Quantum Mechanical Model of Atom including Schrödinger Equation

The quantum mechanical model of the atom is a fundamental concept in modern chemistry and physics. It describes the behavior of electrons within an atom using the principles of quantum mechanics, as opposed to the earlier Bohr model which treated electrons as particles orbiting the nucleus in fixed paths. The quantum mechanical model is based on the wave-like properties of particles and the uncertainty principle.

Schrödinger Equation

At the heart of the quantum mechanical model is the Schrödinger equation, formulated by Erwin Schrödinger in 1926. This equation is a mathematical description of how the quantum state of a quantum system changes with time. For a non-relativistic, single particle, the time-independent Schrödinger equation is given by:

[ \hat{H}\psi(\mathbf{r}) = E\psi(\mathbf{r}) ]

where:

• $\hat{H}$ is the Hamiltonian operator, representing the total energy of the system (kinetic + potential energy).
• $\psi(\mathbf{r})$ is the wave function of the system, which contains all the information about the system's state.
• $E$ is the energy eigenvalue associated with the state described by $\psi$.
• $\mathbf{r}$ represents the position vector.

The Hamiltonian operator in the context of an electron in an atom is given by:

[ \hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) ]

where:

• $\hbar$ is the reduced Planck's constant ($h/2\pi$).
• $m$ is the mass of the electron.
• $\nabla^2$ is the Laplacian operator, which corresponds to the kinetic energy of the electron.
• $V(\mathbf{r})$ is the potential energy function, often the Coulomb potential due to the nucleus.

Quantum Numbers and Atomic Orbitals

The solutions to the Schrödinger equation for the hydrogen atom result in a set of quantum numbers and corresponding atomic orbitals. These quantum numbers are:

1. Principal quantum number ($n$): Determines the energy level and size of the orbital.
2. Angular momentum quantum number ($l$): Determines the shape of the orbital.
3. Magnetic quantum number ($m_l$): Determines the orientation of the orbital in space.
4. Spin quantum number ($m_s$): Describes the intrinsic spin of the electron.

Each orbital is represented by a wave function, and the square of the wave function's magnitude, $|\psi(\mathbf{r})|^2$, gives the probability density of finding an electron at a point in space.

Differences Between Bohr Model and Quantum Mechanical Model

Feature	Bohr Model	Quantum Mechanical Model
Electron Behavior	Electrons orbit the nucleus in fixed paths.	Electrons have wave-like properties and exist in orbitals.
Energy Levels	Quantized, with fixed energy values.	Quantized, with energy values determined by quantum numbers.
Position of Electrons	Well-defined orbits with certain radii.	Probabilistic distribution within orbitals.
Mathematical Basis	Semi-classical physics.	Wave mechanics and the Schrödinger equation.
Limitations	Only accurate for hydrogen-like atoms.	Applicable to all atoms and molecules.

Example: Hydrogen Atom

For the hydrogen atom, the potential energy $V(\mathbf{r})$ is the Coulomb potential due to the proton in the nucleus:

[ V(\mathbf{r}) = -\frac{e^2}{4\pi\epsilon_0 r} ]

where:

• $e$ is the charge of the electron.
• $\epsilon_0$ is the vacuum permittivity.
• $r$ is the distance from the nucleus.

Solving the Schrödinger equation with this potential leads to the well-known hydrogen-like wave functions and energy levels:

[ E_n = -\frac{me^4}{8\epsilon_0^2h^2}\frac{1}{n^2} ]

where $n$ is the principal quantum number.

Conclusion

The quantum mechanical model of the atom, as described by the Schrödinger equation, provides a comprehensive framework for understanding the behavior of electrons in atoms. It explains the quantization of energy levels, the shapes and orientations of orbitals, and the probabilistic nature of electron positions. This model is essential for the study of atomic structure, chemical bonding, and the electronic properties of materials.