Consider a one-dimensional harmonic oscillator with mass m, whose position is denoted by x, and subject to a potential \(V = \frac{1}{2} kx^2.\) The Hamiltonian for this system is \(H = \frac{p^2}{(2m)} + \frac{1}{2} kx^2.\) Now let's consider a transformation in this phase space to new coordinates (Q, P) according to \( Q = 2x + \frac{p}{mω} \) and \(P = 2p - mωx\) where ω is the angular frequency of the oscillator. After this transformation, in terms of (Q,P), what does the Hamiltonian become?
1
\(H = \frac{P^2}{(8m)} + \frac{kQ^2}{8}\)
2
\(H = \frac{P^2}{(4m)} + \frac{kQ^2}{4}\)
3
\(H = \frac{P^2}{(2m)} + \frac{1}{2} kQ^2\)
4
\(H = \frac{P^2}{(4m)} + KQ^2\)
5
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