Landau Quantization: Classical Particle Motion in a Uniform Magnetic Field

[Sonny Liston]

I just integrated v_{x} from 0 to t to get (v_{0})(cos(t\omega) - cos(0)) + x_{0} \longrightarrow (v_{0})cos(t\omega) -v_{0} + x_{0}

although the (-v_{0}) term seems strange to me

 

[Dickfore]

But, x is not a constant:

<br /> x(t) = x_{0} + \int_{t_{0}}^{t}{v_{x}(t&#039;) \, dt&#039;}<br />

 

[Sonny Liston]

Right. Integrating v_{x} = v_{0}cos(t\omega) over [t_{0}, t] gives

v_{0}[\omega(sin(t_{0} \omega) - sin(t \omega))] where (sin(t_{0} \omega) is some constant, unless I'm being an idiot.

Oh jeez, I just saw that I did the wrong integral in my last post. Now I think your comment makes sense.

 

[Dickfore]

No!

<br /> \int{\cos{(a t)} \, dt} = \frac{1}{a} \, \sin{(a \, t)}<br />

 

[Sonny Liston]

Jesus. Embarrassing. Long day.

But we are doing a definite integral, so it should still end up as v_{0}[\frac{1}{\omega}(sin(t_{0} \omega) - sin(t \omega))] shouldn't it? If we set t_{0}=0, then the first sin term vanishes, and we end up with v_{0}[\frac{1}{\omega}(-sin(t \omega))] ?

 

[Dickfore]

Yes, this would be x(t) that you need to substitute in the above expression and integrate again.

 

[Sonny Liston]

Integrating my LagrangianL = (m/2)[(v_{0})^2 +1] - (q/c)B[(x_{0} - \frac{v_{0}}{\omega} sin(t\omega)][-(v_{0})sin(t\omega)]

Gives

S(t) = [(m(v_{0})^2)/2]\Delta t - <br /> <br /> (q/c)B[\frac{(v_{0})x_{0}}{\omega}](cos(\omega t_{1}) - cos(\omega t_{0})) - <br /> <br /> (q/c)B[\frac{(v_{0})^2}{\omega}]((t_{1})/2 - \frac{sin(2\omega t_{1})}{4\omega} - (t_{0})/2) + \frac{sin(2\omega t_{0})}{4\omega}

This looks more complicated than it should be..

 

[Dickfore]

Ok, do this first:

Take the lagrangian:

<br /> L = \frac{m v^{2}_{0}}{2} \, \left(v^{2}_{x} + v^{2}_{y} \right) - \frac{q B}{c} \, x \, v_{y}<br />

substitute:

<br /> \begin{array}{rcl}<br /> v_{x} &amp; = &amp; v_{0} \, \cos{(\omega \, t)} \\<br /> <br /> v_{y} &amp; = &amp; -v_{0} \, \sin{(\omega \, t)} \\<br /> <br /> x &amp; = &amp; \frac{v_{0}}{\omega} \, \sin{(\omega t)}<br />

and perform integration from 0 to t, by keeping in mind that:

<br /> \omega = \frac{q \, B}{m \, c}<br />

What is the lagrangian as a function of t?

What is the integral of this from 0 to t?

 

[Sonny Liston]

I get that L=(v_{0})^2[\frac{m}{2} + (sin(\omega t)^2)

Have I done anything stupid so far?

 

[Dickfore]

First of all, your equation is dimensionally inconsistent. In the large parentheses, the first term has a dimension of mass, and the second term is dimensionless.

 

[Sonny Liston]

All I did was cancel the omega in the x term and the omega in the v_{y} term, which I got from plugging in the provided equations.

Sigh. How did I manage to mess THAT up?

 

[Dickfore]

The vector potential term from the Lagrangian has the product:

<br /> x v_{x} = \frac{v_{0}}{\omega} \, \sin{(\omega t)} \, v_{0} \, \cos{(\omega t)} = \frac{v^{2}_{0}}{\omega} \sin{(\omega t)} \, \cos{(\omega t)}<br />

using the double angle formula:

<br /> \sin{(2 \alpha)} = 2 \, \sin{(\alpha)} \, \cos{(\alpha)}<br />

this product can be further simplified as:

<br /> \frac{v^{2}_{0}}{2 \omega} \, \sin{(2 \omega t)}<br />

However, this product is also multiplied by q \, B/c in the Lagrangian. Using the definition of the cyclotron frequency \omega = q \, B/(m \, c), what is:

<br /> \frac{q B}{\omega c} = ?<br />

The kinetic energy term has the sum of squares:

<br /> v^{2}_{x} + v^{2}_{y} = ?<br />

What is this sum equal to? What is it multiplied by in the Lagrangian?

 

[Sonny Liston]

<br /> \frac{q B}{\omega c} = ?<br />\frac{qB}{\omega c} = \frac{qBmc}{Bc} = qm[/tex]<br /> <br /> The kinetic energy term has the sum of squares:<br /> <br /> &lt;br /&gt; v^{2}_{x} + v^{2}_{y} = ?&lt;br /&gt;<br /> <br /> What is this sum equal to? What is it multiplied by in the Lagrangian?<br /> <br /> &lt;br /&gt; v^{2}_{x} + v^{2}_{y} = ((v_{0})^2)((cos(\omega t))^2 + (sin(\omega t))^2) = ((v_{0})^2)&lt;br /&gt;<br /> <br /> This is then multiplied by \frac{m}{2}

 

[Sonny Liston]

Whaddya mean?

