The quantized value of energy is given by

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Examoogle · Accepted Answer

$E = \frac{h^2}{8mL^2}(n_x^2 + n_y^2 + n_z^2)$

Explanation: The quantized energy of a particle of mass 'm' trapped in a three-dimensional infinite potential well (box) of side length 'L' is given by the expression derived from solving the Schrödinger equation. This model illustrates the confinement of particles, where energy levels are restricted to discrete values based on quantum numbers $n_x, n_y,$ and $n_z$.

Working Principle: For a 3D box, the wave function must vanish at the boundaries ($x,y,z = 0, L$). This boundary condition leads to quantized wave vectors $k_x = \frac{n_x \pi}{L}$, $k_y = \frac{n_y \pi}{L}$, and $k_z = \frac{n_z \pi}{L}$. Substituting these into the energy equation $E = \frac{\hbar^2 k^2}{2m}$, where $k^2 = k_x^2 + k_y^2 + k_z^2$ and $\hbar = \frac{h}{2\pi}$, we arrive at the energy expression $E = \frac{h^2}{8mL^2}(n_x^2 + n_y^2 + n_z^2)$.

Key Formulas: $E = \frac{h^2}{8mL^2}(n_x^2 + n_y^2 + n_z^2)$ — Quantized energy for a 3D potential well; $\Psi(x,y,z) = \sqrt{\frac{8}{V}} \sin(\frac{n_x \pi x}{L}) \sin(\frac{n_y \pi y}{L}) \sin(\frac{n_z \pi z}{L})$ — Normalized wave function

Key Points: The energy levels depend on three independent quantum numbers.; The ground state energy occurs when $n_x=n_y=n_z=1$.; Energy increases as the volume $V = L^3$ of the box decreases.; This model serves as the foundational concept for the electron gas theory in metals.

Advantages: Predicts discrete energy spectra for confined electrons.; Explains phenomena like quantum dots and size-dependent bandgaps.

Disadvantages: Assumes an infinite potential barrier, which is an idealization.; Ignores electron-electron interactions.

Applications: Quantum Dots and Nanoparticles; Semiconductor physics; Statistical mechanics of ideal gases

Additional Info: For a 1D box, the expression simplifies to $E = \frac{n^2 h^2}{8mL^2}$.. Option B is incorrect because it relates only to indices, not energy.. Option C represents the volume calculation in k-space used to derive the density of states.

Answer: A is correct — The formula represents the discrete energy eigenvalues of a particle confined in a three-dimensional cubic potential well of length L.

Exam Tip: Always remember that in quantum confinement, as the physical dimensions (L) shrink, the energy level spacing increases, which is why nanoparticles show different colors compared to bulk materials.

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