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To solve this problem using the Hamilton-Jacobi method, we follow these steps: --- ### Step 1: Hamilton-Jacobi Equation (HJE) The Hamiltonian for a three-dimensional isotropic harmonic oscillator is given by: \[ H = \frac{1}{2m} (p_x^2 + p_y^2 + p_z^2) + \frac{1}{2} k (x^2 + y^2 + z^2) \] Using the Hamilton-Jacobi equation (HJE), we replace \( p_x, p_y, p_z \) with derivatives of Hamilton’s principal function \( W(x,y,z) \): \[ H \left( \frac{\partial W}{\partial x}, \frac{\partial W}{\partial y}, \frac{\partial W}{\partial z}, x, y, z \right) = E \] Substituting the expressions _____ _____ ___ _______ _____ __________ ______ __________ ____ ___ ________ ________.
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