How can I calculate the ionization energy of isotopes using the Bohr model?

To get the ionization energy of hydrogen (with the use of Bohr’s model of atom), I can use formula
$$
E = frac{-mu e^4}{32pi^2epsilon^2_0hbar^2n^2}
$$

where $mu=m_pm_e/(m_p+m_e)$ is reduced mass and $n = 1$. This gives me the correct answer ≈ -13.598 eV.

I want to calculate the ionization energy of deuterium and I thought the only difference is in reduced mass because of the neutron. So I just changed $m_p$ to $m_d$. However, this changes the answer to ≈ -13,602, which is incorrect.

I think that all the input values are correct, so I guess the problem is with the formula.

Are there any modifications of the formula needed to calculate the ionisation energies of isotopes?

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How can I calculate the ionization energy of deuterium?

To get the ionization energy of hydrogen (with the use of Bohr’s model of atom), I can use formula
$$
E = -mu e^4/32pi^2epsilon^2_0hbar^2n^2
$$

where $mu=m_pm_e/(m_p+m_e)$ is reduced mass and $n = 1$. This gives me the correct answer ≈ -13.598 eV.

I want to calculate the ionization energy of deuterium and I thought the only difference is in reduced mass because of the neutron. So I just changed $m_p$ to $m_d$. However, this changes the answer to ≈ -13,602, which is incorrect.

I think that all the input values are correct, so I guess the problem is with the formula.

How can I get the correct answer?

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Violation of energy conservation during collisions of a particle with different sections of a rod

Imagine a homogenous rod with total mass $M$ and length $l$ floating in free space without any force or constraint acting on it. Then, think about two possible scenarios. In the first, a particle with lineal momentum $mv$ impacts right at the center of the rod. In the second, the same particle with same lineal momentum impacts right at one of the ends of the rod.

First case

In the first case there will be no rotation, only translation. We apply conservation of momentum (where I think there are two cases, one with the particle having final lineal momentum $v_f$ and the other with $-v_f$, but i will omit the first one).

$$mv = MV – mv_f quad quad V= frac{m}{M}(v+v_f)$$

Then the final energy is:

$$E_f = frac{1}{2} MV^2 + frac{1}{2} mv_f^2 = frac{m^2}{2M} (v+v_f)^2 + frac{1}{2} mv_f^2$$

Second case

In the second case there will only be rotation (or so I think, but I’m getting really confused so that’s why I’ve come up with this scenarios to try to prove it) without translation. Again we apply conservation of momentum:

$$mv = MV – mv_f quad quad V= frac{m}{M}(v+v_f)$$

Let me explain what $V$ represents in this case. Since the rod rotates all the points in it don’t have the same velocity, but, since the velocity increases linearly with the radius and the rod is homogenous, if the velocity of the ends is $2V$ (which I compell it to be this way) then the average speed of thte whole rod is $V$. Then we proceed (the cross product of $L$ doesn’t matter):

$$I omega = L = pr = MV l/4 = frac{ml(v+v_f)}{4}$$

So the final energy of the whole system is (having $I=frac{1}{12}Ml^2$):

$$E_f= frac{1}{2} frac{(Iomega)^2}{I} + frac{1}{2} mv_f^2 = frac{3m^2}{8M} (v+v_f)^2 + frac{1}{2} mv_f^2$$

So comparing the two cases, there’s a difference in the final energy, specifically a difference of $frac{m^2}{8M} (v+v_f)^2$, which doesn’t make sense.

Conclusions

This difference can mean various things, and I don’t know which of this ones is, or even if it implies something else I haven’t thought about. But before, there’s something I don’t understand and then there would be no need of other explanations.

As I’ve said the average velocity of the rod is $V$ so why should it have different speed when rotating than if the whole rod is advancing? In the end, the product mass times velocity is the same. So that’s one thing I don’t understand.

If that’s not the case and the kinetic energy is different for some reason here’s the explanations I’ve thought about for this difference in the energies:

  1. In the second case there’s also kinetic energy in the form of translation of the rod, and not only rotation, specifically with a translational kinetic energy of $frac{m^2}{8M} (v+v_f)^2$, but why this specific quantity?
  2. Actually, apart from the rotation, the rod has the same translational kinetic energy as in the first case, and $v_f$ in the second case is what causes the difference, being much smaller, so in this case the rod would have much more kinetic energy in the end when having the particle impact at one of the ends than having it impact at the very center.
  3. There’s no translation, which would agree with my hypothesis, and $v_f$ would be bigger in the second case than in the first.

