For a particle in an infinite potential well the separation between energy state

Question

For a particle in an infinite potential well the separation between energy states increases as n increases. But doesn’t the correspondence principle require closer spacing between states as n increases so as to approach a classical (nonquantized) situation? Explain.
For a particle in an infinite potential well the separation between energy states increases as n increases. But doesn’t the correspondence principle require closer spacing between states as n increases so as to approach a classical (nonquantized) situation? Explain.
For a particle in an infinite potential well the separation between energy states increases as n increases. But doesn’t the correspondence principle require closer spacing between states as n increases so as to approach a classical (nonquantized) situation? Explain.

Explanation / Answer

As per theory, Quantum mechanics reduces to classical mechanics in the limit where the principal quantum number n approaches infinity. Let's take a ratio of change in energy to energy for a quantum state

delta E / E = En+1 - En / En where En = n2h2 / 8ma2 . So putting in the values, we get

delta E / E = 2n + 1 / n2

Now, when n ---> infinity , the ratio becomes zero. It shows that energy increases faster as n2 making the spectrum continuous at which point we use correspondece principle which also states that any model describing the behavior of quantum systems must yield the same results as classical physics in the macroscopic limit

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