Schrodinger Time independent equation

Definition:

The time-independent Schrödinger equation is a fundamental equation in quantum mechanics that describes the stationary states of a system whose energy does not change with time.

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Equation:

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

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Explanation of Terms:

• Ĥ (Hamiltonian Operator):
  Represents total energy of the system (kinetic + potential energy)
• ψ (Wave Function):
  Describes the probability amplitude of finding a particle in space
• E (Energy Eigenvalue):
  Represents allowed quantized energy levels

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Interactive Derivation of Time-Independent Schrödinger Equation

Click each step to reveal the derivation gradually 👇

Step 1: Start from Time-Dependent Schrödinger Equation

$$ i\hbar \frac{\partial \Psi(x,t)}{\partial t} = \hat{H}\Psi(x,t) $$

This is the general equation describing quantum systems.

Step 2: Assume separable solution

$$ \Psi(x,t) = \psi(x)T(t) $$

We assume space and time parts can be separated.

Step 3: Substitute into equation

$$ i\hbar \psi(x)\frac{dT}{dt} = T(t)\hat{H}\psi(x) $$

Step 4: Separate variables

$$ i\hbar \frac{1}{T}\frac{dT}{dt} = \frac{1}{\psi}\hat{H}\psi $$

Left side depends on time, right side on space → both = constant E

Step 5: Time equation

$$ i\hbar \frac{dT}{dt} = ET $$

$$ T(t) = e^{-iEt/\hbar} $$

Step 6: Final Time-Independent Schrödinger Equation

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

This describes stationary energy states of a quantum system.

 Physical Meaning:

When the Hamiltonian operator acts on a wave function, it produces the same wave function multiplied by a constant energy value. This means only specific wave functions and energies are allowed.

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Important Features:

• Describes stationary states (time-independent systems)
• Energy levels are quantized
• Wave function gives probability distribution
• Only specific solutions (eigenfunctions) are allowed

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Analogy:

Like a guitar string 🎸, only certain vibrations are allowed. Each vibration corresponds to a specific energy level.

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Applications:

• Atomic structure (Hydrogen atom)
• Quantum chemistry
• Semiconductor physics
• Molecular energy levels

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Conclusion:

The time-independent Schrödinger equation is a key equation in quantum mechanics that determines allowed energy levels and wave functions of microscopic systems.

🧠 Schrödinger Equation Interactive Quiz (15 Questions)

1. Schrödinger equation belongs to which branch?

Classical mechanics Quantum mechanics Thermodynamics Optics

2. Time-independent Schrödinger equation is:

F = ma Ĥψ = Eψ E = mc² V = IR

3. ψ represents:

Wave function Energy Force Mass

4. Hamiltonian operator represents:

Total energy Force Momentum only Temperature

5. Energy in quantum mechanics is:

Quantized Continuous always Random Zero

6. Schrödinger equation describes:

Quantum systems Classical motion Heat transfer Light reflection

7. Time-independent equation is used for:

Stationary states Accelerated motion Heat flow Sound waves only

8. ψ² represents:

Probability density Energy Velocity Force

9. Schrödinger equation is similar to:

Ohm’s law Wave equation Newton’s law Boyle’s law

10. Allowed solutions are called:

Eigenfunctions Random functions Noise functions Chaos functions

11. Energy values are called:

Eigenvalues Constants only Variables Forces

12. Electron behaves like:

Wave and particle Only particle Only wave Solid object

13. Schrödinger equation is used in:

Quantum mechanics Classical physics Thermodynamics Fluid mechanics

14. Stationary state means:

Energy is constant Motion stops Particle disappears Energy changes

15. Main result of Schrödinger equation is:

Energy levels Temperature Force Pressure

⚛️ Schrödinger Equation – Drag & Drop Game

Drag the concepts into correct categories 🧠

ψ → Wave Function

Ĥ → Hamiltonian

E → Energy Eigenvalue

Ĥψ = Eψ

🌊 Wave Function

⚡ Hamiltonian

🔋 Energy

📌 Equation

🎯 Result will appear here