Bohr model

eku_girl83

Here's my problem:
Show that the speed of an electron in the nth Bohr orbit of hydrogen is (alpha*c)/n, where alpha is the fine structure constant. What would be the speed in a hydrogenlike atom with a nuclear charge of Ze?

We didn't talk about the fine structure constant in class, so could someone explain to me what it is? Hints on how to show that speed = alpha c/n would also be appreciated.

Thanks,
eku_girl83

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Kane O'Donnell

Hi,

In the Bohr model we assume that angular momentum is quantised:

$$L = mvr = n\hbar$$

From this you can find the expression for the tangential velocity of the electron. You then need to find the expression for the Bohr radius for a particular value of n, which turns out to be (for Z = 1, for Hydrogen-like atoms just replace e^2 with Z(e^2)):

$$r_{n} = \frac{4\pi\epsilon_{0}\hbar^{2}n^2}{me^2}$$

When you sub in for r you get:

$$v_{n} = \frac{e^2}{4\pi\epsilon_{0}\hbar}$$

From this you should be able to work out what the fine structure constant is - just compare the equation you were given to the one above. In undergrad physics courses the name "fine structure constant" is often applied to a few dimensionless constants that all look similar. It's just a number that happens to arise in a lot of Quantum Mechanical situations. You'll see it a fair bit :)

Cheerio!

Kane

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