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Arranging coefficients in a derivation of the Casimir energy

I’m working on the derivation of the Casimir energy from quantum field theory. From the K-G equation (with $c=1$ and $hbar=1)$ I found the vacuum energy:

$$langle 0|H|0rangle=E_{vac}=Vint_{-infty}^{+infty} d^3p frac{1}{2}E_p$$

where $V$ is an arbitrary large volume (hence $E_{vac}$ is infinite). However, when trying to calculate the vaccum energy for a region between plates separated by a distance $a$, I had trouble finding a solution to the K-G equation with boundary conditions. We have the K-G equation to be

$$(partial^2_t-nabla^2+m^2)psi=0 , .$$

Which has the general solution, in terms of the creation and anhiquilation operators:

$$psi (x,t)=int d^3p frac{1}{sqrt{2E_p}}(hat{a}e^{-ipcdot x}+hat{a}^dagger e^{+ipcdot x}) , .$$

For simplicity, let’s suppose the two plates are situation in $x_1=0$ and $x_1=L$, so our field must satisfy $psi(x_1=0)=psi(x_1=L)=0$. This leads to
begin{align}
psi(x=0,t)
&= int d^3p frac{1}{sqrt{2E_p}}(hat{a}e^{-ip_| cdot x_|}+hat{a}^dagger e^{+ip_|cdot x_|})=0 \
&= int d^3p frac{1}{sqrt{2E_p}}(hat{a}e^{-ip_| cdot x_|}e^{-ip_1L}+hat{a}^dagger e^{+ip_|cdot x_|}e^{ip_1L})=0 , .
end{align}

Where I introduced the notation $p_|cdot x_|=E_pt-p_2x_2-p_3x_3$ to denote the directions parallel to the plates (not sure if this helps). However, I cannot find a way to arrange the coefficients so I can get an expression (out of intuition I think some sinus should appear, but I can’t justify it). How can I do that?

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Derivation of the Casimir Energy

I’m working on the derivation of the Casimir energy from quantum field theory. From the K-G equation (with $c=1$ and $hbar=1)$ I found the vacuum energy:

$langle 0|H|0rangle=E_{vac}=Vint_{-infty}^{+infty} d^3p frac{1}{2}E_p$

Where $V$ is an arbitrary large volume (hence $E_{vac}$ is infinite). However, when trying to calculate the vaccum energy for a region between plates separated by a distance $a$, I had trouble finding a solution to the K-G equation with boundary conditions. We have the K-G equation to be:

$(partial^2_t-nabla^2+m^2)psi=0$

Which has the general solution, in terms of the creation and anhiquilation operators:

$psi (x,t)=int d^3p frac{1}{sqrt{2E_p}}(hat{a}e^{-ipcdot x}+hat{a}^dagger e^{+ipcdot x})$

For simplicity, let’s suppose the two plates are situation in $x_1=0$ and $x_1=L$, so our field must satisfy $psi(x_1=0)=psi(x_1=L)=0$. This leads to:

$psi(x=0,t)=int d^3p frac{1}{sqrt{2E_p}}(hat{a}e^{-ip_| cdot x_|}+hat{a}^dagger e^{+ip_|cdot x_|})=0$

$psi(x=0,t)=int d^3p frac{1}{sqrt{2E_p}}(hat{a}e^{-ip_| cdot x_|}e^{-ip_1L}+hat{a}^dagger e^{+ip_|cdot x_|}e^{ip_1L})=0$

Where I introduced the notation $p_|cdot x_|=E_pt-p_2x_2-p_3x_3$ to denote the directions parallel to the plates (not sure if this helps). However, I cannot find a way to arrange the coefficients so I can get an expression (out of intuition I think some sinus should appear, but I can’t justify it). Do you have any idea or tips on how to proceed, or any book where I can find the full derivation?

