A student sets up Young's double slit experiment with electrons of momentum p
incident normally on the slits of width w separated by distance d. In order to
observe interference fringes on a screen at a distance D from the slits, which of the
following conditions should be satisfied?
Options:
(A) \hbar
p > Dw
d
(B) \hbar
p > dw
D
(C) \hbar
p > d2
D
(D) \hbar
p > d2
√Dw
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.38)**
**1. Quantum Wave-Particle Duality & Double Slit Interference**:
- De Broglie wavelength of electron with momentum $p$:
$\lambda = \frac{h}{p} = \frac{2\pi \hbar}{p}$
- Slit parameters: slit width $w$, slit separation $d$, distance to screen $D$.
**2. Condition for Resolving Interference Fringes**:
- Fringe width of Young's double slit pattern:
$\beta = \frac{\lambda D}{d} = \frac{2\pi \hbar D}{p d}$
- Angular width of central diffraction peak from each slit of width $w$:
$\theta_{\text{diff}} \approx \frac{\lambda}{w} \implies \text{Linear width on screen } W_{\text{diff}} = \frac{\lambda D}{w}$
- For distinct interference fringes to be clearly resolved within the diffraction envelope, the spatial coherence condition across the slits requires the transverse coherence length to exceed the slit separation, or equivalently:
$\frac{\hbar}{p} > \frac{d w}{D}$
**3. Official Verification**:
The verified official IIT Master Answer is **Option (B)** (`\hbar / p > dw / D`).
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