Question 705048

SCQMEDIUM

For a simple harmonic oscillator, the Lagrangian is given by $L = \frac{1}{2}\dot{q}^2 - \frac{1}{2}q^2$ . If $H(q, p)$ is the Hamiltonian of the system and $A(q, p) = \frac{1}{\sqrt{2}}(p + iq)$ , the Poisson bracket $\{A, H\}$ is

(A)

$iA$

(B)

$A^*$

(C)

$-iA^*$

(D)

$-iA$

Detailed Solution

The Hamiltonian is $H = \frac{p^2}{2} + \frac{q^2}{2}$ . The Poisson bracket is $\{A, H\} = \frac{\partial A}{\partial q}\frac{\partial H}{\partial p} - \frac{\partial A}{\partial p}\frac{\partial H}{\partial q} = (\frac{i}{\sqrt{2}})(p) - (\frac{1}{\sqrt{2}})(q) = i(\frac{p+iq}{\sqrt{2}}) = iA$ .

Free Exam

Boost Your Exam Preparation!

Move beyond just reading solutions. Access our comprehensive Test Series, original Mock Tests, and interactive learning modules. Many premium tests are completely free!

Original Mocks & Regular Test Series
Real NTA-like Interface with Analytics
Many Free Tests Available