Which of the following is the most correct expression for Heisenberg's uncerainty principle?

Options

A. `trianglex.trianglep=(h)/(4pi)`
B. `trianglex.trianglepge(h)/(4pi)`
C. `trianglex.triangleple(h)/(4pi)`
D. `trianglex.trianglev=(h)/(4pi)`

Answer

To solve the question regarding Heisenberg's uncertainty principle, we will follow these steps:

### Step 1: Understand Heisenberg's Uncertainty Principle
Heisenberg's uncertainty principle states that it is impossible to simultaneously know both the position and momentum of a particle with absolute precision. The more accurately we know one of these quantities, the less accurately we can know the other.

### Step 2: Identify the Mathematical Expression
The principle can be mathematically expressed as:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where:
- \(\Delta x\) is the uncertainty in position,
- \(\Delta p\) is the uncertainty in momentum,
- \(h\) is Planck's constant.

### Step 3: Relate Momentum to Velocity
Momentum \(p\) is defined as:
\[ p = mv \]
where \(m\) is the mass and \(v\) is the velocity of the particle. Therefore, the uncertainty in momentum can be expressed as:
\[ \Delta p = m \cdot \Delta v \]
where \(\Delta v\) is the uncertainty in velocity.

### Step 4: Substitute into the Uncertainty Principle
Substituting \(\Delta p\) into the uncertainty principle gives us:
\[ \Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi} \]

### Step 5: Analyze the Given Options
Now, we need to analyze the provided options to determine which one correctly represents the Heisenberg uncertainty principle:
1. If an option shows only an equality (e.g., \(\Delta x \cdot \Delta p = \frac{h}{4\pi}\)), it is incorrect because the principle states an inequality.
2. If an option states \(\Delta x \cdot \Delta p < \frac{h}{4\pi}\), it is also incorrect for the same reason.
3. If an option omits mass when discussing momentum, it is incorrect.

### Conclusion
After analyzing the options based on the steps above, the correct expression for Heisenberg's uncertainty principle is:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
Thus, the correct answer is option B.

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