\backslash frac\{\backslash hat\{p\}^2\}\{2m\_0\}\; =\; -\backslash frac\{\backslash hbar^2\}\{2m\_0\}\; \backslash nabla^2\; =$$

- \frac{\hbar^2}{2m_0\,r^2} \left[ \frac{\partial}{\partial r} \left(r^2 \frac{\partial}{\partial r}\right) - \hat{L}^2 \right].The spherical harmonics satisfy$$$$

\hat{L}^2 Y_{\ell m}(\theta,\phi)\equiv \left\{ -\frac{1}{\sin^2\theta} \left[

\sin\theta \frac{\partial}{\partial\theta} \left(\sin\theta \frac{\partial}{\partial\theta}\right)

+\frac{\partial^2}{\partial \phi^2}\right]\right\} Y_{\ell m}(\theta,\phi)

= \ell(\ell+1)Y_{\ell m}(\theta,\phi).

Substituting this into the Schrödinger equation we get a one-dimensional eigenvalue equation,$$\backslash frac\{1\}\{r^2\}\backslash frac\{d\}\{dr\}\; \backslash left(r^2\backslash frac\{dR\}\{dr\}\backslash right)\; -\; \backslash frac\{\backslash ell(\backslash ell+1)\}\{r^2\}\; R\; +\; \backslash frac\{2m\_0\}\{\backslash hbar^2\}\; \backslash left[E-V(r)\backslash right]R\; =\; 0.$$This equation can be reduced to an equivalent 1-D Schrödinger equation by substituting $R(r)\; =\; u(r)/r$, where $u(r)$ satisfies$$\{d^2\; u\; \backslash over\; dr^2\}\; +\; \backslash frac\{2m\_0\}\{\backslash hbar^2\}\; \backslash left[E-V\_\{\backslash mathrm\{eff\}\}(r)\backslash right]\; u\; =\; 0$$which is precisely the one-dimensional Schrödinger equation with an effective potential given by$$V\_\{\backslash mathrm\{eff\}\}(r)\; =\; V(r)\; +\; \{\backslash hbar^2\; \backslash ell(\backslash ell+1)\; \backslash over\; 2m\_0r^2\},$$where $r\; \backslash in\; [0,\backslash infty)$. The correction to the potential V(r) is called the centrifugal barrier term.

If $\backslash lim\_\{r\; \backslash to\; 0\}\; r^2\; V(r)\; =0$, then near the origin, $R\; \backslash sim\; r^\backslash ell$.

Since the Hamiltonian is spherically symmetric, it is said to be invariant under rotation, ie:

$\{U(R)\}^\backslash dagger\; \backslash hat\{H\}\; U(R)=\backslash hat\{H\}$

Since angular momentum operators are generators of rotation, applying the Baker-Campbell-Hausdorff Lemma we get:

$\{U(R)\}^\backslash dagger\; \backslash hat\{H\}\; U(R)\; =\; e^\{\backslash frac\{i\backslash hat\{n\}\backslash cdot\; \backslash vec\{L\}\backslash theta\}\{\backslash hbar\}\}\; \backslash hat\{H\}\; e^\{\backslash frac\{-i\backslash hat\{n\}\backslash cdot\; \backslash vec\{L\}\backslash theta\}\{\backslash hbar\}\}\; =\backslash hat\{H\}\; +\; i\backslash frac\{\backslash theta\}\{\backslash hbar\}\backslash left[\; \backslash hat\{n\}\backslash cdot\; \backslash vec\{L\},\; \backslash hat\{H\}\backslash right]\; +\backslash frac\; 1\; \{2!\}\backslash left(\; i\backslash frac\{\backslash theta\}\{\backslash hbar\}\backslash right)^2\backslash left[\; \backslash hat\{n\}\backslash cdot\; \backslash vec\{L\},\backslash left[\; \backslash hat\{n\}\backslash cdot\; \backslash vec\{L\},\; \backslash hat\{H\}\backslash right]\backslash right]\; +\; \backslash cdots\; =\; \backslash hat\; H$

Since this equation holds for all values of $\backslash theta$, we get that $[\; \backslash hat\{n\}\backslash cdot\; \backslash vec\{L\},\; \backslash hat\{H\}]\; =0$, or that every angular momentum component commutes with the Hamiltonian.

