Case when the potential energy of the 1st excited state is zero

[PSN03]

Homework Statement

For a neutral hydrogen atom where the ground state energy of the electron is taken as E=-13.6eV (with standard reference). Now if the potential energy of the first excited state is taken as zero reference potential energy then the potential energy of the third excited state is.

Relevant Equations

E=-13.6/n² where n is the orbit number.
Standard potential energy at infinity is zero.

I know how to solve this problem when the energy at ground state is zero but I don't know how to deal with 1st excited state energy as zero.
According to me since the potential energy is zero therefore the kinetic energy must be 13.6eV according to conservation of energy.
I also know that the energy is inversely proportional to the square of orbit number.
I don't know how to proceed further after this. Any kind of help or even a hint would be appreciated.

 

[PeroK]

This is just a change of reference point for potential energy. You're overthinking things.

 

[PSN03]

Can u please give me some hints cause I can't think in other direction

 

[PeroK]

Can u please give me some hints cause I can't think in other direction

It says in the question:

Homework Statement:: For a neutral hydrogen atom where the ground state energy of the electron is taken as E=-13.6eV (with standard reference).

What's the "standard reference"?

 

[PSN03]

It says in the question:What's the "standard reference"?

The standard reference is infinity I guess. Cause at infinity the potential energy becomes zero so any potential energy below that is negative

 

[PeroK]

The standard reference is infinity I guess. Cause at infinity the potential energy becomes zero so any potential energy below that is negative

Yes.

I suggest you first write down the energies of the relevant states in the standrad reference.

 

[PSN03]

Yes.

I suggest you first write down the energies of the relevant states in the standrad reference.

Yes I have done that.
At standard state the energy of 2nd excited state is -13.6/3²=-1.51 and 3rd excited state is -0.85eV

 

[PeroK]

Yes I have done that.
At standard state the energy of 2nd excited state is -13.6/3²=-1.51 and 3rd excited state is -0.85eV

I thought you wanted the first excited state and to take that as you new zero potential?

 

[PSN03]

I thought you wanted the first excited state and to take that as you new zero potential?

Yes that's exactly what has to be done with the new frame of reference. But I was answering it with respect to the old frame of potential energy. I don't know how to deal it with new frame of reference.

 

[PeroK]

Yes that's exactly what has to be done with the new frame of reference. But I was answering it with respect to the old frame of potential energy. I don't know how to deal it with new frame of reference.

Why didn't you calculate the energy of the first excited state?

 

[PSN03]

Why didn't you calculate the energy of the first excited state?

I don't know how to find that in the new frame of reference. In the old frame it's -3.4eV

 

[PeroK]

I don't know how to find that in the new frame of reference. In the old frame it's -3.4eV

Have you learned about things like the Bohr Radius?

 

[PSN03]

Have you learned about things like the Bohr Radius?

Yes I have learned about it.

 

[PeroK]

Yes I have learned about it.

Do you know the radius applicable to the energy states?

 

[PSN03]

Do you know the radius applicable to the energy states?

According to my knowledge energy is inversely proportional to the square of radius.
E=k/n²

 

[PeroK]

According to my knowledge energy is inversely proportional to the square of radius.
E=k/n²

Okay, but you need to find the potential energy for each of these states. To do that you need to define the potential energy for each of these states. I assume you are supposed to calculate the potential for the 3rd Bohr radius and use that as the new zero. That being the second excited state.

 

[PSN03]

Okay, but you need to find the potential energy for each of these states. To do that you need to define the potential energy for each of these states. I assume you are supposed to calculate the potential for the 3rd Bohr radius and use that as the new zero. That being the second excited state.

According to me me we are given that the energy of the 2nd excited state is zero and we are supposed to find the subsequent energy of the third excited state.

 

[PeroK]

According to me me we are given that the energy of the 2nd excited state is zero and we are supposed to find the subsequent energy of the third excited state.

That's one interpretation. Then the problem is very easy.

