17/07/2025
Schrödinger Equation
The Schrödinger equation is the fundamental equation in quantum mechanics, describing how the wave function of a quantum system evolves over time. It was developed by the Austrian physicist Erwin Schrödinger in 1925.
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💡 The Basic Differential Equation (Non-Relativistic Form)
📌 Time-Independent Schrödinger Equation:
\hat{H} \psi(x) = E \psi(x)
: The Hamiltonian operator, representing total energy (kinetic + potential).
: The wave function.
: The total energy of the system.
For a particle in one dimension:
- \frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x) \psi(x) = E \psi(x)
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📌 Time-Dependent Schrödinger Equation:
i \hbar \frac{\partial \psi(x,t)}{\partial t} = \hat{H} \psi(x,t)
: The reduced Planck’s constant.
: The imaginary unit.
: The time derivative of the wave function.
: The Hamiltonian operator.
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📘 What Does the Wave Function Represent?
: Represents the probability of finding the particle at position at time .
The equation doesn’t give the exact location, but rather the probability of the particle being there.
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🧠 Importance of Schrödinger’s Equation:
It is the foundation of quantum mechanics.
Used to understand atoms, electrons, molecules, and all quantum systems.
Explains phenomena that classical physics cannot, such as:
Discrete energy levels in atoms.
Quantum tunneling.
Quantum superpositions
