Particle in Magnetic Field

a) Give a relationship between Hamilton’s equations under a canonical transformation. Verify that the transformation

is canonical.

b) Find Hamilton’s equations of motion for a particle moving in a plane in a magnetic field described by the vector potential

in terms of the new variables Q₁, Q₂, P₁, P₂ introduced above, using ω = eH/mc.

SOLUTION

a) A canonical transformation preserves the form of Hamilton’s equations:

where H = H(Q, P) is the transformed Hamiltonian. It can be shown that Poisson brackets are invariant under such a transformation. In other words, for two functions f, g

(1)

where q, p and Q, P are the old and new variables, respectively. Since we have the following equations for Poisson brackets:

(2)

(1) and (2) combined give equivalent conditions for a transformation to be canonical:

Let us check for our transformation (we let

and

Similarly

and so on.

For a particle in a magnetic field described by the vector potential A =(–YH/2,XH/2,0), which corresponds to a constant magnetic field we should use the generalized momentum P in the Hamiltonian

so the Hamiltonian

So the Hamiltonian H does not depend on Q₁, Q₂ and

where α is the initial phase. Also

Where X₀ and Y₀ are defined by the initial conditions. We can write this solution in terms of the variables X, Y, p_x, p_y:

Similarly

so this is indeed the solution for a particle moving in one plane in a constant magnetic field perpendicular to the plane.

مواضيع ذات صلة

مصطلحات الفيزياء النووية

إزاحة حمراء REDSHIFT

المركبات النانوية (Nanocomposites)

مولد فان دي جراف الالكتروستاتي

التطويرات التقنية الحديثة في المعجلات الالكتروستاتية

معجلات الالكترونات الالكتروستاتية

معجلات الفان دي جراف الترادفية

الحد من زيادة الطاقة في معجل فان دي جراف

أنبوبة التعجيل قي معجلات الجسيمات من نوع فان دي جراف

ارتكاز معجل فان دي جراف

مبدأ تشغيل معجل الفان دي جراف

القضايا الأخلاقية المتعلقة بتكنولوجيا النانو