Dirac particle in a spherical potential box

[samuelsanches]

Hello, I'm studyng relativistic quantum mechanics by the book Relativistic quantum mechanics. Wave equations - Greiner, W. and I'm trying to derive the energy eingenvalues for s1/2 states, so I have the equation that I uploaded with the name eq1.jpg. In the text the author says, "If we assume R0 to be very small mcR/h<<1, we may solve this equation approximately by expanding in terms of mcR/h" (the h is h bar), leading to the table attach with the name table1.jpg, but I tried expanding the tan, and I can't solve that.

Could anyone help me?

A lot of thanks

 

[eljose79]

to anyone who can help with this!

Hello,

Thank you for your post. It looks like you are working on deriving the energy eigenvalues for s1/2 states in relativistic quantum mechanics. The equation you have uploaded (eq1.jpg) is a bit difficult to read, but I assume it is the Dirac equation for a free particle.

In order to solve this equation for the energy eigenvalues, the author suggests using an approximation when R0 is very small (mcR/h<<1). This is a common technique in physics, known as a perturbation expansion, where we start with a simple solution and then add small corrections to it.

In this case, the author suggests expanding the equation in terms of mcR/h. This means that we can write the solution as a series, where each term is multiplied by increasing powers of mcR/h. The table you have attached (table1.jpg) shows the first few terms in this expansion. As you can see, the first term is just a constant, the second term involves a square root, and the third term involves a tangent function.

Now, you mentioned that you have tried expanding the tangent function and have encountered difficulties. This is because the tangent function is not a simple polynomial, so we cannot simply multiply it by increasing powers of mcR/h. Instead, we have to use a different technique called series reversion, which allows us to expand a function in terms of its inverse function.

I would recommend looking into series reversion techniques, or consulting with your professor or a colleague who may be familiar with this method. It can be a bit tricky to grasp at first, but with practice, you should be able to expand the tangent function and solve for the energy eigenvalues.

I hope this helps. Best of luck with your studies!