Since $L\_z$ and $L^2$ are such mutually commuting operators that also commute with the Hamiltonian, the wavefunctions can be expressed as $|\backslash alpha;\backslash ell,m\backslash rangle$ or $\backslash psi\_\{\backslash alpha;\backslash ell,m\}(r,\backslash theta,\backslash phi)$ where $\backslash alpha$ is used to label different wavefunctions.

Since $L\_\backslash pm\; =\; L\_x\; \backslash pm\; i\; L\_y$ also commutes with the Hamiltonian, the energy eigenvalues in such cases are always independent of $m$.

$$\backslash hat\; H\; |\backslash alpha;\; \backslash ell,m\backslash rangle\; =\; E\_\{\backslash alpha;\; \backslash ell\}|\backslash alpha;\; \backslash ell,m\backslash rangle$$

Combined with the fact that $L\_\backslash pm$ differential operators only act on the functions of $\backslash theta$ and $\backslash phi$, it shows that if the solutions are assumed to be separable as $\backslash psi(r,\; \backslash theta,\; \backslash phi)\; =\; R(r)\; Y(\backslash theta,\backslash phi)$, the radial wavefunction $R(r)$ can always be chosen independent of $m$ values. Thus the wavefunction is expressed as:{{Cite web |last=Littlejohn |first=Robert G |title=Physics 221A: Central Force Motion |url=https://bohr.physics.berkeley.edu/classes/221/notes/cenforce.pdf |url-status=live |archive-url=https://web.archive.org/web/20231208010523/https://bohr.physics.berkeley.edu/classes/221/notes/cenforce.pdf |archive-date=8 December 2023 |access-date=18 February 2024}}

$$\backslash psi\_\{\backslash alpha;\backslash ell,\; m\}(r,\; \backslash theta,\; \backslash phi)\; =\; R(r)\_\{\backslash alpha;\backslash ell\}Y\_\{\backslash ell\; m\}(\backslash theta,\backslash phi).$$

There are five cases 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. $V(r)=V\_0$ (finite) for
3. $V(r)\; =\; 0$ for and infinite elsewhere, the spherical equivalent of the square well, useful to describe bound states in a nucleus or quantum dot.
4. $V(r)\; \backslash propto\; r^\{2\}$for the three-dimensional isotropic harmonic oscillator.
5. $V(r)\; \backslash propto\; \backslash frac\{1\}\{r\}$ to describe bound states of hydrogen-like atoms.

The solutions are outlined in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box.

Let us now consider $V(r)\; =\; 0$. Introducing the dimensionless variables$$\backslash rho\backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; kr,\; \backslash qquad\; k\backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash sqrt\{2m\_0E\backslash over\backslash hbar^2\},$$the equation becomes a Bessel equation for $J(\backslash rho)\backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash sqrt\backslash rho\; R(r)$:$$\backslash rho^2\backslash frac\{d^2\; J\}\{d\backslash rho^2\}+\backslash rho\; \backslash frac\{dJ\}\{d\backslash rho\}+\backslash left[\backslash rho^2-\backslash left(\backslash ell+\backslash frac\{1\}\{2\}\backslash right)^2\; \backslash right]\; J\; =\; 0$$where regular solutions for positive energies are given by so-called Bessel functions of the first kind $J\_\{\backslash ell+1/2\}(\backslash rho)$ so that the solutions written for $R(r)$ are the so-called spherical Bessel function

$R(r)\; =\; j\_l(kr)\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash sqrt\{\backslash pi/(2kr)\}\; J\_\{\backslash ell+1/2\}(kr)$.

The solutions of the Schrödinger equation in polar coordinates in vacuum are thus labelled by three quantum numbers: discrete indices ℓ and m, and k varying continuously in $[0,\backslash infty)$:$$\backslash psi(\backslash mathbf\{r\})\; =\; j\_\backslash ell(kr)\; Y\_\{\backslash ell\; m\}(\backslash theta,\backslash phi)$$These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves $\backslash exp(i\; \backslash mathbf\{k\}\backslash cdot\backslash mathbf\{r\})$.

