21.5.1. Schrodinger model - Mathematical details of the analytical solution

Schrodinger equation for the hydrogen atom in space basis reads

\[i \hbar \Psi(\mathbf{r},t) = \left[ - \frac{\hbar^2}{2 m} \nabla^2 - \frac{q^2}{4 \pi \varepsilon} \frac{1}{r} \right] \Psi(\mathbf{r},t) \ . \]

The eigenproblem of the Hermitian operator (projected onto space basis, as well) reads

\[E_n \Psi_n(\mathbf{r}) = - \frac{\hbar^2}{2 m} \nabla^2 \Psi_n(\mathbf{r}) - \frac{q^2}{4 \pi \varepsilon r} \Psi_n(\mathbf{r}) \ .\]

Given the symmetries of the system, the solution of the eigenproblem is evaluated using spherical coordinates \(\{ r, \theta, \varphi \}\), and the method of separation of variables. Using spherical coordinates,

\[- \frac{\hbar^2}{2m} \left\{ \frac{1}{r^2} \partial_r \left( r^2 \partial_r \right) + \frac{1}{r^2 \sin\theta} \partial_\theta \left( \sin\theta \partial_\theta \right) + \frac{1}{r^2 \sin^2\theta} \partial_{\varphi\varphi} \right\} \Psi_n - \frac{q^2}{4\pi \varepsilon_0 r} \Psi_n = E_n \Psi_n\]

Separation of variables, \(\ \Psi_{nlm}(r, \theta, \varphi) = R_n(r) Y_{lm}(\theta, \varphi) \ \)

With the method of separation of variables, we look for solutions in product form:

\[\Psi_{nlm}(r, \theta, \varphi) = R_n(r) Y_{lm}(\theta, \varphi) \ ,\]

and substituting into the Schrödinger equation:

\[\left( E_n + \frac{q^2}{4\pi \varepsilon_0 r} \right) R_n Y_{lm} = - \frac{\hbar^2}{2m} \left[ \frac{1}{r^2} \left( r^2 R_n' \right)' Y_{lm} + R_n \left( \frac{1}{r^2 \sin\theta} \partial_\theta \left( \sin\theta \partial_\theta Y_{lm} \right) + \frac{1}{r^2 \sin^2\theta} \partial_{\varphi\varphi} Y_{lm} \right) \right]\]

Dividing by \(R_n Y_{lm}\), and multiplying by \(r^2 \frac{2m}{\hbar^2}\),

\[0 = \underbrace{\frac{1}{R_n} (r^2 R_n')' + \frac{2m r^2}{\hbar^2} \left( E_n + \frac{q^2}{4\pi \varepsilon_0 r} \right)}_{f(r)} + \underbrace{\frac{1}{Y_{lm}} \left\{ \frac{1}{\sin\theta} \partial_\theta (\sin\theta \partial_\theta Y_{lm}) + \frac{1}{\sin^2\theta} \partial_{\varphi\varphi} Y_{lm} \right\}}_{g(\theta, \varphi)}\]

Since \(f(r) + g(\theta, \varphi) = 0\), both terms must equal equal-and-opposite constant \(C\) for all valid domains \(r \in (0, +\infty)\), \(\theta \in (0, \pi)\), \(\varphi \in (0, 2\pi)\)

\[C = f(r) = - g(\theta, \varphi) \ .\]

Radial equation, and energy levels

The radial component determines the energy level,

(21.2)\[E_n = - \frac{m q^4}{32 \pi^2 \varepsilon_0^2 \hbar^2} \frac{1}{n^2} \ .\]

being \(n \in \{ 1, 2, \dots \}\).

The radial distribution of the energy state functions is

\[R(\rho) = e^{-\rho/2} \rho^{\ell} L^{2 \ell+1}_{n-\ell-1}(\rho) \ ,\]

with \(\rho = r \frac{\sqrt{-8 m E_n}}{\hbar}\) the non dimensional radius, \(\ell \in \{ 0, 1, \dots, n-1 \}\), and \(L_{p}^{k}(x)\) the solution of the associated Laguerre differential equation \(x y'' + ( k + 1 - x ) y' + p y = 0\).

