Prove that the normalization constant of the 1 s radial wave...

Prove that the normalization constant of the 1 s radial wave function of the hydrogen atom is $\left(\pi a_{\mathrm{B}}^{3}\right)^{-1 / 2},$ as given in Equation 42.7 Hint: A useful definite integral is $$\int_{0}^{\infty} x^{n} e^{-a x} d x=\frac{n !}{\alpha^{n+1}}$$

Key Concepts

Quantum Mechanical Wave Function Normalization

In quantum mechanics, the normalization of a wave function ensures that the total probability of finding a particle in all space is equal to one. This process involves integrating the absolute square of the wave function over the entire coordinate space and setting the result equal to unity. It is a fundamental requirement connected to the probabilistic interpretation of quantum states.

Hydrogen Atom Radial Wave Function

The hydrogen atom, being the simplest atomic system, has analytically solvable wave functions. The radial part of the wave function, especially for the 1s state, is derived by solving the Schrödinger equation in a spherically symmetric potential. The proper normalization of this radial component is essential for accurate predictions of atomic properties and electron distributions.

Integration in Spherical Coordinates

When dealing with systems exhibiting spherical symmetry, the integration for normalization is performed in spherical coordinates. This requires incorporating the Jacobian of the transformation, which includes a factor of r^2 for the radial part. The volume element in spherical coordinates is crucial when integrating the square of the wave function over the entire space.

Use of Exponential Integrals

Exponential integrals play a key role in normalizing functions that involve an exponential decay, such as those found in atomic wave functions. A standard result, ??? x? e^(??x) dx = n!/?^(n+1), simplifies the computation of integrals encountered during normalization. This result is particularly useful in evaluating the radial integrals that emerge from the spherically symmetric solutions of the Schrödinger equation.

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