Dynamics of electron in Bohr Hydrogen atom

In summary: The Attempt at a SolutionWe 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^
  • #1
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
 
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  • #2
seems like the classical picture
 
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  • #3
sgd37 said:
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.
 
  • #4
EDIT: Looks like I misunderstood the problem statement, so ignore this post.

ani4physics said:
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.
 
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  • #5
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
 
  • #6
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.
 
  • #7
Redbelly98 said:
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?
 
  • #8
ani4physics said:
so what I did is right?
I still have to say no, despite my previous misunderstanding. Read on.
ani4physics said:
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?
 

1. What is the Bohr model of the Hydrogen atom?

The Bohr model of the Hydrogen atom is a simplified representation of the structure of an atom proposed by Danish physicist Niels Bohr in 1913. It states that the electrons orbit the nucleus in specific energy levels, and that the energy of the electrons is quantized.

2. How does the Bohr model explain the dynamics of electrons in the Hydrogen atom?

The Bohr model explains the dynamics of electrons in the Hydrogen atom by stating that electrons can only exist in certain energy levels, and that they can transition between these levels by absorbing or emitting energy. The energy levels are determined by the distance of the electron from the nucleus, with higher levels having more energy and being further from the nucleus.

3. How is the energy of an electron in the Bohr model calculated?

The energy of an electron in the Bohr model is calculated using the formula E = -13.6/n^2, where n is the energy level. This formula takes into account the distance of the electron from the nucleus and the quantized energy levels.

4. Why is the Bohr model of the Hydrogen atom important?

The Bohr model of the Hydrogen atom is important because it was the first model to successfully explain the spectral lines of Hydrogen, which had been previously observed but not understood. It also laid the foundation for further advancements in quantum mechanics and our understanding of atomic structure.

5. How accurate is the Bohr model in describing the dynamics of electrons in the Hydrogen atom?

The Bohr model is a simplified representation of the Hydrogen atom and does not account for all factors, such as electron spin and the uncertainty principle. However, it is still accurate in explaining the basic dynamics of electrons in the Hydrogen atom and is a useful tool for understanding atomic structure.

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