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

An electron is trapped in a one-dimensional infinite potential well. For what (a) higher quantum number and (b) lower quantum number is the corresponding energy difference equal to the energy of the $n=5$ level? (c) Show that no pair of adjacent levels has an energy difference equal to the energy of the $n=6$ level.

Key Concepts

Infinite Potential Well

The infinite potential well is a fundamental quantum mechanical model where a particle is confined in a region with infinitely high potential barriers at the boundaries. This confinement leads to discrete (quantized) energy levels since the wavefunction must vanish at the boundaries, and the allowed wavelengths and energies are determined by integer multiples of a fundamental mode.

Quantized Energy Levels and Quantum Numbers

In an infinite potential well, the allowed energy levels are quantized and depend on the quantum number n. The energy of each level typically follows a relation proportional to n². This means that as the quantum number increases, the energy grows as the square of the integer, which is central to understanding differences between energy levels and how they scale.

Energy Level Differences

The energy difference between two states in a quantized system is determined by the difference in their respective quantum numbers. Using the relation where energy is proportional to n², the difference between two energy levels can be expressed as a difference of squares. Analyzing these differences allows one to find pairs of quantum numbers for which this difference matches the energy of a given level, while also demonstrating when it is impossible for such a match to occur (for example, between adjacent levels).

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