Which of the following is/are correct form of Heisenberg's indeterminacy principle?

Options

A. `DeltapxxDeltaxgeh/2`
B. `DeltavxxDeltaxgeh/(2m)`
C. `DeltaExxDetatgeh/2`
D. `DeltapxxDeltaEgeh/(2m)`

Answer

To solve the question regarding the correct forms of Heisenberg's indeterminacy principle, we will go through the key concepts and equations step by step.

### Step 1: Understanding Heisenberg's Uncertainty Principle
The Heisenberg Uncertainty Principle states that there is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously. Mathematically, it is 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 2: Relating Momentum to Velocity
Momentum (\(p\)) can be expressed in terms of mass (\(m\)) and velocity (\(v\)):

\[ p = mv \]

Thus, the uncertainty in momentum can be expressed as:

\[ \Delta p = m \cdot \Delta v \]

where \(\Delta v\) is the uncertainty in velocity.

### Step 3: Substituting into the Uncertainty Principle
Substituting \(\Delta p\) into the uncertainty principle gives:

\[ \Delta x \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi} \]

This can be rearranged to show that:

\[ m \cdot \Delta v \cdot \Delta x \geq \frac{h}{4\pi} \]

### Step 4: Energy-Time Uncertainty Relation
Another form of the uncertainty principle relates energy and time. The uncertainty in energy (\(\Delta E\)) and the uncertainty in time (\(\Delta t\)) can be expressed as:

\[ \Delta E \cdot \Delta t \geq \frac{h}{4\pi} \]

### Step 5: Summary of Forms
From the above steps, we can summarize the forms of Heisenberg's uncertainty principle as follows:

1. \(\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\)
2. \(m \cdot \Delta v \cdot \Delta x \geq \frac{h}{4\pi}\)
3. \(\Delta E \cdot \Delta t \geq \frac{h}{4\pi}\)

### Conclusion
The correct forms of Heisenberg's indeterminacy principle include the relationships between uncertainties in position and momentum, as well as energy and time.