Qntm p

 The Schrödinger equation is the fundamental equation of quantum mechanics, describing how the quantum state of a physical system evolves over time.

It was formulated in 1925 by Erwin Schrödinger.

1️⃣ Time-Dependent Schrödinger Equation (TDSE)

This version describes how a quantum system changes with time:

i\hbar \frac{\partial \Psi(\mathbf{r},t)}{\partial t} = \hat{H}\Psi(\mathbf{r},t)

Where:

• i = imaginary unit
• \hbar = reduced Planck’s constant
• \Psi(\mathbf{r},t) = wavefunction
• \hat{H} = Hamiltonian operator (total energy operator)

For a single particle in a potential V(\mathbf{r},t):

i\hbar \frac{\partial \Psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t) \right]\Psi

2️⃣ Time-Independent Schrödinger Equation (TISE)

Used when the potential does not depend on time.

\hat{H}\psi = E\psi

or

-\frac{\hbar^2}{2m}\nabla^2 \psi + V(\mathbf{r})\psi = E\psi

This is an eigenvalue equation:

• \psi = stationary state
• E = allowed energy level

3️⃣ Physical Meaning

• The wavefunction \Psi contains all measurable information about a system.
• The probability density of finding a particle at position \mathbf{r} is:

|\Psi(\mathbf{r},t)|^2

• The equation replaces Newton’s laws at microscopic scales.

4️⃣ Why It’s Important

The Schrödinger equation explains:

• Atomic structure
• Chemical bonding
• Electron behavior in atoms
• Semiconductors
• Quantum tunneling

For example, solving it for hydrogen correctly predicts its energy levels — a major success of early quantum theory.

If you’d like, I can:

• Derive it step-by-step
• Solve it for a particle in a box
• Solve it for hydrogen
• Explain it conceptually without math
• Show how it connects to classical mechanics

Just tell me your level (high school / undergrad / advanced).