Change of variables, and asymptotic behavior

Defining \(C = \ell(\ell+1)\), and dividing the equation \(f(r) = C\) by \(r^2\), for the states with \(E_n <0\),

\[\frac{1}{r^2} \frac{d}{dr} \left( r^2 R' \right) + \left[ \frac{2m}{\hbar^2} \left( E + \frac{q^2}{4\pi \varepsilon_0 r} \right) - \frac{\ell(\ell + 1)}{r^2} \right] R = 0\]

Defining the dimensionless radial variable \(\rho = \kappa r\), so \(\frac{d}{dr} = \kappa \frac{d}{d\rho}\), and factoring out \(\kappa^2\)

\[0 = \kappa^2 \left\{ \frac{1}{\rho^2} \frac{d}{d\rho} \left( \rho^2 \frac{dR}{d\rho} \right) + \left( \frac{2m E}{\hbar^2 \kappa^2} + \frac{2m q^2}{\hbar^2 \kappa \cdot 4\pi \varepsilon_0 \rho} - \frac{\ell(\ell + 1)}{\rho^2} \right) \right\} R\]

By choosing \(\kappa^2 = -\frac{8 m E}{\hbar^2}\) (where \(\kappa = \frac{\sqrt{-8 m E}}{\hbar}\) for \(E < 0\)), the first term inside the parentheses becomes \(-1\). Defining the constant parameter \(\rho_0\):

\[\rho_0 = \frac{2m q^2}{\hbar^2 \kappa \cdot 4\pi \varepsilon_0} = \frac{2m q^2}{\hbar^2 \sqrt{-8 m E} \cdot 4\pi \varepsilon_0} = \frac{q^2}{8\pi \varepsilon_0 \hbar} \sqrt{\frac{2m}{-E}} \ ,\]

the dimensionless radial equation simplifies to:

\[0 = \frac{1}{\rho^2} \frac{d}{d\rho} \left( \rho^2 \frac{dR}{d\rho} \right) + R \left[ - \frac{\ell(\ell + 1)}{\rho^2} - \frac{1}{4} + \frac{\rho_0}{\rho} \right]\]

Boundary conditions required for regular wavefunctions:

\[R(\rho = 0) < \infty, \quad R(\rho \to +\infty) = 0\]

Asymptotic behavior for \(\ r \rightarrow 0\), and \(\ r \rightarrow +\infty\)

Large Distance Limit (\(\rho \to +\infty\)). For large \(\rho\), terms proportional to \(\frac{1}{\rho}\) and \(\frac{1}{\rho^2}\) become negligible:

\[\frac{1}{\rho^2} \frac{d}{d\rho} \left( \rho^2 R' \right) = R'' + \frac{2}{\rho} R' \sim R''\]

The equation reduces to:

\[R'' - \frac{1}{4} R \sim 0\]

General solution:

\[R(\rho) = A e^{-\rho/2} + B e^{\rho/2}\]

To satisfy the non-divergent boundary condition at infinity, set \(B = 0\):

\[R(\rho) \sim e^{-\rho/2} \quad \text{as } \rho \to +\infty\]

Small Distance Limit (\(\rho \to 0\)). For \(\rho \to 0\), the centrifugal term \(\frac{\ell(\ell+1)}{\rho^2}\) dominates over the constant and \(\frac{1}{\rho}\) terms:

\[R'' + \frac{2}{\rho} R' - \frac{\ell(\ell + 1)}{\rho^2} R \sim 0\]

Proposing a power-law ansatz \(R(\rho) \sim \rho^s\):

\[s(s - 1) \rho^{s-2} + 2s \rho^{s-2} - \ell(\ell + 1) \rho^{s-2} = 0\]

\[s^2 + s - \ell(\ell + 1) = 0\]

Solving the quadratic characteristic equation for \(s\):

\[s_{1,2} = \frac{-1 \mp \sqrt{1 + 4\ell(\ell + 1)}}{2} = \frac{-1 \mp (1 + 2\ell)}{2}\]

\[s_1 = \ell, \quad s_2 = -(\ell + 1)\]

Since \(\ell \ge 0\), \(s_2 = -(\ell + 1)\) leads to divergence at the origin \(\rho \to 0\). Retaining only the physically acceptable solution \(s = \ell\):

\[R(\rho) \sim \rho^\ell \quad \text{as } \rho \to 0\]

Transformation & Differential Equation for \(v(\rho)\). Combining both asymptotic behaviors, we represent the full radial solution as:

\[R(\rho) = \rho^\ell e^{-\rho/2} v(\rho)\]

Writing out derivatives of \(R(\rho)\):

\[R' = \left( \ell \rho^{\ell-1} - \frac{1}{2} \rho^\ell \right) e^{-\rho/2} v + \rho^\ell e^{-\rho/2} v'\]

After expanding \(\frac{1}{\rho^2} \frac{d}{d\rho} \left( \rho^2 R' \right) + R \left[ - \frac{\ell(\ell + 1)}{\rho^2} - \frac{1}{4} + \frac{\rho_0}{\rho} \right] = 0\):

\[= e^{-\rho/2} \Big\{ \rho^\ell v'' + v' \left[ 2\ell \rho^{\ell-1} - \frac{1}{2} \rho^\ell + 2 \rho^{\ell-1} \right] + v \left[ \rho^\ell \ell(\ell+1) - \rho^{\ell+1} \left( \ell + 1 \right) + \frac{1}{4} \rho^{\ell+2} - \ell(\ell+1) \rho^{\ell} - \frac{1}{4} \rho^{\ell+2} + \rho_0 \rho^{\ell+1} \right] \Big\} = 0\]

