Schrödinger Wave Equation | Structure of Atom Class 11 Notes

Schrödinger Wave Equation

Quantum Mechanical Model of Atom | Structure of Atom Class 11

1. Introduction to Quantum Mechanics

Erwin Schrödinger developed a model for the atom in 1926 based on the wave nature of electrons. Unlike the Bohr model, which treated electrons as particles in fixed paths, this model describes electrons as three-dimensional waves in the electric field of the nucleus.

2. The Schrödinger Equation

For a system (such as an atom), the Schrödinger equation is given by:

$$ \hat{H} \Psi = E \Psi $$

Where:

$\hat{H}$ = Hamiltonian Operator (Total Energy Operator).
$\Psi$ (Psi) = Wave Function (Amplitude of the electron wave).
$E$ = Total Energy of the system (Eigenvalue).

Differential Form (Time-Independent):

$$ \frac{\partial^2\Psi}{\partial x^2} + \frac{\partial^2\Psi}{\partial y^2} + \frac{\partial^2\Psi}{\partial z^2} + \frac{8\pi^2m}{h^2}(E-V)\Psi = 0 $$

Where $x, y, z$ are coordinates, $m$ is mass of electron, and $V$ is potential energy.

3. Significance of $\Psi$ and $\Psi^2$

Wave Function ($\Psi$)

It represents the amplitude of the electron wave. By itself, $\Psi$ has no physical significance.

Probability Density ($\Psi^2$)

                    $\Psi^2$ (or $|\Psi|^2$) represents the probability of finding the electron at a specific point in space around the nucleus.
                    If $\Psi^2$ is large, probability of finding the electron is high.
If $\Psi^2$ is zero, probability is zero (Node).

                

4. Concept of Atomic Orbitals

The solution to the Schrödinger equation gives us Orbitals. An orbital is a three-dimensional region in space around the nucleus where the probability of finding an electron is maximum ($>90\%$).

Difference: Orbit vs Orbital

Orbit (Bohr Model)	Orbital (Quantum Model)
Well-defined circular path.	3D region of high probability.
Planar motion.	3D motion.
Violates Uncertainty Principle.	Agrees with Uncertainty Principle.

5. Nodes and Nodal Planes

Regions where the probability of finding an electron is zero ($\Psi^2 = 0$).

Radial Nodes (Spherical): Number = $n - l - 1$
Angular Nodes (Planar): Number = $l$
Total Nodes: Number = $n - 1$

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