Consider the potential $V(r)=V\_0$ for square potential.A. Messiah, Quantum Mechanics, vol. I, p. 78, North Holland Publishing Company, Amsterdam (1967). Translation from the French by G.M. Temmer

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. 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.

The resolution essentially follows that of the vacuum case above 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. The following constraints must hold for a normalizable, physical wavefunction:

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 and Hankel functions are singular at the origin. The physical requirement that $\backslash psi$ must be defined everywhere selected Bessel function of the first kind over the other possibilities in the vacuum case. For the same reason, the solution will be of this kind inside the sphere:Note that for bound states, . Bound states bring the novelty as compared to the vacuum case now that . This, along with the third constraint, selects the 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):$$R(r)\; =\; Bh^\{(1)\}\_\backslash ell\backslash left(i\backslash sqrt\{\backslash frac\{-2\; m\_0\; E\}\{\backslash hbar^2\}\}r\backslash right),\; \backslash qquad\; r>r\_0$$The second constraint on continuity of $\backslash psi$ at $r=r\_0$ along with normalization allows the determination of constants $A$ and $B$. Continuity of the derivative (or logarithmic derivative for convenience) requires quantization of energy.

In case where the potential well is infinitely deep, so that we can take $V\_0=0$ inside the sphere and $\backslash infty$ 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 $u\_\{\backslash ell,k\}$ the k^th zero of $j\_\backslash ell$, we have:$$E\_\{kl\}\; =\; \backslash frac\{u\_\{\backslash ell,k\}^2\; \backslash hbar^2\}\{2\; m\_0\; r\_0^2\}$$so that the problem is reduced to the computations of these zeros $u\_\{\backslash ell,k\}$, typically by using a table or calculator, as these zeros are not solvable for the general case.

In the special case $\backslash ell\; =\; 0$ (spherical symmetric orbitals), the spherical Bessel function is $j\_0(x)\; =\; \backslash frac\{\backslash sin\; x\}\; \{x\}$, which zeros can be easily given as $u\_\{0,k\}\; =\; k\; \backslash pi$. Their energy eigenvalues are thus:

$$E\_\{k0\}\; =\; \backslash frac\{(k\; \backslash pi)^2\backslash hbar^2\}\{\; 2\; m\_0\; r\_0^2\}\; =\; \backslash frac\{k^2\; h^2\}\{8\; m\_0\; r\_0^2\}$$

===3D isotropic harmonic oscillator===

The potential of a 3D isotropic harmonic oscillator is

$$V(r)\; =\; \backslash frac\{1\}\{2\}\; m\_0\; \backslash omega^2\; r^2.$$

An N-dimensional isotropic harmonic oscillator has the energies

$$E\_n\; =\; \backslash hbar\; \backslash omega\backslash left(\; n\; +\; \backslash frac\{N\}\{2\}\; \backslash right)\; \backslash quad\backslash text\{with\}\backslash quad\; n=0,1,\backslash ldots,\backslash infty,$$

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

$$\backslash left[-\backslash frac\{\backslash hbar^2\}\{2m\_0\}\; \backslash frac\{d^2\}\{dr^2\}\; +\; \{\backslash hbar^2\; \backslash ell(\backslash ell+1)\; \backslash over\; 2m\_0r^2\}+\backslash frac\{1\}\{2\}\; m\_0\; \backslash omega^2\; r^2\; -\; \backslash hbar\backslash omega\backslash left(n+\backslash tfrac\{3\}\{2\}\backslash right)\; \backslash right]\; u(r)\; =\; 0.$$

Introducing

$$\backslash gamma\; \backslash equiv\; \backslash frac\{m\_0\backslash omega\}\{\backslash hbar\}$$

and recalling that $u(r)\; =\; r\; R(r)$, we will show that the radial Schrödinger equation has the normalized solution,

$$R\_\{n\backslash ell\}(r)\; =\; N\_\{n\backslash ell\}\; \backslash ,\; r^\{\backslash ell\}\; \backslash ,\; e^\{-\backslash frac\{1\}\{2\}\backslash gamma\; r^2\}\backslash ;\; L^\{\backslash scriptscriptstyle\backslash left(\backslash ell+\backslash frac\{1\}\{2\}\backslash right)\}\_\{\backslash frac\{1\}\{2\}(n-\backslash ell)\}(\backslash gamma\; r^2),$$

where the function $L^\{(\backslash alpha)\}\_k(\backslash gamma\; r^2)$ is a generalized Laguerre polynomial in $yr^2$ of order $k$.