But, it does say "potential" energy. Which suggests that they want you to calculate the potential energy for the 2nd excited state. That would involve putting an appropriate radius into the Coulomb potential.

Perhaps you might want to confirm which it is?

 

[PSN03]

That's one interpretation. Then the problem is very easy.

But, it does say "potential" energy. Which suggests that they want you to calculate the potential energy for the 2nd excited state. That would involve putting an appropriate radius into the Coulomb potential.

Perhaps you might want to confirm which it is?

Actually you are right. The problem says potential energy is zero and not the total energy.
But why should I calculate the potential energy for 2nd excited state?

 

[PeroK]

Actually you are right. The problem says potential energy is zero and not the total energy.
But why should I calculate the potential energy for 2nd excited state?

I believe the expected value of the Coulomb potential involves ##r_n = an^2##, where ##a## is the Bohr radius. I would use that.

 

[PSN03]

I believe the expected value of the Coulomb potential involves ##r_n = an^2##, where ##a## is the Bohr radius. I would use that.

Yes that's absolutely right but it only gives us the radius of nth orbit. How to deal it with the new frame of reference?

 

[PeroK]

Yes that's absolutely right but it only gives us the radius of nth orbit. How to deal it with the new frame of reference?

There no frame of reference involved. There is only a zero point for potential energy.

 

[PSN03]

There no frame of reference involved. There is only a zero point for potential energy.

Oh actually I meant that 2nd orbital should be new reference point for the potential energy...right?

 

[PeroK]

Oh actually I meant that 2nd orbital should be new reference point for the potential energy...right?

Yes.

 

[PSN03]

Yes.

What to do after this. I mean how will this new frame affect the energy of the electron?

 

[PeroK]

What to do after this. I mean how will this new frame affect the energy of the electron?

Try to work out the potential energy of the first excited state. Until you have done that you can't go any further.

 

[PSN03]

Try to work out the potential energy of the first excited state. Until you have done that you can't go any further.

The potential energy of first excited state would be -13.6*2/4=6.8eV in old reference frame. In new reference frame it's given to be zero.

 

[PeroK]

The potential energy of first excited state would be -13.6*2/4=6.8eV in old reference frame. In new reference frame it's given to be zero.

If you mean ##-6.8 eV## that's right.

 

[PSN03]

If you mean ##-6.8 eV## that's right.

Yes -6.8eV ...but after this.
The new energy is 0 therefore +6.8eV has been supplied and the change in energy is of 6.8-(-6.8)=13.6eV.

 

[PeroK]

Yes -6.8eV ...but after this.
The new energy is 0 therefore +6.8eV has been supplied and the change in energy is of 6.8-(-6.8)=13.6eV.

It's the (potential) energy of the third excited state you want.

 

[PeroK]

Yes -6.8eV ...but after this.
The new energy is 0 therefore +6.8eV has been supplied and the change in energy is of 6.8-(-6.8)=13.6eV.

This calculation is wrong. You need some notation. Let's have: ##E_n, V_n## for the original energy and potential energy of the hydrogen energy states; and. ##E'_n, V'_n## for the energies with the changed zero potential to be at the first excited state.

I suggest you should write them all down so you can see the pattern.

 

[PSN03]

This calculation is wrong. You need some notation. Let's have: ##E_n, V_n## for the original energy and potential energy of the hydrogen energy states; and. ##E'_n, V'_n## for the energies with the changed zero potential to be at the first excited state.

I suggest you should write them all down so you can see the pattern.

Ohk...
So
##E_1##=-13.6
##E_2##=-3.4
##E_3##=-1.51
##E_4##=-0.85
##V_2##=-6.8
##V_3##=-3.02
##V_4##=-1.7

##V'_2##=0

 

[PeroK]

It's ##V'_4## you want, isn't it?

 

[PSN03]

It's ##V'_4## you want, isn't it?

Yes but how to go about it?

 

[PeroK]

Yes but how to go about it?

You're so close!

Try this question: what is the new potential ##V'## at infinity now?