Context: reducing a two-body problem to a one body problem (each body is a quantum particle, although this is not relevant to the parts of the problem I am stuck on(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Griffiths QM, Problem 5.1

Typically the interaction potential depends only on the vector [itex] \mathbf{r} = \mathbf{r_1} - \mathbf{r_2} [/itex] between the two particles. In that case, the Schrodinger equation separates if we change variables from [itex] \mathbf{r_1}, \mathbf{r_2} [/itex] to [itex] \mathbf{r} [/itex] and [itex] \mathbf{R} \equiv (m_1\mathbf{r_1} + m_2\mathbf{r_2})/m_1+m_2 [/itex] (the centre of mass).

(a) Show that [itex] \mathbf{r_1} = \mathbf{R} + (\mu/m_1)\mathbf{r} [/itex], [itex] \mathbf{r_2} = \mathbf{R} - (\mu/m_2)\mathbf{r} [/itex], and [itex] \nabla_1 = (\mu/m_2)\nabla_R + \nabla_r [/itex], [itex] \nabla_2 = (\mu/m_1)\nabla_R - \nabla_r [/itex], where

[tex] \mu \equiv \frac{m_1m_2}{m_1 + m_2} [/tex]

is thereduced massof the system.

2. Relevant equations

Already given

3. The attempt at a solution

(a)

(i) [tex] \mathbf{r_1} = \mathbf{r} + \mathbf{r_2} = \mathbf{r} + \left(\frac{m_1 + m_2}{m_2}\mathbf{R} - \frac{m_1}{m_2}\mathbf{r_1}

\right)[/tex]

[tex] \mathbf{r_1}\left(1 + \frac{m_1}{m_2}\right) = \mathbf{r_1}\left(\frac{m_1 + m_2}{m_2}\right) = \left(\frac{m_1 + m_2}{m_2}\right)\mathbf{R} + \mathbf{r} [/tex]

[tex] \longRightarrow \mathbf{r_1} = \mathbf{R} + \frac{m_2}{m_1 + m_2}\mathbf{r} = \mathbf{R} + \frac{\mu}{m_1}\mathbf{r} [/tex]

(ii) [tex] \mathbf{r_2} = \mathbf{r_1} - \mathbf{r} = \mathbf{R} + \frac{\mu}{m_1}\mathbf{r} - \mathbf{r} [/tex]

[tex] = \mathbf{R} + \left(\frac{\mu}{m_1} - 1 \right) \mathbf{r} = \mathbf{R} + \left(\frac{m_2}{m_1 + m_2} - \frac{m_1 + m_2}{m_1 + m_2} \right)\mathbf{r} [/tex]

[tex] = \mathbf{R} - \left(\frac{m_1}{m_1 + m_2}\right)\mathbf{r} = \mathbf{R} - \frac{\mu}{m_2}\mathbf{r} [/tex]

(iii) Now here's where I got stuck. I thought to start in cartesian coordinates:

[tex] \nabla_1 \equiv \mathbf{\hat{x}} \frac{\partial}{\partial x_1} + \mathbf{\hat{y}} \frac{\partial}{\partial y_1} + \mathbf{\hat{z}} \frac{\partial}{\partial z_1} [/tex]

and then based on the relationships between [itex] x_1 [/itex], [itex] x_R [/itex], and [itex] x_r [/itex] obtained in the first part, I could find the answer. But if I do that, I just get:

[tex] \nabla_1 = \nabla_R + \frac{\mu}{m_1}\nabla_r [/tex]

which is clearly not the right answer. Any tips for this section?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Change of Coordinates (Two-Body Problem)

**Physics Forums | Science Articles, Homework Help, Discussion**