Dividing out non-zero factor \(\rho^{\ell-1} e^{-\rho/2}\):

\[\rho v''(\rho) + [2(\ell + 1) - \rho] v'(\rho) + (\rho_0 - \ell - 1) v(\rho) = 0\]

Polynomial expansion, energy quantization \(\ E_n \ \) and radial distribution \(\ R_n(\rho)\)

Starting from the transformed radial equation:

\[\rho v''(\rho) + [2(\ell + 1) - \rho] v'(\rho) + (\rho_0 - \ell - 1) v(\rho) = 0\]

Looking for a power series solution:

\[\begin{split}\begin{aligned} v(\rho) & = \sum_{k=0}^{\infty} c_k \rho^k \\ v'(\rho) & = \sum_{k=0}^{\infty} k c_k \rho^{k-1} \\ v''(\rho) & = \sum_{k=0}^{\infty} k(k-1) c_k \rho^{k-2} \\ \end{aligned}\end{split}\]

Substituting into the differential equation:

\[\sum_{k=0}^{\infty} k(k-1) c_k \rho^{k-1} + \sum_{k=0}^{\infty} 2(\ell + 1) k c_k \rho^{k-1} - \sum_{k=0}^{\infty} k c_k \rho^k + (\rho_0 - \ell - 1) \sum_{k=0}^{\infty} c_k \rho^k = 0\]

Shift indices to group terms by power \(\rho^k\):

\[\sum_{k=0}^{\infty} \Big\{ (k+1)(k) c_{k+1} + 2(\ell + 1)(k+1) c_{k+1} - k c_k + (\rho_0 - \ell - 1) c_k \Big\} \rho^k = 0\]

Recurrence Relation. Equating coefficients to zero yields the recurrence relation for \(c_k\):

\[c_{k+1} = c_k \frac{k + \ell + 1 - \rho_0}{(k + 1)(k + 2\ell + 2)}\]

Asymptotic Behavior & Series Termination. If the power series does not terminate, as \(k \to +\infty\):

\[\frac{c_{k+1}}{c_k} \longrightarrow \frac{1}{k}\]

This asymptotic behavior matches that of the exponential expansion \(e^\rho \sim \sum \frac{\rho^k}{k!}\) and thus \(\frac{c_{k+1}}{c_k} = \frac{1}{k+1} \sim \frac{1}{k}\). Thus, an infinite series \(v(\rho) \sim e^\rho\), would lead to \(R(\rho) \sim e^{-\rho/2} e^\rho = e^{\rho/2}\), which diverges as \(\rho \to +\infty\). Thus, in order to maintain physical normalizability, the series must terminate at a maximum power \(k_{\max} = N\):

\[c_{N+1} = 0 \implies N + \ell + 1 - \rho_0 = 0 \implies \rho_0 = N + \ell + 1 \equiv n\]

where:

• \(N \ge 0\) is the radial quantum number.

• \(n = N + \ell + 1\) is the Principal Quantum Number (\(n \ge 1\)).

• \(\forall n\), the orbital angular momentum quantum number satisfies \(0 \le \ell \le n - 1\).

Energy Quantization. Recalling the definition of \(\rho_0\), \(n = \rho_0 = \sqrt{\frac{2m}{-E}} \frac{q^2}{4\pi \varepsilon_0 \hbar}\), and solving for the bound state energies \(E_n\) (\(E < 0\)), the relation between the energy \(E_n\) of the \(n^{th}\) energy level and the principal quantum number \(n\) follows

(21.3)\[E_n = - \frac{m q^4}{32 \pi^2 \varepsilon_0^2 \hbar^2} \frac{1}{n^2} \ .\]

Remark. The value of energy \(E_n\) of Schrodinger model coincides with the value (21.1) of the energy produced by Bohr model.

Radial solution. Putting everything together, the radial part of the solution reads

\[R(\rho) = e^{-\rho/2} \rho^{\ell} L^{2 \ell+1}_{n-\ell-1}(\rho) \ ,\]

being \(L_{p}^{k}(x)\) the solution of the associated Laguerre differential equation \(x y'' + ( k + 1 - x ) y' + p y = 0\).

