I am trying to analyze a particle in a box with a rather strange potential inside the box: $V(x) \propto x^{3/2}$ I've tried using the WKB approximation, but I get some strange results and I don't know how physical they are. Basically, I get results of the expected order of magnitude for a "very big" size of the box - the regime in which I'm actually interested in. (actually I get 2 roots for the energy but they are identical to 4-5 decimal places) However, if I squeeze down the box, the 2 roots remain of the same order of magnitude, but the difference between them increases as I squeeze the box. Furthermore, squeezing down the box reduces the value of the energy, which is not what I would expect (but for a very small size of the box the energy blows up as expected). The other thing is that the values of the energy go down with $n$, I would have expected them to go up. However in the case of very large $n$ energy tends to the classical expected value which I believe it's okay from the point of view of the WKB approximation.

Do these results make sense? And what could be the cause of this strange behaviour?

Edit: the infinite walls are at $x=0$ and $x=b$, where $b$ is some positive value. I assume (and it seems that it actually checks out) that $E>V(x)$ everywhere inside the box.

Edit 2: after doing some more calculations I found out that in the regime where the size of the box is small the energy values actually go up with $n$, as expected. However in the regime of a very big size of the box the energy goes down with $n$ towards the classical value for large values of $n$.


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