Show that the wave function for a hydrogen atom in the 2 s...

Show that the wave function for a hydrogen atom in the $2 s$ state $$\psi_{2 s}(r)=\frac{1}{4 \sqrt{2 \pi}}\left(\frac{1}{a_{0}}\right)^{3 / 2}\left(2-\frac{r}{a_{0}}\right) e^{-r / 2 a_{0}}$$ satisfies the spherically symmetric Schrödinger equation given in Problem 16.

Key Concepts

Hydrogen Atom Model

The model of the hydrogen atom in quantum mechanics involves a single electron moving under the influence of a Coulomb potential created by the proton. The hydrogen atom serves as a prototypical system for understanding quantum behavior. Its exact solutions, which are characterized by quantum numbers and specific radial and angular functions, exemplify the interplay of kinetic and potential energies in a bound system, and demonstrate how quantization emerges naturally from the Schrödinger equation.

Radial Wave Function

In the presence of spherical symmetry, the overall wave function can be expressed as a product of a radial part and an angular part. The radial wave function satisfies a specific form of the Schrödinger equation where terms arise due to the radial dependence of the potential energy as well as angular momentum contributions. Analysis of the radial equation provides insight into the spatial distribution of electrons in atoms, and checking that a given function satisfies this equation confirms its validity as a solution.

Time-independent Schrödinger Equation

The time-independent Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes with respect to spatial coordinates. It involves the kinetic and potential energy operators and allows for the determination of energy eigenvalues and corresponding eigenfunctions by solving a differential equation. This equation is central to analyzing atomic systems such as the hydrogen atom.

Spherical Symmetry

Spherical symmetry means that the potential energy in the system depends only on the radial distance from a central point, and not on the angular coordinates. This symmetry simplifies the Schrödinger equation by allowing it to be separated into radial and angular parts. Recognizing spherical symmetry is crucial because it guides the method of separation of variables, leading to distinct equations for the radial and angular components of the wave function.

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