Let the wavefunction of the stationary states of the...

Let the wavefunction of the stationary states of the gross-structure Hamiltonian of hydrogen be $\langle\mathbf{x} \mid n, l, m\rangle=u_{n}^{l}(r) \mathrm{Y}_{l}^{m}(\theta, \phi) .$ Show that $$ \int_{0}^{\infty} \mathrm{d} r r^{2} u_{n}^{l}(r) u_{n^{\prime}}^{l}(r)=\delta_{n n^{\prime}} $$ By considering an appropriate Sturm-Liouville equation, or otherwise, show further that $$ \int_{0}^{\infty} \mathrm{d} r u_{n}^{l}(r) u_{n}^{l^{\prime}}(r)=C_{l} \delta_{l l^{\prime}} $$

Let the wavefunction of the stationary states of the gross-structure Hamiltonian of hydrogen be $\langle\mathbf{x} \mid n, l, m\rangle=u_{n}^{l}(r) \mathrm{Y}_{l}^{m}(\theta, \phi) .$ Show that
$$
\int_{0}^{\infty} \mathrm{d} r r^{2} u_{n}^{l}(r) u_{n^{\prime}}^{l}(r)=\delta_{n n^{\prime}}
$$
By considering an appropriate Sturm-Liouville equation, or otherwise, show further that
$$
\int_{0}^{\infty} \mathrm{d} r u_{n}^{l}(r) u_{n}^{l^{\prime}}(r)=C_{l} \delta_{l l^{\prime}}
$$

Key Concepts

Wavefunction Separation

In quantum systems with central potentials, the full wavefunction can be factorized into a radial part and an angular part. This separation simplifies the analysis by allowing each part to be treated independently, with the angular part typically expressed in terms of spherical harmonics and the radial part satisfying its own differential equation.

Orthogonality of Eigenfunctions

Eigenfunctions of a Hamiltonian in quantum mechanics, which is a Hermitian operator, are orthogonal with respect to an appropriate inner product. For the radial solutions, orthogonality is expressed by integrals over the radial variable with a weighting factor associated with the measure (such as r^2), ensuring that different eigenfunctions do not overlap.

Sturm-Liouville Theory

Many radial differential equations in quantum mechanics are of Sturm-Liouville type. This theory guarantees that the eigenfunctions corresponding to different eigenvalues are orthogonal with respect to a specified weight function. The weighting function plays a crucial role in inner product definitions, leading to relations like ? dr r^2 u_n^l(r) u_{n'}^l(r) = ?_{nn'} and similar expressions for other quantum numbers.

Normalization Conditions

Normalization of the quantum state is essential for probabilistic interpretation. The integrals of the square of the wavefunction (or its components, when separated) over the appropriate domain must equal one or the delta function when comparing different states. These normalization conditions ensure that the set of eigenfunctions forms a complete and orthonormal basis.

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