Total Energy Of Particle In Cubic Box Calculator

Energy of Particle in 3D Square Box Formula:

\[ E = \frac{(h_P)^2 \cdot (n_x^2 + n_y^2 + n_z^2)}{8 \cdot m \cdot l^2} \]

Energy Levels along X axis (nₓ):

Energy Levels along Y axis (nᵧ):

Energy Levels along Z axis (n_z):

Mass of Particle:

Length of 3D Square Box:

Total Energy of Particle in Cubic Box:

Unit Converter ▲

Unit Converter ▼

From:	To:

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

The energy of a particle in a 3D square box is a fundamental concept in quantum mechanics that describes the quantized energy levels available to a particle confined within a three-dimensional potential well with infinite barriers.

2. How Does the Calculator Work?

The calculator uses the quantum mechanical formula:

\[ E = \frac{(h_P)^2 \cdot (n_x^2 + n_y^2 + n_z^2)}{8 \cdot m \cdot l^2} \]

Where:

\( h_P \) — Planck's constant (6.626070040 × 10⁻³⁴ J·s)
\( n_x \) — Quantum number along X axis
\( n_y \) — Quantum number along Y axis
\( n_z \) — Quantum number along Z axis
\( m \) — Mass of the particle (kg)
\( l \) — Length of the cubic box (m)

Explanation: The energy is quantized and depends on the quantum numbers in all three dimensions, the particle's mass, and the size of the confinement box.

3. Importance of Quantum Energy Calculation

Details: This calculation is crucial for understanding quantum confinement effects in nanomaterials, semiconductor physics, and the behavior of electrons in potential wells. It forms the basis for many quantum mechanical applications.

4. Using the Calculator

Tips: Enter positive integer values for quantum numbers (nₓ, nᵧ, n_z ≥ 1), positive values for mass (kg) and box length (m). All values must be greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: Why are the energy levels quantized?
A: Due to the wave nature of particles and boundary conditions in the infinite potential well, 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 state of the particle in the box.

Q3: Can the quantum numbers be zero?
A: No, quantum numbers must be positive integers starting from 1. A zero quantum number would imply zero energy, which is not physically possible.

Q4: How does box size affect the energy?
A: Energy decreases as the square of the box length increases. Larger boxes have more closely spaced energy levels.

Q5: What real-world systems does this model represent?
A: This models electrons in quantum dots, particles in nanoscale containers, and provides insights into many quantum confinement phenomena.