Subtopic: Wavefunction, Probability Density & Uncertainty Principle
Question Statement:
An electromagnetic pulse has a pulse width of 10-3 s. The uncertainty in the
momentum of the corresponding photon is of the order of 10- kg m s-1, where N
is an integer. The value of N is ____________ (speed of light = 3 × 108 m s-1,
ℎ = 6.6 × 10-^34 J s)
Official Answer Key:
39 to 40
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.33)**
**1. Quantum Mechanics Uncertainty Principle**:
- For an electromagnetic wave pulse of duration $\Delta t$, the uncertainty in energy is given by:
$\Delta E \Delta t \sim \hbar \quad \text{or} \quad h$
- For photons, energy and momentum are related by $E = pc \implies \Delta E = c \Delta p$.
**2. Detailed Step-by-Step Derivation**:
- Substitute $\Delta E = c \Delta p$:
$c \Delta p \Delta t \sim h$
$\Delta p \sim \frac{h}{c \Delta t}$
- Substitute the numerical values:
- $h = 6.6 \times 10^{-34}$ J s
- $c = 3 \times 10^8$ m s$^{-1}$
- $\Delta t = 10^{-3}$ s
- Evaluating:
$\Delta p \sim \frac{6.6 \times 10^{-34}}{(3 \times 10^8)(10^{-3})} = \frac{6.6 \times 10^{-34}}{3 \times 10^5} = 2.2 \times 10^{-39} \text{ kg m s}^{-1}$
- The order of magnitude is $10^{-N}$ with integer $N = 39$ (or $N = 40$ using $\hbar/2$).
**3. Official Verification**:
The verified official IIT Master Answer range is **39 to 40**.
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