December
- 6
- 0
[SOLVED] Macroscopic quantum model
Hi!
I'm supposed to calculate the quantum number of a macroscopic system (The Earth and a satellite).
I should assume that the satellite is moving in a circular motion around earth, and that it fulfills the same quantization conditions as the Bohr model of the Hydrogen atom.
So far, I started by calculating the total energy of the system, but using a gravitational potential instead of the Coulomb potential, which gives me a total energy of:
[tex]E=mr^{2}\omega^{2}-G\frac{mM}{r}[/tex]
Then, by using the centripetal force and the gravitational force (to find an equilibrium between the two), I got:
[tex]G\frac{mM}{r^{2}}=mr\omega^{2}[/tex]
Substituting this into the energy expression gives:
[tex]E=\frac{1}{2}G\frac{mM}{r}-G\frac{mM}{r}[/tex]
The total energy then becomes:
[tex]E=-G\frac{mM}{2r}[/tex]
... After this, I'm stuck. As far as I can see (using the book that I have in this course), Bohr postulated that the emitted radiation from the hydrogen atom has a frequency which is given by:
[tex]E_{n}-E_{n'}=hf[/tex]
... Where [tex]E_{n}[/tex] is given by:
[tex]E_{n}=-\frac{Rhc}{n^{2}}[/tex]
My thought was that the total energy expression which I calculated must be for a specific value of n, so what I tried was to set my expression equal to [tex]E_{n}[/tex] and then calculate the quantum number n from this relationship. This yielded:
[tex]n=\sqrt{\frac{2rRhc}{GmM}}[/tex]
...However, I'm not even sure that this is a reasonable approach. What especially bothers me is the Rydberg Constant. Can I use a standardized value on this, or do I have to recalculate it so that it too depends on a gravitational force?
I'm really stuck on this one (I think)... Any help is truly appreciated!
Hi!
I'm supposed to calculate the quantum number of a macroscopic system (The Earth and a satellite).
I should assume that the satellite is moving in a circular motion around earth, and that it fulfills the same quantization conditions as the Bohr model of the Hydrogen atom.
So far, I started by calculating the total energy of the system, but using a gravitational potential instead of the Coulomb potential, which gives me a total energy of:
[tex]E=mr^{2}\omega^{2}-G\frac{mM}{r}[/tex]
Then, by using the centripetal force and the gravitational force (to find an equilibrium between the two), I got:
[tex]G\frac{mM}{r^{2}}=mr\omega^{2}[/tex]
Substituting this into the energy expression gives:
[tex]E=\frac{1}{2}G\frac{mM}{r}-G\frac{mM}{r}[/tex]
The total energy then becomes:
[tex]E=-G\frac{mM}{2r}[/tex]
... After this, I'm stuck. As far as I can see (using the book that I have in this course), Bohr postulated that the emitted radiation from the hydrogen atom has a frequency which is given by:
[tex]E_{n}-E_{n'}=hf[/tex]
... Where [tex]E_{n}[/tex] is given by:
[tex]E_{n}=-\frac{Rhc}{n^{2}}[/tex]
My thought was that the total energy expression which I calculated must be for a specific value of n, so what I tried was to set my expression equal to [tex]E_{n}[/tex] and then calculate the quantum number n from this relationship. This yielded:
[tex]n=\sqrt{\frac{2rRhc}{GmM}}[/tex]
...However, I'm not even sure that this is a reasonable approach. What especially bothers me is the Rydberg Constant. Can I use a standardized value on this, or do I have to recalculate it so that it too depends on a gravitational force?
I'm really stuck on this one (I think)... Any help is truly appreciated!