Question

Show that the speed of an electron in the nth bohr orbit of hydrogen is αc/n, where α is the fine structure constant. what would be the speed in a hydrogen-like atom with a nuclear charge of ze?

Answer 1

In the Bohr model we assume that angular momentum is quantised: L = mvr = nℏ 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)): rn=4πϵ0ℏ2n2/me2 When you sub in for r you get: vn=e2/4πϵ0ℏ 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.