The normalization constant $N\_\{nl\}$ is,

$$N\_\{n\backslash ell\}\; =\; \backslash left[\backslash frac\{\; 2^\{n+\backslash ell+2\}\; \backslash ,\backslash gamma^\{\backslash ell+\backslash frac\{3\}\{2\}\}\; \}\; \{\backslash pi^\{\backslash frac\{1\}\{2\}\}\}\; \backslash right]^\{\backslash frac\{1\}\{2\}\}$$

\left[\frac{ \left[\frac{1}{2} (n-\ell)\right]!\;\left[\frac{1}{2} (n+\ell)\right]!}{(n+\ell+1)!} \right]^{\frac{1}{2}} .

The eigenfunction $R\_\{nl\}(r)$ is associated with energy $E\_n$, where

$$\backslash ell\; =\; n,\; n-2,\; \backslash ldots,\; \backslash ell\_\backslash min\backslash quad$$

\text{with}\quad \ell_\min = \begin{cases}

1 & \text{if}~ n~ \text{odd} \\

0 & \text{if}~ n~ \text{even}

\end{cases}

This is the same result as the quantum harmonic oscillator, with $\backslash gamma\; =\; 2\; \backslash nu$.

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 {{math|r² dr}}.

First we scale the radial coordinate

$$y\; =\; \backslash sqrt\{\backslash gamma\}r\; \backslash quad\; \backslash text\{with\}\backslash quad\; \backslash gamma\; \backslash equiv\; \backslash frac\{m\_0\backslash omega\}\{\backslash hbar\},$$

and then the equation becomes

$$\backslash left[\backslash frac\{d^2\}\{dy^2\}\; -\; \backslash frac\{\backslash ell(\backslash ell+1)\}\{y^2\}\; -\; y^2\; +\; 2n\; +\; 3\; \backslash right]\; v(y)\; =\; 0$$

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

Consideration of the limiting behavior of {{math|v(y)}} at the origin and at infinity suggests the following substitution for {{math|v(y)}},

$$v(y)\; =\; y^\{\backslash ell+1\}\; e^\{-y^2/2\}\; f(y).$$

This substitution transforms the differential equation to

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

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

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

$$\backslash frac\{d\}\{dy\}\; =\; \backslash frac\{dx\}\{dy\}\backslash frac\{d\}\{dx\}\; =\; 2\; y\; \backslash frac\{d\}\{dx\}\; =\; 2\; \backslash sqrt\{x\}\; \backslash frac\{d\}\{dx\},$$ and

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

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

$$x\backslash frac\{d^2g\}\{dx^2\}\; +\; \backslash left(\; \backslash left(\backslash ell+\backslash frac\{1\}\{2\}\backslash right)\; +\; 1\; -\; x\backslash right)\; \backslash frac\{dg\}\{dx\}\; +\; \backslash frac\{1\}\{2\}\; (n\; -\; \backslash ell)\; g(x)\; =\; 0$$

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

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

$$g(x)\; =\; L\_k^\{\backslash scriptscriptstyle\backslash left(\backslash ell+\backslash frac\{1\}\{2\}\backslash right)\}(x).$$

From the conditions on $k$ follows: (i) $n\; \backslash ge\; \backslash ell$ and (ii) $n$ and $l$ are either both odd or both even. This leads to the condition on $l$ given above.

Remembering that $u(r)\; =\; r\; R(r)$, we get the normalized radial solution:

$$R\_\{n\backslash ell\}(r)\; =\; N\_\{n\backslash ell\}\; \backslash ,\; r^\{\backslash ell\}\; \backslash ,\; e^\{-\backslash frac\{1\}\{2\}\backslash gamma\; r^2\}\backslash ;\; L^\{\backslash scriptscriptstyle\backslash left(\backslash ell+\backslash frac\{1\}\{2\}\backslash right)\}\_\{\backslash frac\{1\}\{2\}(n-\backslash ell)\}(\backslash gamma\; r^2).$$

The normalization condition for the radial wave function is:

$$\backslash int^\backslash infty\_0\; r^2\; |R(r)|^2\; \backslash ,\; dr\; =\; 1.$$

