Positronium ; Expecting TL;DRs ...

[IdealFugacity]

This question DOES concern a homework problem (in case you have moral qualms), though I am trying to keep it as 'honest' as possible.

I have been trying to work this positronium problem for probably 4 or more hours now today in the library. The first two questions, after explaining what positronium is (mass m and charge e- and mass m and charge e+), concern the energy difference between the ground and first excited states of Ps, and the ionization energy of Ps, the latter of which is widely publicized as 6.80 eV, exactly 1/2 that of Hydrogen with 13.60 eV.

I am almost positive it has something to do with 'reduced mass', particularly because wikipedia says so. However, whenever I try to derive the equivalent of, for hydrogen
En =
(- mass_electron * e^2 ) /
( 8 * epsilon^2 * h^2 * n^2 )

I am not getting an answer that is 1/2 of that. (that equation reduces to En = -13.60 eV / n^2)

Any tips to get me started on the right path without giving away the answer completely would be very helpful. These are the three equations I believe the derivation should start with (both from notes and wikipedia, for some reason i can't find an analagous section in the book), one or more of which must be changed to reflect the new mass.

http://en.wikipedia.org/wiki/Bohr_mo...ls_in_hydrogen

1) mvr = n * h/2pi (don't think that'll change)

2) Etot = PE + KE =
1/2 * m_electron * v^2 - e^2 / 4*pi*epsilon*r

3) The centripetal/coulomb force relation, which he did not explain in class at all, I wouldn't understand this even slightly if it wasn't for wikipedia, and i suspect that has something to do with my troubles:
e^2 / 4*pi*epsilon*r^2 = m_electron * v^2 / r

Whenever i try to change one of the m_e to (1/2)m_e, it doesn't seem to work out. Anyone have a hint as to how to get started here? I would really appreciate it, and like I said, I don't want a full answer; I will even post my complete answer if your tip helps me to crack open the problem!

[IdealFugacity]

In case I wasn't clear, this equation is the one I am having trouble deriving, though I can work my way through (without an extremely clear understanding) the one for hydrogen which it very nearly resembles:

[flipdeadshot22]

[ QUOTE ]
This question DOES concern a homework problem (in case you have moral qualms), though I am trying to keep it as 'honest' as possible.

I have been trying to work this positronium problem for probably 4 or more hours now today in the library. The first two questions, after explaining what positronium is (mass m and charge e- and mass m and charge e+), concern the energy difference between the ground and first excited states of Ps, and the ionization energy of Ps, the latter of which is widely publicized as 6.80 eV, exactly 1/2 that of Hydrogen with 13.60 eV.

I am almost positive it has something to do with 'reduced mass', particularly because wikipedia says so. However, whenever I try to derive the equivalent of, for hydrogen
En =
(- mass_electron * e^2 ) /
( 8 * epsilon^2 * h^2 * n^2 )

I am not getting an answer that is 1/2 of that. (that equation reduces to En = -13.60 eV / n^2)

Any tips to get me started on the right path without giving away the answer completely would be very helpful. These are the three equations I believe the derivation should start with (both from notes and wikipedia, for some reason i can't find an analagous section in the book), one or more of which must be changed to reflect the new mass.

http://en.wikipedia.org/wiki/Bohr_mo...ls_in_hydrogen

1) mvr = n * h/2pi (don't think that'll change)

2) Etot = PE + KE =
1/2 * m_electron * v^2 - e^2 / 4*pi*epsilon*r

3) The centripetal/coulomb force relation, which he did not explain in class at all, I wouldn't understand this even slightly if it wasn't for wikipedia, and i suspect that has something to do with my troubles:
e^2 / 4*pi*epsilon*r^2 = m_electron * v^2 / r

Whenever i try to change one of the m_e to (1/2)m_e, it doesn't seem to work out. Anyone have a hint as to how to get started here? I would really appreciate it, and like I said, I don't want a full answer; I will even post my complete answer if your tip helps me to crack open the problem!

[/ QUOTE ]

tl;dr, JK [img]/images/graemlins/smile.gif[/img]

don't really understand the confusion, just replace the reduced mass of 1/2m_e into the equivalent energy levels for hydrogen:

En =
(- mass_electron * e^2 ) /
( 8 * epsilon^2 * h^2 * n^2 )

and you'll get the 1/2 factor you're looking for.

[IdealFugacity]

I don't know how to explain that via derivation though...

*Why* are we using 1/2 m_e here?



edit: Sorry if I seem to be asking a question that I don't realize is clearer than I think it is, given your post... I just want to understand this before I just write it down, you know?

[flipdeadshot22]

[ QUOTE ]
I don't know how to explain that via derivation though...

*Why* are we using 1/2 m_e here?



edit: Sorry if I seem to be asking a question that I don't realize is clearer than I think it is, given your post... I just want to understand this before I just write it down, you know?

