Total Energy of Particle in 2D Square Box Calculator

Formula Used:

\[ E = \frac{[hP]^2 \times (n_x^2 + n_y^2)}{8 \times m \times l^2} \]

Energy Levels along X axis (n_x):

Energy Levels along Y axis (n_y):

Mass of Particle:

Length of 2D Square Box:

Total Energy of Particle in 2D Square Box:

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1. What is the Total Energy of Particle in 2D Square Box?

The Total Energy of Particle in 2D Square Box represents the quantized energy levels available to a particle confined within a two-dimensional square potential well. This is a fundamental concept in quantum mechanics that demonstrates the quantization of energy in confined systems.

2. How Does the Calculator Work?

The calculator uses the quantum mechanical formula:

\[ E = \frac{[hP]^2 \times (n_x^2 + n_y^2)}{8 \times m \times l^2} \]

Where:

\( [hP] \) — Planck's constant (6.626070040 × 10^-34 J·s)
\( n_x \) — Quantum number along X axis
\( n_y \) — Quantum number along Y axis
\( m \) — Mass of the particle (kg)
\( l \) — Length of the 2D square box (m)

Explanation: The formula calculates the discrete energy levels available to a quantum particle confined in a two-dimensional infinite potential well.

3. Importance of Quantum Energy Calculation

Details: Understanding quantized energy levels is crucial for studying quantum systems, semiconductor physics, nanoscale devices, and fundamental quantum mechanical principles.

4. Using the Calculator

Tips: Enter quantum numbers (positive integers), particle mass in kilograms, and box length in meters. All values must be positive and non-zero.

5. Frequently Asked Questions (FAQ)

Q1: Why are energy levels quantized in a 2D box?
A: Due to the boundary conditions imposed by the infinite potential walls, the particle can only exist in specific standing wave patterns with discrete energy values.

Q2: What are typical values for particle mass and box length?
A: For electrons, mass is approximately 9.11 × 10^-31 kg. Box lengths are typically nanoscale (10^-9 to 10^-10 m) for quantum effects to be significant.

Q3: Can the quantum numbers be zero?
A: No, quantum numbers must be positive integers (n = 1, 2, 3...) as they represent the number of half-wavelengths that fit in the box.

Q4: What happens when n_x = n_y?
A: When quantum numbers are equal, the energy states are degenerate, meaning different states have the same energy value.

Q5: How does this relate to real-world applications?
A: This model is fundamental for understanding quantum dots, quantum wells in semiconductors, and confined electron systems in nanotechnology.