In the Sommerfeld's modification of Bohr's theory, the trajectory of an electron in a hydrogen atom is

Options

A. a perfect ellipse
B. a closed ellipse like curve, narrower at the perihelion position and flatter at the aphelion position
C. a closed loop on the spherical surface
D. a rosette

Answer

To solve the question regarding the trajectory of an electron in a hydrogen atom according to Sommerfeld's modification of Bohr's theory, we can follow these steps:

### Step 1: Understand Bohr's Theory
Bohr's original model of the hydrogen atom proposed that electrons move in circular orbits around the nucleus. However, this model had limitations, particularly in explaining the fine structure of spectral lines.

**Hint:** Recall that Bohr's model was limited to circular orbits and did not account for more complex motion.

### Step 2: Introduction to Sommerfeld's Modification
Sommerfeld modified Bohr's theory by introducing elliptical orbits. He proposed that electrons do not just move in circular paths but can also move in elliptical paths, which can explain the observed spectral lines more accurately.

**Hint:** Remember that Sommerfeld expanded on Bohr's model by allowing for elliptical orbits.

### Step 3: The Nature of Electron Motion
In Sommerfeld's model, the trajectory of an electron is described as a precessing ellipse. This means that while the electron moves in an elliptical orbit, the orientation of this ellipse changes over time, creating a rosette pattern.

**Hint:** Focus on the term "precessing ellipse" to understand the unique motion of the electron.

### Step 4: Identify the Correct Option
Given the options:
- A) Perfect eclipse
- B) Closed eclipse
- C) Closed loop on the spherical surface
- D) Rosette

Since Sommerfeld described the electron's path as a precessing ellipse, the correct answer is option D, which refers to the rosette pattern created by the precessing motion of the ellipse.

**Hint:** Look for keywords in the options that relate to the concept of precession and elliptical motion.

### Conclusion
Thus, the trajectory of an electron in a hydrogen atom according to Sommerfeld's modification of Bohr's theory is best described as a rosette.

**Final Answer:** Option D (Rosette) is correct.