Substituting $q\; =\; \backslash gamma\; r^2$, gives $dq\; =\; 2\; \backslash gamma\; r\; \backslash ,\; dr$ and the equation becomes:

$$\backslash frac\{N^2\_\{n\backslash ell\}\}\{2\backslash gamma^\{\backslash ell+\backslash frac\{3\}\{2\}\}\}$$

\int^\infty_0 q^{\ell + {1 \over 2}} e^{-q} \left [ L^{\scriptscriptstyle\left(\ell+\frac{1}{2}\right)}_{\frac{1}{2}(n-\ell)}(q) \right ]^2 \, dq = 1.

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

$$\backslash frac\{N^2\_\{n\backslash ell\}\}\{2\backslash gamma^\{\backslash ell+\backslash frac\{3\}\{2\}\}\}\; \backslash cdot\; \backslash frac\{\backslash Gamma\{\backslash left[\backslash frac\{1\}\{2\}\; (n+\backslash ell+1)+1\backslash right]\}\}\{\backslash left[\backslash frac\{1\}\{2\}(n-\backslash ell)\backslash right]!\}\; =\; 1.$$

Hence, the normalization constant can be expressed as:

$$N\_\{n\backslash ell\}\; =\; \backslash sqrt\{\backslash frac\{2\; \backslash ,\; \backslash gamma^\{\backslash ell+\backslash frac\{3\}\{2\}\}\; \backslash ,\; \backslash left(\backslash frac\{n-\backslash ell\}\{2\}\backslash right)!\; \}\{\backslash Gamma\{\backslash left(\backslash frac\{n+\backslash ell\}\{2\}+\backslash frac\{3\}\{2\}\backslash right)\}\}\}$$

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:

$$\backslash Gamma\; \backslash left[\backslash frac\{1\}\{2\}\; +\; \backslash left(\; \backslash frac\{n+\backslash ell\}\{2\}\; +\; 1\; \backslash right)\; \backslash right]$$

= \frac{\sqrt{\pi}(n+\ell+1)!!}{2^{\frac{n+\ell}{2}+1}}

= \frac{\sqrt{\pi}(n+\ell+1)!}{2^{n+\ell+1}\left[\frac{1}{2}(n+\ell)\right]!},

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

$$N\_\{n\backslash ell\}$$

= \left [ \frac{2^{n+\ell+2} \,\gamma^{\ell+{3 \over 2}}\left[{1 \over 2}(n-\ell)\right]!\left[{1 \over 2}(n+\ell)\right]!}{\;\pi^{1 \over 2} (n+\ell+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-\ell)!!}{(n+\ell+1)!!}}.

{{main|hydrogen-like atom}}

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:

$$V(r)\; =\; -\backslash frac\{1\}\{4\; \backslash pi\; \backslash varepsilon\_0\}\; \backslash frac\{Ze^2\}\{r\}$$

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.

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

$$E\_\backslash textrm\{h\}\; =\; \backslash mu\; \backslash left(\; \backslash frac\{e^2\}\{4\; \backslash pi\; \backslash varepsilon\_0\; \backslash hbar\}\backslash right)^2\; \backslash quad\backslash text\{and\}\backslash quad\; a\_0\; =\; \{\{4\backslash pi\backslash varepsilon\_0\backslash hbar^2\}\backslash over\{\backslash mu\; e^2\}\}.$$

where $\backslash mu\; \backslash approx\; m\_e$ is the reduced mass in the $m\_e\; \backslash ll\; m\_\{nucleus\}$ limit. Substitute $y\; =\; Zr/a\_0$ and $W\; =\; E/(Z^2\; E\_\backslash textrm\{h\})$ into the radial Schrödinger equation given above. This gives an equation in which all natural constants are hidden,

$$\backslash left[\; -\backslash frac\{1\}\{2\}\; \backslash frac\{d^2\}\{dy^2\}\; +\; \backslash frac\{1\}\{2\}\; \backslash frac\{\backslash ell(\backslash ell+1)\}\{y^2\}\; -\; \backslash frac\{1\}\{y\}\backslash right]\; u\_\backslash ell\; =\; W\; u\_\backslash ell\; .$$

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. Considering only class (i) solutions restricts the solutions to wavefunctions which are bound states, in contrast to the class (ii) solutions that are known as scattering states.

