Change of Coordinates (Two-Body Problem)

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  • #1
cepheid
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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

Homework Statement



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 the reduced mass of the system.

Homework Equations



Already given

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?
 
Last edited:

Answers and Replies

  • #2
quasar987
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It works fine over here. For instance,

[tex]\frac{\partial}{\partial x_1}=\frac{\partial R_x}{\partial x_1}\frac{\partial}{\partial R_x}+\frac{\partial r_x}{\partial x_1}\frac{\partial}{\partial r_x}=\frac{\mu}{m_2}\frac{\partial}{\partial R_x}+\frac{\partial}{\partial r_x}[/tex]
 
  • #3
cepheid
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You're right. It turns out I was just being dumb. Thanks.
 

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