An electron is trapped in a one-dimensional infinite...

An electron is trapped in a one-dimensional infinite potential well. Show that the energy difference $\Delta E$ between its quantum levels $n$ and $n+2$ is $\left(h^{2} / 2 m L^{2}\right)(n+1)$ .

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Key Concepts

Quantum Numbers

Quantum numbers are integers that index the discrete energy levels available to a particle in a confined system like the infinite potential well. They determine the allowed wavelengths and energies of the particle’s wavefunction. The difference in energy between levels, which is calculated using these quantum numbers, provides insight into the transition dynamics and spectral properties of quantum systems.

Infinite Potential Well

The infinite potential well is a fundamental model in quantum mechanics where a particle is confined within perfectly rigid boundaries, meaning the potential energy is zero inside the well and infinite outside. This model provides a clear example of how confinement leads to the quantization of energy levels because the boundary conditions force the particle’s wavefunction to vanish at the walls.

Energy Quantization

Energy quantization in confined systems arises because only certain wavefunctions, which satisfy the boundary conditions, are allowed. For a particle in an infinite potential well, the energy levels are proportional to the square of an integer quantum number. This quantization is a hallmark of quantum mechanics and contrasts sharply with the continuous energy values permitted in classical physics.

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