Review of Hamiltonians Lecture (20 minutes)

Central Forces Notes Section 13

This lecture reviews what students have learned about Schrödinger's equation and the Hamiltonian from previous classes.

  • A lecture discussion helps students to recall the form and meaning of the Hamiltonians for the particle in a box ($\hat{H}=-\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial x^2} + V(x)$) and for the spin of an electron in a magnetic field ($\hat{H} = \frac{e B_0}{m_e c} S_z$).
  • Students are asked, as a small whiteboard question to write down what they think the Hamiltonian will be for two particles interacting via a central force. While we don't expect students to know the correct answer, this question stimulates students to think about the meaning of the parts of the Hamiltonian and serves as a starting point for an interractive discussion that leads to writing down the proper hamiltonian.

$$\hat{H}= \frac{\hbar^2}{2 m_p} \nabla^2_p + \frac{\hbar^2}{2 m_e} \nabla^2_e -\frac{1}{4 \pi \epsilon_0} \frac{e^2}{|r_p-r_e|}$$ where $$\nabla^2_p = \frac{1}{r_p^2} \frac{\partial}{\partial r_p} \left(r_p^2 \frac{\partial}{\partial r_p} \right)+ \frac{1}{r_p^2 \sin(\theta_p)} \frac{\partial}{\partial \theta_p} \left(\sin(\theta_p) \frac{\partial}{\partial \theta_p} \right)+ \frac{1}{r_p^2 \sin^2(\theta_p)}\frac{\partial^2}{\partial \phi_p^2}$$ and $$\nabla^2_e = \frac{1}{r_e^2} \frac{\partial}{\partial r_e} \left(r_e^2 \frac{\partial}{\partial r_e} \right)+ \frac{1}{r_e^2 \sin(\theta_e)} \frac{\partial}{\partial \theta_e} \left(\sin(\theta_e) \frac{\partial}{\partial \theta_e} \right)+ \frac{1}{r_e^2 \sin^2(\theta_e)}\frac{\partial^2}{\partial \phi_e^2}$$

in spherical coordinates. This then leads into an optional derivation in which this Hamiltonian is simplified using the idea of reduced mass.

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