 

[Dickfore]

<br /> \frac{q B}{\omega c} = ?<br />


\frac{qB}{\omega c} = \frac{qBmc}{Bc} = qm[/tex]<br />

<br /> This is incorrect.<br /> <br /> <blockquote data-attributes="" data-quote="Sonny Liston" data-source="post: 2985775" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Sonny Liston said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> The kinetic energy term has the sum of squares:<br /> <br /> &lt;br /&gt; v^{2}_{x} + v^{2}_{y} = ?&lt;br /&gt;<br /> <br /> What is this sum equal to? What is it multiplied by in the Lagrangian?<br /> <br /> &lt;br /&gt; v^{2}_{x} + v^{2}_{y} = ((v_{0})^2)((cos(\omega t))^2 + (sin(\omega t))^2) = ((v_{0})^2)&lt;br /&gt;<br /> <br /> This is then multiplied by \frac{m}{2} </div> </div> </blockquote>This is correct.

 

[Sonny Liston]

I didn't see the charge term in the original cyclotron frequency. In that case, we don't get qm, we get m.

 

[Dickfore]

I didn't see the charge term in the original cyclotron frequency. In that case, we don't get qm, we get m.

Yes. So, what is the final expression for the Lagrangian L?

 

[Sonny Liston]

L = \frac{m(v_{0})^2}{2} - \frac{m(v_{0})^2}{2\omega}sin(2t\omega)

 

[Dickfore]

L = \frac{m(v_{0})^2}{2} - \frac{m(v_{0})^2}{2\omega}sin(2t\omega)

Where did the extra \omega come from in the second term? C'mon man. Concentrate on your algebra. This is taking too long.

 

[Sonny Liston]

S(t) = [\frac{m((v_{0})^2)}{2}]t + [\frac{m((v_{0})^2)}{2}]\frac{(cos(2t\omega) + 1}{2\omega}

Extra omega eliminated here. Sorry about that.

 

[Dickfore]

(Again, you kept an extra \omega in the second term.) I see you corrected that.

Also:

<br /> \int_{0}^{t}{\sin{(2 \omega t)} \, dt} = -\left.\frac{1}{2 \omega} \, \cos{(\omega t)}\right|^{t}_{0} = \frac{1}{2\omega} \, \left[1 - \cos{(2 \omega t)}\right]<br />

Finally, what will be the change in S during one period (take t = 2\pi/\omega in the above formula)?

 

[Sonny Liston]

1 - cos (2 \omega \frac{(2\pi}{\omega}) = 0, so it is appropriately periodic.

 

[Dickfore]

No, the first term is proportional to t and is not periodic.

 

[Sonny Liston]

Assuming that's satisfactory, may I ask you one last quick question (you are a saint, by the way. I have been unbelievably useless throughout this.) How would I go about evaluating the magnetic flux through the classical trajectory as a function of the energy?

In case you're wondering about my remarkable ignorance, I'm a math student who never took classical mechanics and is in a quantum course, so all the classical mechanics stuff is new to me. Thanks again.

 

[Dickfore]

What does the trajectory look like? What are its dimensions in terms of the quantities we have already defined? What is the magnetic flux from a homogeneous magnetic field perpendicular through a contour?

 

[Sonny Liston]

In my notes, I have that it is given by \frac{1}{2\pi}\oint A dx, and the baby E&M book I am looking at says that it can be given by the Area of the surface dotted with the B field times cos(theta), for theta the angle between the B field and the surface.

But since the B field is homogeneous in the z-direction, \theta = \frac{\pi}{2} and cos=0, so something must be really flawed in my reasoning.

 

[Dickfore]

In my notes, I have that it is given by \frac{1}{2\pi}\oint A dx, and the baby E&M book I am looking at says that it can be given by the Area of the surface dotted with the B field times cos(theta), for theta the angle between the B field and the surface.

But since the B field is homogeneous in the z-direction, \theta = \frac{\pi}{2} and cos=0, so something must be really flawed in my reasoning.

Yes, they are both equal due to Stokes' Theorem:

<br /> \phi_{m} = \int_{S}{\left(\mathbf{B} \cdot \hat{\mathbf{n}}\right) \, da} = \oint{\mathbf{A} \cdot d\mathbf{l}}<br />

So, there is no factor of 1/(2\pi) if you use the circulation of the vector potential.

 

[Sonny Liston]

So there is nothing wrong in my reasoning and the flux actually is zero? It seems odd to ask me to evaluate it "as a function of the energy" then.

 

[Dickfore]

No, it is not zero. I didn't see the part where you say it's zero. Since \mathbf{A} has only a zero component, you need to evaluate:

<br /> \oint{\mathbf{A} \cdot d\mathbf{l}} = \oint{A_{y} \, dy} = B \int_{0}^{T}{x(t) \, v_{y}(t)}<br />

where T = 2\pi/\omega is the period of the trajectory and i used A_{y} = B \, x.

 

[Sonny Liston]

B\int x(t) v_{y} (t) = B \frac{-(v_{0})^2}{\omega} \int (sin(\omega t)^2)

Using (sin(\omega t)^2) = \frac{1 - cos (2\omega t)}{2} and doing the integral, I get

\Phi = B\pi(- \frac{v_{0}}{\omega})^2