Choose the correct statement.

Options

A. Syrnmetry of nature w.r.t translation in time is equivalent to law of conservation of energy
B. Symmetry of nature w.r.t. translation in space is equivalent to law of conservation of linear momentum
C. Isotropy of space is equivalent lo law of conservation of angular momentum
D. All of these

Answer

To solve the question, we need to analyze the statements provided and determine which one is correct. The statements relate to fundamental principles in physics, particularly concerning conservation laws and symmetries.

### Step-by-Step Solution:

1. **Understanding the First Statement:**
- The first statement discusses the symmetry of nature with respect to translation in time, which is related to the law of conservation of energy.
- According to Noether's theorem, symmetries in physics correspond to conservation laws. The translation in time leads to the conservation of energy.
- Therefore, this statement is **correct**.

2. **Understanding the Second Statement:**
- The second statement mentions the relationship between momentum and position, expressed as \( \Delta p \Delta x \equiv \hbar \).
- This is a formulation of the Heisenberg Uncertainty Principle, which states that the more precisely the position (x) of a particle is known, the less precisely its momentum (p) can be known, and vice versa.
- Thus, this statement is also **correct**.

3. **Understanding the Third Statement:**
- The third statement relates isotropy of space to the conservation of angular momentum, expressed as \( \Delta L \Delta \theta \equiv \hbar \).
- Isotropy of space means that the laws of physics are the same in all directions. This symmetry leads to the conservation of angular momentum.
- Hence, this statement is also **correct**.

4. **Analyzing the Fourth Statement:**
- The fourth statement claims that all of the previous statements are correct.
- Since we have established that the first, second, and third statements are indeed correct, the fourth statement must also be **correct**.

### Conclusion:
All four statements are correct, making option four the right choice.

### Final Answer:
**All of these are correct.**

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