Hydrogen Atom and the Schrödinger Equation

2.6a Hydrogen Atom and the Schrödinger Equation

When the Schrödinger equation is solved for the hydrogen atom, the solution yields the possible energy levels the electron can occupy and the corresponding wave function $\psi$ for each energy level. These quantized energy states and their wave functions are characterized by a set of three quantum numbers:

• Principal quantum number $n$
• Azimuthal quantum number $l$
• Magnetic quantum number $m_l$

These quantum numbers arise naturally from the mathematics of the solution — they are not imposed from outside. They are a direct consequence of the wave nature of the electron.

When an electron is in any particular energy state, the wave function $\psi$ corresponding to that state contains all the information about the electron. The wave function itself is a mathematical function whose value depends on the coordinates of the electron in the atom. It carries no direct physical meaning on its own.

Such wave functions for hydrogen or hydrogen-like species (any one-electron system, like $\text{He}^+$, $\text{Li}^{2+}$) are called atomic orbitals. These are one-electron systems.

The probability of finding an electron at a particular point within an atom is proportional to $|\psi|^2$ at that point. This quantity $|\psi|^2$ is called the probability density and is always positive.

The quantum mechanical results for the hydrogen atom successfully predict all aspects of the hydrogen atom spectrum — including phenomena that the Bohr model could not explain.

Multi-Electron Atoms

Applying the Schrödinger equation to multi-electron atoms presents a difficulty: the equation cannot be solved exactly. This problem is overcome using approximate methods. Modern computer calculations show that orbitals in atoms other than hydrogen do not differ radically from hydrogen orbitals. The principal difference is the increased nuclear charge, which causes all orbitals to be somewhat contracted.

Important Features of the Quantum Mechanical Model of Atom

The quantum mechanical model of the atom is the picture that emerges from applying the Schrödinger equation to atoms. Here are its essential features, each explained fully.

1. Quantized Energy

The energy of electrons in atoms is quantized — it can only have certain specific values. This happens when electrons are bound to the nucleus. The electron cannot have any arbitrary energy; it must occupy one of the allowed energy levels.

2. Wave Nature as the Source of Quantization

The existence of quantized electronic energy levels is a direct result of the wave-like properties of electrons. These quantized levels are the allowed solutions of the Schrödinger wave equation. Just as a guitar string can only vibrate at certain frequencies (its harmonics), an electron bound to a nucleus can only exist in certain energy states.

3. Heisenberg Uncertainty Principle and the Abandonment of "Paths"

Both the exact position and exact velocity of an electron in an atom cannot be determined simultaneously. This is the Heisenberg uncertainty principle. Consequently, the path (or trajectory) of an electron in an atom can never be determined or known accurately. This is why we speak only of the probability of finding the electron at different points in an atom — not of a definite orbit.

4. Atomic Orbitals as Wave Functions

An atomic orbital is the wave function $\psi$ for an electron in an atom. Whenever an electron is described by a wave function, we say that the electron occupies that orbital. Since many such wave functions are possible for an electron, there are many atomic orbitals in an atom.

These "one-electron orbital wave functions" (or simply orbitals) form the basis of the electronic structure of atoms. In each orbital, the electron has a definite energy. An orbital cannot contain more than two electrons.

In a multi-electron atom, electrons are filled in various orbitals in order of increasing energy. For each electron in a multi-electron atom, there is an orbital wave function characteristic of the orbital it occupies. All information about the electron in an atom is stored in its orbital wave function $\psi$, and quantum mechanics makes it possible to extract this information from $\psi$. …