I How to find the wavefunction in this case?

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Kashmir
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We've a two interacting particle system, with Hamiltonian as:
##H_{s y s}=\frac{\mathbf{p}_{1}^{2}}{2 m_{1}}+\frac{\mathbf{p}_{2}^{2}}{2 m_{2}}+V\left(\mathbf{r}_{1}, \mathbf{r}_{2}\right)##

we reduce it to two non interacting fictitious particles,one moving freely other in a central field, thus the system has Hamiltonian: ##H_{s y s}=\frac{\mathbf{P}^{2}}{2 M}+\frac{\mathbf{p}_{r e l}^{2}}{2 \mu}+V(\mathbf r)## with the operator relations as:

##\mathbf{R}=\frac{m_{1} \mathbf{r}_{1}+m_{2} \mathbf{r}_{2}}{m_{1}+m_{2}}####\mathbf{r}=\mathbf{r}_{2}-\mathbf{r}_{1}##

##\mathbf{p}_{r e l}=\frac{m_{1} \mathbf{p}_{2}-m_{2} \mathbf{p}_{1}}{m_{1}+m_{2}}##

##\mathbf{P}=\mathbf{p}_{1}+\mathbf{p}_{2}##


We solve ##H|E\rangle=E|E\rangle## for the two non interacting particles and find the wavefunction ##\psi_{s y s}(\mathbf{R}, \mathbf{r})=\psi_{C M}(\mathbf{R}) \psi_{r e l}(\mathbf{r})##

How do we find the wavefunction in terms of ##\mathbf r_1,\mathbf r_2##? We can't just invert ##\mathbf{R}=\frac{m_{1} \mathbf{r}_{1}+m_{2} \mathbf{r}_{2}}{m_{1}+m_{2}}####\mathbf{r}=\mathbf{r}_{2}-\mathbf{r}_{1}## and use them because they are operator relations.
 
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Why do you think that? The wave function in terms of ##\vec{r}_1## and ##\vec{r}_2## is just the wave function you've written down with ##\vec{R}## and ##\vec{r}## expressed in terms of ##\vec{r}_1## and ##\vec{r}_2##.
 
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vanhees71 said:
Why do you think that? The wave function in terms of ##\vec{r}_1## and ##\vec{r}_2## is just the wave function you've written down with ##\vec{R}## and ##\vec{r}## expressed in terms of ##\vec{r}_1## and ##\vec{r}_2##.
The eigenstate is
##\int|R\rangle \otimes|r\rangle
\psi(R, r) d R d r=|E\rangle##

In ##r_1,r_2## representation I need to find ##\left\langle r_{1}| \otimes\left\langle r_{2}|\left|\int\right| R\right\rangle \otimes \mid r\right\rangle \psi(R, r) d R d r##
 
Not sure what you mean,

(1) Not sure what you mean, in the position representation the position operators act just as numbers.
(2) Even in the operator sense, r1 and r2 most definitely can be given in terms of RCM and rrel. Try to find the inverse transformation.

I'll take the opportunity to point out to this channel
 
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andresB said:
Not sure what you mean,

(1) Not sure what you mean, in the position representation the position operators act just as numbers.
(2) Even in the operator sense, r1 and r2 most definitely can be given in terms of RCM and rrel. Try to find the inverse transformation.

I'll take the opportunity to point out to this channel

I do use this channel. Professor M is nice :). Thank you
 
vanhees71 said:
Why do you think that? The wave function in terms of ##\vec{r}_1## and ##\vec{r}_2## is just the wave function you've written down with ##\vec{R}## and ##\vec{r}## expressed in terms of ##\vec{r}_1## and ##\vec{r}_2##.
So I simply write ##R, r## in terms of ##r_1,r_2## in ##\psi_{s y s}(\mathbf{R}, \mathbf{r})=\psi_{C M}(\mathbf{R}) \psi_{r e l}(\mathbf{r})## and get my wavefunction in terms of ##r_1,r_2##. Right?
 
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Right!
 
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