If a hydrogen atom at rest, emits a photon of wavelength `lambda`, the recoil speed of the atom of mass m is given by :

Options

A. `(h)/(lambda)`
B. `(mh)/(lambda`
C. `mhlambda`
D. `(hlambda)/(m)`

Answer

To solve the problem of finding the recoil speed of a hydrogen atom after it emits a photon, we can follow these steps:

### Step 1: Understand the Conservation of Momentum
When the hydrogen atom emits a photon, the total momentum before and after the emission must be conserved. Initially, the hydrogen atom is at rest, so the initial momentum is zero.

### Step 2: Set Up the Momentum Equation
Let:
- \( m \) be the mass of the hydrogen atom.
- \( v \) be the recoil speed of the hydrogen atom after emitting the photon.
- \( \lambda \) be the wavelength of the emitted photon.
- \( h \) be Planck's constant.

The momentum of the photon can be expressed using the de Broglie wavelength formula:
\[ p_{\text{photon}} = \frac{h}{\lambda} \]

Since the atom is at rest initially, its initial momentum is zero. After emitting the photon, the momentum of the hydrogen atom and the photon must equal zero:
\[ p_{\text{atom}} + p_{\text{photon}} = 0 \]
This can be rearranged to:
\[ p_{\text{atom}} = -p_{\text{photon}} \]

### Step 3: Substitute the Momentum of the Photon
Substituting the expression for the momentum of the photon into the equation gives:
\[ m v = -\frac{h}{\lambda} \]

### Step 4: Solve for the Recoil Speed \( v \)
To find the recoil speed \( v \), we rearrange the equation:
\[ v = -\frac{h}{m \lambda} \]
Since speed is a scalar quantity, we can ignore the negative sign (it indicates direction):
\[ v = \frac{h}{m \lambda} \]

### Conclusion
Thus, the recoil speed of the hydrogen atom after emitting a photon of wavelength \( \lambda \) is given by:
\[ v = \frac{h}{m \lambda} \]

### Final Answer
The correct option is:
**Option A: \( \frac{h}{m \lambda} \)**

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