Dynamics of electron in Bohr Hydrogen atom

[ani4physics]

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.

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

 

[sgd37]

seems like the classical picture

 

[ani4physics]

seems like the classical picture

Yeah it is. Could someone please tell me if I did it right. If yes, then I will post the next part of the problem. Thanks guys.

 

[Redbelly98]

EDIT: Looks like I misunderstood the problem statement, so ignore this post.

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.

In the term "-a₀y", isn't that a y-hat (unit vector in y-direction)?

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

You are treating r(t) like it is a scalar, but it is really a vector.

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

You're way off track. The Bohr model has the electron moving in a circle, at a constant angular speed ω. What would be an expression for the x-coordinate of the electron, x(t)? Hint: it involves a well-known function, and it is 0 at t=0.

 

[sgd37]

yeah but that requires quantum mechanics. Classically the thing is supposed to spin itself into the proton which that equation does. And anyway you can't really expect to derive the spherical harmonics from that setup

 

[Redbelly98]

It looks like I misunderstood the problem. If it's to calculate a classical trajectory for the electron, initially at a distance = Bohr radius from the nucleus, then what I said earlier was wrong. My apologies.

 

[ani4physics]

It looks like I misunderstood the problem. If it's to calculate a classical trajectory for the electron, initially at a distance = Bohr radius from the nucleus, then what I said earlier was wrong. My apologies.

so what I did is right?

 

[Redbelly98]

so what I did is right?

I still have to say no, despite my previous misunderstanding. Read on.

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.

What class is this for -- electrodynamics, quantum mechanics, classical mechanics, or something else?

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).

I think not. v is (mostly) in the tangential direction. So it does not equal the rate of change of the radius of the orbit.

Again, can you tell us what class this is for? Also, do you know anything about the power radiated by an accelerating charge? And -- does the problem ask you to find the vector, r(t), or simply the radius of the orbit vs. time?