Total Energy Of Particle In 3D Box Calculator

Total Energy of Particle in 3D Box Formula:

\[ E = \frac{(n_x)^2 \cdot (h)^2}{8 \cdot m \cdot (l_x)^2} + \frac{(n_y)^2 \cdot (h)^2}{8 \cdot m \cdot (l_y)^2} + \frac{(n_z)^2 \cdot (h)^2}{8 \cdot m \cdot (l_z)^2} \]

Energy Levels along X axis (nₓ):

Energy Levels along Y axis (nᵧ):

Energy Levels along Z axis (n_z):

Mass of Particle:

Length of Box along X axis:

Length of Box along Y axis:

Length of Box along Z axis:

Total Energy of Particle in 3D Box:

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1. What is the Total Energy of Particle in 3D Box?

The Total Energy of Particle in 3D Box is a quantum mechanical concept that describes the quantized energy levels of a particle confined within a three-dimensional box. This model is fundamental in quantum mechanics and helps understand the behavior of particles at atomic and subatomic levels.

2. How Does the Calculator Work?

The calculator uses the quantum mechanical formula for a particle in a 3D box:

\[ E = \frac{(n_x)^2 \cdot h^2}{8 \cdot m \cdot (l_x)^2} + \frac{(n_y)^2 \cdot h^2}{8 \cdot m \cdot (l_y)^2} + \frac{(n_z)^2 \cdot h^2}{8 \cdot m \cdot (l_z)^2} \]

Where:

\( E \) — Total energy of the particle (Joules)
\( n_x, n_y, n_z \) — Quantum numbers along x, y, z axes (positive integers)
\( h \) — Planck's constant (6.626070040 × 10⁻³⁴ J·s)
\( m \) — Mass of the particle (kilograms)
\( l_x, l_y, l_z \) — Lengths of the box along x, y, z axes (meters)

Explanation: The equation shows that energy is quantized and depends on the quantum numbers and dimensions of the confining box.

3. Importance of Quantum Energy Calculation

Details: This calculation is crucial for understanding quantum confinement effects, semiconductor physics, nanomaterial properties, and fundamental quantum mechanical principles.

4. Using the Calculator

Tips: Enter positive integer values for quantum numbers (nₓ, nᵧ, n_z), positive values for mass and box dimensions. All values must be greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: Why are energy levels quantized in a box?
A: Due to the wave nature of particles and boundary conditions, only specific discrete energy levels are allowed.

Q2: What happens when all quantum numbers equal 1?
A: This represents the ground state - the lowest possible energy level for the particle.

Q3: How does box size affect energy levels?
A: Smaller box dimensions result in larger energy spacing between levels (quantum confinement effect).

Q4: What types of particles does this model apply to?
A: This model applies to any quantum particle with wave-like properties, including electrons, protons, and other fundamental particles.

Q5: What are the limitations of this model?
A: The model assumes infinite potential walls and doesn't account for particle interactions or relativistic effects.