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I have been faced with the Hamiltonian $$H = P_1^2/2m_1 + P_2^2/2m_2 + (k/2) x_1^2 + (k/2)x_2^2 + (K/2)(x_1-x_2)^2$$ I'm trying to find a systematic way to decouple it other than guessing. So, I wrote it in the form $$H = P^T\begin{bmatrix}1/2m_1 & 0 \\ 0 & 1/2m_2 \end{bmatrix}P + X^T\begin{bmatrix}k/2 + K/2 & -K/2 \\ -K/2 & k/2 + K/2\end{bmatrix}X $$ Now, how can I proceed? I also checked the commutator of the two matrices and found out that it doesn't vanish, does this tell me anything? I will appreciate it if anyone guided me to the right reference.

Ammar Amgad
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1 Answers1

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You proceed in the following way:

(1) Consider the Hamiltonian to be a Hamilton function $H=T+V$ in classical mechanics and express the momenta in $T$ and $V$ ( both matrices are real and symmetric) by the respective coordinate velocities $P_i=m_iv_i$.

(2) Consider the Lagrange function $L=T-V$ which contains only the coordinates and their time derivatives (velocities).

(3) Make a coordinate scaling transformation of the Lagrange function so that the diagonal elements of the T-matrix become equal. This results in a new potential energy (V-)matrix

(4) Then perform a further, orthogonal transformation (rotation) on the new coordinates so that the real symmetric V-matrix becomes diagonal. This transformation doesn't change the T-matrix from having the same diagonal elements. This results in the Lagrangian $$L'(x'_i, \dot x_i')=T(x_i', \dot x_i')-V(x_i', \dot x_i')$$

(5) Now you obtain the conjugate momenta $$p_i'=\frac {\partial L'}{\partial \dot x_i'}$$

(6) With this you obtain the new Hamilton function $$H'(x_i', p_i')=\sum_i p_i'\dot x_i'-L'$$ with decoupled kinetic and potential energy terms of harmonic oscillators$$

(7) From this Hamilton function in the generalized coordinates $x_i'$ and generalized conjugate momenta $p_i'$ follow the the Schrödinger equation (or Heisenberg equations) by interpreting the generalized coordinates and momenta as operators.

freecharly
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