Uncertainty and double slits

kof9595995

1. Homework Statement
Show that in order to be able to determine through which slit a double slit system each photon passes without destroying the double silt diffraction pattern, the condition
$$\delta y\delta {p_y} \ll \frac{h}{{4\pi }}$$
must be satisfied. Since this condition violated the uncertainty principle, it can not be met

2. Homework Equations
$$d\sin \theta = \lambda$$
$$\sin \theta = \frac{{\delta {p_y}}}{p}$$
$$\lambda = \frac{h}{p}$$

3. The Attempt at a Solution
In order not to destroy the pattern, the angle should not be large enough to shift one maximum to its adjacent maximum. So $$\sin \theta \ll \frac{\lambda }{d}$$, and then we have
$$\frac{{\delta {p_y}}}{p} \ll \frac{\lambda }{d}$$
To figure out which slit the photon passes through, we must have
$$\delta y \ll \frac{d}{2}$$
Combine these two and use de broglie's relation $$\lambda = \frac{h}{p}$$
We can get
$$\delta y\delta {p_y} \ll \frac{h}{2}$$

But it seems to me the extra $$2\pi$$ just comes out from nowhere. I'm really pulling my hair off on this quetsion
1. Homework Statement

2. Homework Equations

3. The Attempt at a Solution

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