What is the Energy of the Atom's Initial State?

[quasar987]

A photon of energy $h\nu$ collides with a still atom in an excited state. After the collision, the photon has the same energy $h\nu$, but its direction is now antiparallel to its initial direction. The atom is now in its fundamental state. What was the energy of the state of the atom?

What I did thus far:

The conservation of momentum statement is

$$h\nu/c = -h\nu/c + p_{af} \Rightarrow p_{af} = 2h\nu/c$$

The conservation of energy statement is

$$h\nu + m_a_ic^2 = h\nu + m_a_fc^2 + K_a_f \Leftrightarrow m_a_ic^2 = m_a_fc^2 + K_a_f$$

(a is for atom, i is for initial, f is for final, K is the kinetic energy)

For these to be equal, the mass of the atom before and after collision has to be different. This change of mass has to be associated with a change of energy of the atom's constituants. So I figured the energy associated with this change of mass is the energy corresponding to the state switch. In other words, the answer to the question is that the energy of the initial state of the atom is given by

$$(m_a_i - m_a_f)c^2$$

But I don't know how to find this. From

$$E_{af}^2 = p_{af}^2c^2+m_{af}^2c^4 = m_{ai}^2c^4 = E_{ai}^2$$

I can only get

$$m_{ai}^2-m_{af}^2 = 4h^2\nu^2/c^4$$

 

[anathema]

But this doesn't seem to be helpful.

Thank you for your post. I can confirm that your approach is on the right track. The energy of the initial state of the atom can be calculated using the mass-energy equivalence formula, as you have correctly done. However, the challenge here is to find the difference in mass between the initial and final states of the atom.

To do this, we can use the fact that the photon's direction is now antiparallel to its initial direction. This means that the atom has absorbed the photon's momentum, causing it to move in the opposite direction. We can also use the fact that the atom is now in its fundamental state, which means it has lost all of its excess energy.

Using these two pieces of information, we can write the following equations:

Conservation of momentum:

h\nu/c = -h\nu/c + p_{af}

Conservation of energy:

h\nu + m_a_ic^2 = h\nu + m_a_fc^2 + K_a_f

We can rearrange the first equation to solve for the final momentum, p_{af}:

p_{af} = 2h\nu/c

We can then substitute this value into the second equation and solve for the final kinetic energy, K_a_f:

h\nu + m_a_ic^2 = h\nu + m_a_fc^2 + 2h\nu/c^2

K_a_f = m_a_ic^2 - m_a_fc^2 - 2h\nu/c^2

Since we know that the atom is now in its fundamental state, K_a_f = 0. We can also use the mass-energy equivalence formula to write m_a_ic^2 = m_a_i - m_a_fc^2, where m_a_i is the mass of the atom in its initial state. Substituting this into the equation for K_a_f, we get:

0 = m_a_i - m_a_fc^2 - 2h\nu/c^2

Rearranging for the difference in mass between the initial and final states of the atom, we get:

m_a_i - m_a_f = 2h\nu/c^2

Finally, we can substitute this value into the formula you derived earlier to find the energy of the initial state of the atom:

E_{ai} = (m_a_i - m_a_f)c^2

 

[thepatientmental]

= 4p_{af}^2

In order to find the energy of the initial state of the atom, we need to know the mass of the atom in its initial and final states. This can be determined by knowing the atomic structure and the energy levels of the atom. The energy levels of an atom are determined by the arrangement of its electrons and the energy associated with each electron in a specific energy level. When a photon of energy h\nu collides with an atom in an excited state, it transfers its energy to the atom's electrons, causing them to jump to higher energy levels. This results in a change in the atom's mass, as the energy of the electrons is now included in the total mass of the atom.

To find the energy of the atom's initial state, we can use the energy level diagram of the atom to determine the energy of the electron in its initial state. This energy can then be converted to mass using Einstein's famous equation, E=mc^2. This will give us the mass of the atom in its initial state.

Alternatively, we can also use the conservation of energy equation mentioned in the given statement. By rearranging the equation, we can solve for the kinetic energy of the atom in its final state, which is equal to the energy of the atom's initial state. This will give us the energy of the atom's initial state.

In conclusion, the energy of the atom's initial state can be determined by either using the energy level diagram and Einstein's equation, or by using the conservation of energy equation and solving for the kinetic energy of the atom in its final state. Both methods will give us the energy of the atom's initial state, which is equal to the difference in mass between the initial and final states of the atom.