[/ QUOTE ]

I'm not sure of your level of knowledge of classical mechanics, so feel free to ask any questions: that factor of 1/2 is actually coming from a concept known in physics as a "reduced mass." Usually, when you are dealing with a problem involving an interaction between a very massive object, and a much less massive object (say the gravitational interaction between the Earth and the Moon) you tend to consider the Earth as being stationary, and the moon undergoing the rotational motion. This model applies to the hydrogen atom as well, since the mass of the electron is negligible compared to that of the proton. Physically, this isn't the complete truth however, since they are both rotating around the "center of mass" of the moon-earth system, and the Earth is actually undergoing motion as well. In order to derive the reduced mass, we have to consider Newton's second law as applied to the proton-electron system (for this examples purposes); here's a little schematic of what the system looks like( m symbolizes the smaller electron, while M is the proton, and X is the center of mass. The arrows F1, F2, a1, a2 are the forces and corresponding accelerations of the bodies):

F1 = F F2 = -F
------> <------

(m) -----------X---(M)

----> <----
a1 a2

To account for the motion of the proton physically, it is easiest to consider the motion of the electron relative to the proton (since the Coulomb force is a central force). Doing this means the electron's relative acceleration is proportional to the force (see the above figure). Working out the relative acceleration, we find it to be:

a_relative = a1 - a2 = (F/m) - (-F/M) = F(1/m + 1/M)

This above equation once re-arranged, is simply newton's second law with a funny looking new mass term:

F = 1/(1/m + 1/M)*a_relative or

F = (mM/(m+M))*a_relative

This funny mass term is the reduced mass, which I hope has some physical significance to you now. To get a feel for it's significance, let's play with some limits. In the limit that the larger mass goes to infinity (just like the assumption in the beginning that the Earth is so large it undergoes no motion) the reduced mass term looks like:

lim mM/(m+M) = m
M->inf

So in this case, it seems that the larger masses contribution to the motion becomes negligible, which is in accordance with the fixed central mass approximation. when the masses are equal (such as that for the Positronium system):

mM/(m+M) = m^2/2m = 1/2(m)

edit: sorry the formattings all screwed up for the diagrams

[IdealFugacity]

Right...once again, I understand that we are using a reduced mass. However, if I merely replace each m_e in the original derivation [such as the 3 base equations i listed in the OP) with (1/2)m_e, it does not work out to the final answer (I think). I see what you are saying but I still don't see how to make it work outside of merely replacing the final solution with a (1/2) m_e term. I am more or less looking for the means not the end...

Your post is further convincing me it has to do with that coulomb potential equation, though.

[flipdeadshot22]

[ QUOTE ]
Right...once again, I understand that we are using a reduced mass. However, if I merely replace each m_e in the original derivation [such as the 3 base equations i listed in the OP) with (1/2)m_e, it does not work out to the final answer (I think). I see what you are saying but I still don't see how to make it work outside of merely replacing the final solution with a (1/2) m_e term. I am more or less looking for the means not the end...

Your post is further convincing me it has to do with that coulomb potential equation, though.

[/ QUOTE ]

I see what you're saying now. Copied this from your OP:

2) Etot = PE + KE =
1/2 * m_electron * v^2 - e^2 / 4*pi*epsilon*r

I'm assuming you'll be needing this equation to derive your final result. Now, in your derivation, did you simply replace m_electron in this equation with 1/2m_electron, and plug on from there? If so, this could be your problem right there (I won't tell you why just yet, think about it some more; and as far as it having something to do with the coulomb force, no, as I had said in my long post, the coulomb force is a central force, so the relative motion of constituent particles is irrelevant.)

Here's a hint though: The energy is the TOTAL energy of the system, all particles included.

[IdealFugacity]

I did do the replacement you just mentioned; giving it a little more thought before going to sleep. I might not be able to post again on this until after this weekend, but then again I might make a dozen posts. Will definitely be working on it even if I'm incommunicado, and I appreciate your comments so far, thanks

[IdealFugacity]

Next thing I tried (at 1:55 AM...This isn't due till next Weds, wtf [img]/images/graemlins/tongue.gif[/img] ) was skipping right to the Etot term which eventually gets solved for
Etot =
-1/2 m_electron * v^2

and doubling it, so it is -1 instead of -1/2, to account for the kinetic and potential energy of the positron/electron system, which I suppose in an equal-mass system would be spinning around each other as opposed to an electron orbiting a nucleus.

However, there's got to be something else to it, because working through the rest of the derivation starting with an Etot of (E_tot_hydrogen * 2) actually results in the same answer as for hydrogen except with 2 in the bottom of the denominator, where I am worse off than i started (Now the ionization energy comes out to be 4 times higher than hydrogen's, obviously; 54.4 eV - instead of half as much!)

Also never took into account, during the work described in this post, the reduced mass term. Argh! sleep time...


(Thanks for the tip about central force too. Didn't realize that part was mass-independent)

[IdealFugacity]

Thank you for the replies. I just talked to the professor, and well, it turns out, the problem was intended to show us that Bohr's equations that we learned in class don't really apply well to other atoms, because (duh) they will give the same result every time.

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