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ani4physics
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1. Consider Bohr Hydrogen atom with counter-clockwise electron orbit in the xy plane with intial position r(0)=-a0y. 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 a0 , w, t, x, and y.
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
= -a0 y + [w * squareroot(x^2+y^2)] * t
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
= -a0 y + [w * squareroot(x^2+y^2)] * t