I'm currently taking an introductory quantum mechanics course and we just finished learning about the infinite square square well scenario. I understand all the maths used for calculating the eigenvalue energy levels. I don't understand what these energies actually mean or what they physically represent? Is there an intuitive way to think about these?
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Just as the energy levels in a hydrogen atom are associated with the hydrogen absorption or emission spectrum, you can think of the energy levels in an infinite well as associated with the absorption or emission spectrum of a quantum dot (quantization in 3-D), quantum wire (quantization in 2-D), or quantum well (quantization in 1-D):
The difference between two energy levels $E_2 > E_1$ gives the energy $\hbar\nu = E_2 - E_1$ of the photon that can excite an electron from a state of energy $E_1$ to a state of energy $E_2$ or of the photon emitted when the electron relaxes from energy level $E_2$ to energy level $E_1$.
If you'd like more, Googling "quantum dot potential well" brings up this lecture that gives a really nice and accessible overview of real-world applications of the infinite potential well in electronic devices:
Plenty other materials available too.
udrv
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