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Connection between Noether’s Theorem and classical definitions of energy / momentum

In classical mechanics, change in momentum $Delta mathbf p$ and change in kinetic energy $Delta T$ of a particle are defined as follows in terms of the net force acting on the particle $mathbf F_text{net}$, where in each case the integrations are done over the path taken by the particle through spacetime.
$$begin{align}
Delta T &= int mathbf F_text{net} cdot dmathbf x \
Delta mathbf p &= int mathbf F_text{net} dt
end{align}$$

This suggests some sort of correspondence.

$$begin{align}
mathbf x &longleftrightarrow T \
t & longleftrightarrow mathbf p
end{align}$$

Noether’s theorem provides an association between physical symmetries and conserved quantities.

$$begin{align}
text{symmetry in time} &longleftrightarrow text{conservation of energy} \
text{symmetry in position} &longleftrightarrow text{conservation of momentum}
end{align}$$

Additionally, when studying special relativity, there is a similar suggested correspondence between the components of the position four-vector $mathbf X$ and the energy-momentum four-vector $mathbf P$. Here, $E$ represents total energy $E = mc^2 + T + mathcal O left( v^3/c^3 right)$

$$begin{array}
mathbf X = begin{bmatrix}
ct \ x \ y \ z
end{bmatrix}
&
mathbf P = begin{bmatrix}
E/c \ p_x \ p_y \ p_z
end{bmatrix}
end{array}$$

Thus, comparing components, and discarding factors of $c$ the following correspondence is suggested.
$$
begin{align}
t &longleftrightarrow E \
mathbf x &longleftrightarrow mathbf p
end{align}
$$

Is there an underlying relationship between these correspondences I have pointed out? And, if so, why are the ones for the classical definitions of kinetic energy and momentum swapped compared to the ones arising from Noether’s theorem and special relativity?

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Theoretical Photon Energy Converter

So I’m creating a theoretical photon energy thingy where you shoot photons into a supercooled cloud of rubidium and they clump together and act like atoms in a physical form (there an MIT study on it), anyway I’m trying to figure out how much energy I can get out of it, so here’s what I know,

take x-rays with a frequency of 3×10^17

how much energy does each photon have, and how can I calculate how many photons I need to reach a certain number of watts and/or joules, also how would I figure out how to produce that number of photons???

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How to measure the probability off having no level between two energy intervals? [on hold]

I have two levels E and E+S where the interval I=(E,E+S). I divide this interval I into m parts and want to calculate the probability of having no level in any of these m parts. How can I do that? The notes that I am following provide this information

I = (E, E + S) = (E, E + S/m] U (E + S/m, E + 2S/m] U . . .
U (E + (m − 2)S/m, E + (m − 1)S/m] U (E + (m − 1)S/m, E + S)

They also have mentioned that levels are independent so probability of having no level in (E,E+S) is product of probabilities. Using this assumption they have derived this formula

`p(no level in I)= (1-ro*S/m)^m`

in the limit of large m

`p(np level in I) = exp(-ro*S)`

I am looking how they get above two equations. I want to understand this.

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If a single electron fills a degenerate energy level, does it fill all degenerate orbitals at once?

For example, if a single electron fills the 2p orbitals. Since all orbitals are degenerate, there is no reason why there would be any preference for px, py, pz. Is the electron hence in a superposition of these 3 orbitals?

And what if we consider hydrogen for n=2, where 2s, 2px, 2py and 2pz are degenerate?

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Electric potential energy between three point charges [on hold]

The Question:

Consider the group of three $4.6 text{ nC}$ point charges shown in
the figure. What is the electric potential energy of this system of charges relative to
infinity?

enter image description here

My Attempt:

I used the equation $U=frac{q_0}{4piepsilon_0} cdot left ( sum_{i=1}^n frac{q_i}{r_i} right )$.

$$U=frac{4.6cdot 10^{-9}}{4piepsilon_0} cdot left ( frac{4.6 cdot 10^{-9}}{0.03} + frac{4.6 cdot 10^{-9}}{0.04}right )$$
$$=frac{left (4.6 cdot 10^{-9} right )^2}{4piepsilon_0} cdot left ( frac{1}{0.03} + frac{1}{0.04} right )$$
$$approx 1.109 cdot 10^{-5}$$

I can’t see why this is incorrect (the answer is $1.5 cdot 10^{-5}$) unless the equation does not apply here.

I also reread the text and found out that “relative to infinity” just means that $U=0$ for a very large distance $r$ between the point charges, but I don’t see how that would change my solution.

My question is: What is the potential energy of a system of charges, and how does it differ from the summation of potential energy between sets of charges?

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