Azimuthal and polar equations

Further separation of variables, \(\ Y_{\ell m}(\theta, \varphi) = \Theta(\theta) \Phi(\varphi) \ \)

With \(Y(\theta, \varphi) = \Theta(\theta) \Psi(\varphi)\), the equation

\[ \frac{1}{Y_{lm}} \left\{ \frac{1}{\sin\theta} \partial_\theta (\sin\theta \partial_\theta Y_{lm}) + \frac{1}{\sin^2\theta} \partial_{\varphi\varphi} Y_{lm} \right\} = - \ell ( \ell + 1 ) \ . \]

becomes - after multipyling by \(\sin^2 \theta\),

\[\underbrace{\frac{\sin \theta}{\Theta} \left( \sin \theta \, \Theta' \right)' + \ell ( \ell + 1 ) \, \sin^2 \theta}_{F(\theta)} = -\underbrace{\frac{\Phi''}{\Phi}}_{G(\varphi)} \ ,\]

and thus

\[F(\theta) = - G(\varphi) = m_\ell^2 \ ,\]

with a non-negative constant \(m_{\ell}^2\) in order to match periodic conditions for \(\Phi(\varphi = 0) = \Phi(\varphi = 2 \pi)\).

Azimuthal equation

\[\Phi'' + m_\ell^2 \Phi = 0 \ ,\]

whose solution is \(\Phi(\varphi) \propto \exp \left( \mp i m_{\ell} \varphi \right)\). In order to satisfy periodic conditoins, \(m_{\ell} \in \mathbb{Z}\). This parameter will be furthered constrained later, in the solution of the polar equation, to be \(m_{\ell} \in \{ - \ell, -\ell+1, \dots, \ell \}\), with \(\ell \in \mathbb{N}\). Thus, selecting only the independent solutions, the azimuthal distribution of the eigenfunctions of the Hamiltonian is

\[\Phi(\varphi) \propto \exp\left( i m_{\ell} \varphi \right) \ , \quad m_{\ell} \in \{ - \ell, -\ell+1, \dots \ell \} \ .\]

Polar equation

Multiplying the equation \(F(\theta) = m_{ell}^2\) by \(\Theta\), the polar equation becomes,

\[ \sin \theta \left( \sin \theta \, \Theta' \right)' + \left[ \ell ( \ell + 1 ) \, \sin^2 \theta - m_{\ell}^2 \right] \Theta = 0 \ .\]

Polar equation as an associated Legendre equation

Dividing by \(\sin^2 \theta= 1 - \cos^2 \theta\), the second term can be immediately recast as the second term in the associated Legendre equation, if \(x = \cos \theta\).

As \(\frac{d x}{d \theta} = - \sin \theta\), then \(\frac{d}{d\theta} = \frac{d x}{d \theta} \frac{d}{d x} = - \sin \theta \frac{d}{d x}\), the first term - divided by \(\sin^2 \theta\) - becomes

\[-\frac{1}{\sin^2 \theta} \sin^2 \theta \dfrac{d}{dx} \left( - \sin^2 \theta \dfrac{d}{dx} \Theta \right) = \dfrac{d}{dx}\left[ ( 1 - x^2) \dfrac{d}{dx} \Theta \right]\]

Thus the polar equation can be recast as

\[0 = \frac{d}{dx} \left[ ( 1 - x^2 ) \dfrac{d \Theta}{dx} \right] + \left[ \ell(\ell+1) - \frac{m_\ell^2}{1 - x^2} \right] \Theta \ ,\]

i.e. as the associated Legendre equation whose solution (see below) is \(\Theta(x) = P_\ell^{m_{\ell}}(x)\), for \(m_{\ell} \in \{ - \ell, \dots, \ell\}\), \(\ell \in \mathbb{N}\).

Legendre differential equation and associated Legendre equation

Legendre differential equations read

\[\begin{split}\begin{aligned} 0 & = (1- x^2) P_n''(x) - 2 x P_n'(x) + n(n+1) P_n(x) = \quad , \qquad x \in [-1, 1] \\ & = \dfrac{d}{dx} \left[ ( 1-x^2 ) P_n'(x) \right] + n(n+1) P_n(x) \ , \end{aligned}\end{split}\]

with the solution \(P_n(x)\) be the Legendre polynomial. The associated Legendre differential equations are defined as

\[\begin{split}\begin{aligned} 0 & = \dfrac{d}{dx} \left[ ( 1-x^2 ) P_n'(x) \right] + \left[ n(n+1) - \frac{m^2}{1-x^2} \right] P_n(x) \quad , \qquad x \in [-1, 1] \\ \end{aligned}\end{split}\]

and the solution reads \(P_{n}^{m}(x) = (-1)^{m} \left( 1 - x^2 \right)^{\frac{m}{2}} \dfrac{d^m}{dx^m} P_{n}(x)\), for \(m \ge 0\). The soutions for \(m < 0\) are defined via symmetry conditions, or Rodrigues’ formula, as \(P_{\ell}^{-m} = (-1)^{m} \frac{(\ell-m)!}{(\ell+m)!} P_{\ell}^{m}(x)\), see Wikipedia: associated Lagendre polynomials.