PHY4221_Homework_Assignment_1_Solution

PHY4221_Homework_Assignment_1_Solution - PHY4221 Homework...

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PHY4221 Homework Assignment 1 Solution (Written by: Leung Shing Chi) 1. (In this problem, we use Einstein’s summation convention, i.e. , also we use the following notation: 3 1 ii i AA , i ij j A A x ) Given the Lagrangian to be 1 (,) 2 q Lm v v q r t v A r t c   , we apply the Euler-Lagrange Equation (ELE) to L : 0 dL L dt v x   where 1, 2, 3 i (3 equations in total). , i i jj i Lq Pm v A vc qv xx c    A Therefore, we find  , i i i A dL d q mv v A dt v dt c t   j i j (It should be noted that this result arises as the usage of ordinary derivative) and we take the ELE to be: ,, , 0 1 i i j i j i j i j A dq q mv v A q v A dt c t c A mv q v A v A dt c t c    , j i j j i This, in fact, is the solution by observing the following relation among the scalar and vector potential and the E and B fields.   , 1 i jkj k i i m nm n k i j k ji j j j i i A E ct vB vB vA vA vA   , Certainly, to achieve this result, we need some relations between and , namely: ijk mnk im jn in jm  Finally, it is now obvious to see the equation of motion (EoM) to be:
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 ii i dq mv qE v B dt c mv qE v B dt c   Next, we would see if the Hamiltonian could help us get the identical result as above. First, we write down the Hamiltonian according to the definition: 2 1 (,) 2 1 2 2 HP vL qq mv v v A mv v q r t v A r t cc mv v q r t q PA c qr t m       We intend to write the Hamiltonian in term of P, rather than the velocity, as to maintain the consistency of the agreed usage of variable. However, P itself is no loner the mechanical momentum, only q c is the mechanical momentum. We need this modified momentum in order to obtain the correct equation of motion. For the equation of motion, we have altogether 6 equations, 3 for momentum space and 3 for position space. , , jj ji i i i q qA H c Pq xmc q H c x Pm  One may get the same EoM if one plugs in P and x back to the above equations.
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2. Given a force field (0,0, ) F  K z we may immediately write down the potential to be:
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This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

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PHY4221_Homework_Assignment_1_Solution - PHY4221 Homework...

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