Consider an excited hydrogen atom in state n moving with a velocity `upsilon(upsilon lt lt c)`. It emits a photon in the direction of its motion and changes its state to a lower state m. Apply momentum and energy conservation principle to calculate the frequency v of the emitted radiation, compare this with the frequency `v_(0)` emitted if the atom were at rest.

Explanation

To solve the problem of an excited hydrogen atom emitting a photon while moving, we will apply the principles of conservation of momentum and energy.

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Let the excited state of the hydrogen atom be represented by the quantum number \( n \).
- The atom is moving with a velocity \( \upsilon \) and emits a photon in the direction of its motion, transitioning to a lower energy state represented by the quantum number \( m \).

2. **Conservation of Momentum:**
- Before the emission of the photon, the total momentum of the system (atom + photon) is given by:
\[ p_{\text{initial}} = m \upsilon \]
- After the emission of the photon, the momentum of the atom becomes:
\[ p_{\text{final}} = m \upsilon' + \frac{E_{\text{photon}}}{c} \]
- Here, \( \upsilon' \) is the new velocity of the atom after emitting the photon, and \( E_{\text{photon}} = h \nu \) is the energy of the emitted photon, where \( \nu \) is the frequency of the emitted radiation.

3. **Apply Conservation of Momentum:**
- Setting the initial momentum equal to the final momentum:
\[ m \upsilon = m \upsilon' + \frac{h \nu}{c} \]

4. **Conservation of Energy:**
- The energy of the atom before the emission is given by:
\[ E_{\text{initial}} = E_n \]
- After the emission, the energy of the atom is:
\[ E_{\text{final}} = E_m + h \nu \]
- Here, \( E_n \) and \( E_m \) are the energies of the states \( n \) and \( m \) respectively.

5. **Apply Conservation of Energy:**
- Setting the initial energy equal to the final energy:
\[ E_n = E_m + h \nu \]

6. **Relate the Frequencies:**
- Rearranging the energy equation gives:
\[ h \nu = E_n - E_m \]
- Thus, the frequency of the emitted photon is:
\[ \nu = \frac{E_n - E_m}{h} \]

7. **Compare with the Rest Frame Frequency:**
- If the atom were at rest, the frequency \( \nu_0 \) of the emitted radiation would simply be:
\[ \nu_0 = \frac{E_n - E_m}{h} \]
- Since the atom is moving, we need to account for the Doppler effect. The frequency observed in the lab frame (moving frame) is given by:
\[ \nu' = \nu \left(1 + \frac{\upsilon}{c}\right) \]
- Therefore, the frequency of the emitted radiation when the atom is moving is:
\[ \nu = \nu_0 \left(1 + \frac{\upsilon}{c}\right) \]

### Final Result:
The frequency of the emitted radiation when the atom is moving is:
\[ \nu = \nu_0 \left(1 + \frac{\upsilon}{c}\right) \]