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

$$\backslash left[\; \backslash frac\{d^2\}\{dx^2\}\; -\backslash frac\{\backslash ell(\backslash ell+1)\}\{x^2\}\; +\; \backslash frac\{2\}\{\backslash alpha\; x\}\; -\; \backslash frac\{1\}\{4\}\; \backslash right]\; u\_\backslash ell\; =\; 0,$$

\quad \text{with } x \ge 0.

For $x\; \backslash rightarrow\; \backslash infty$ the inverse powers of x are negligible and the normalizable (and therefore, physical) solution for large $x$ is $\backslash exp[-x/2]$. Similarly, for $x\; \backslash rightarrow\; 0$ the inverse square power dominates and the physical solution for small $x$ is x^ℓ+1.

Hence, to obtain a full range solution we substitute

$$u\_l(x)\; =\; x^\{\backslash ell+1\}\; e^\{-x/2\}f\_\backslash ell(x).$$

The equation for $f\_l(x)$ becomes,

$$\backslash left[\; x\backslash frac\{d^2\}\{dx^2\}\; +\; (2\backslash ell+2-x)\; \backslash frac\{d\}\{dx\}\; +(n\; -\; \backslash ell\; -\; 1)\backslash right]\; f\_\backslash ell(x)\; =\; 0\; \backslash quad\backslash text\{with\}\backslash quad\; n\; =\; (-2W)^\{-1/2\}=\backslash frac\{2\}\{\backslash alpha\}.$$

Provided $n-\backslash ell-1$ is a non-negative integer, this equation has polynomial solutions written as

$$L^\{(2\backslash ell+1)\}\_\{k\}(x),\backslash qquad\; k=0,1,\backslash ldots\; ,$$

which are generalized Laguerre polynomials of order $k$. The energy becomes$$W\; =\; -\backslash frac\{1\}\{2n^2\}\backslash quad\; \backslash text\{with\}\backslash quad\; n\; \backslash equiv\; k+\backslash ell+1\; .$$

The principal quantum number $n$ satisfies $n\; \backslash ge\; \backslash ell+1$. Since $\backslash alpha\; =\; 2/n$, the total radial wavefunction is

$$R\_\{n\backslash ell\}(r)\; =\; \backslash frac\{u\; (r)\}\{r\}\; =\; N\_\{n\backslash ell\}\; \backslash left(\backslash frac\{2Zr\}\{n\; a\_0\}\backslash right)^\{\backslash ell\}\backslash ;\; e^\{-\{\backslash tfrac\{Zr\}\{n\; a\_0\}\}\}\backslash ;\; L^\{(2\backslash ell+1)\}\_\{n-\backslash ell-1\}\backslash left(\backslash frac\{2Zr\}\{n\; a\_0\}\backslash right),$$

with normalization which absorbs extra terms from $\backslash frac\; \{1\}\{r\}$

$$N\_\{n\backslash ell\}\; =\; \backslash left[\backslash left(\backslash frac\{2Z\}\{n\; a\_0\}\backslash right)^3\; \backslash cdot\; \backslash frac\{(n-\backslash ell-1)!\}\{2n[(n+\backslash ell)!]^3\}\backslash right]^\{1\; \backslash over\; 2\},$$

viaH. 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 $\backslash bar\{L\}^\{(k)\}\_\{n+k\}\; =\; (-1)^k\; (n+k)!\; L^\{(k)\}\_n$.

$$\backslash int\_0^\backslash infty\; x^\{2\backslash ell+2\}\; e^\{-x\}\; \backslash left[\; L^\{(2\backslash ell+1)\}\_\{n-\backslash ell-1\}(x)\backslash right]^2\; dx\; =$$

\frac{2n (n+\ell)!}{(n-\ell-1)!} .

The corresponding energy is

$$E\; =\; -\; \backslash frac\{Z^2\}\{2n^2\}E\_\backslash textrm\{h\},\backslash qquad\; n=1,2,\backslash ldots\; .$$

{{DEFAULTSORT:Particle In A Spherically Symmetric Potential}}

Category:Partial differential equations

Category:Quantum models