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**1. Consider Bohr Hydrogen atom with counter-clockwise electron orbit in the xy plane with intial position**

**r(0)**=-a_{0}**y**. The angular frequency of the orbit is w. Derive an expression for the position of electron at a later time t,**r(t)**in terms of a_{0}, w, t, x, and y.## Homework Equations

## The Attempt at a Solution

We know, angular frequency, w = 2*pi/T, where T is the time period of completing the orbit. Now, if the velocity of the electron is v, then T = 2*pi*R/v, where R is the radius of the Bohr orbit. So, w = v/R = v/squareroot(x^2 + y^2).

This gives, v = w * squareroot(x^2+y^2).

So, dr(t)/dt = w * squareroot(x^2+y^2).

dr(t) = [w * squareroot(x^2+y^2)] dt

Integrate both sides:

r(t) - r(0) = [w * squareroot(x^2+y^2)] * t

So, r(t) = r(0) + [w * squareroot(x^2+y^2)] * t

= -a

_{0}

**y**+ [w * squareroot(x^2+y^2)] * t