The Inability to Know the Exact Coordinate

and Impulse Simultaneously (The Uncertainty Principle)

If the matter really reveals the behavior of particle and wave, it's not possible to determine the impulse and coordinate simultaneously any more (in the classical description the impulse and coordinate are defined in each point). It means that if we want, for instance to determine the impulse more precisely we would definitely lose the accuracy of coordinate. .

If we send a particle through the slit Δx, we can make this slit Δx more narrow in order to more precisely determine the particle position.

The first minimum is when the angle θ = λ/Δx (interference condition). (photons of different wavelengths will create different interference pictures). The photon impulse in the x-direction is as follows:

Δpx = p · q = pλ/Δx

Since p = h/λ , we can obtain

Δx · Δpx≈ h

Heisenberg drew the same conclusion ideally watching an electron under his microscope:

One can see the light coming from the electron and under angle 2α on the lens is falling the impulse.

Δpx » p · sinα ≈ h/λ.d/2y (sinα = d/2y)

The exact position of electron is unknown according to the diffraction in the microscope objective (the limited resolution).

Δx ≈ 2y sinθ ≈ 2y · λ/d

Δx · Δpx » h

Note:
"The largest error" was revealed, for instance when there would the first diffraction minimum in the slit appear. If we'll more definitely determine the statistical error of the (optimal) distribution then we will obtain the following:

The Heisenberg Uncertainty principle Δx · Δpx ≥ h/4π = h/2

The magnitude h (one should speak h crossed) is only for short way of writing formulas in order to get rid of the factor 2π.The relation Δx · Δp ³ h/2 is obtained for optimal line shapes (Gauss) in the dependencies x and p.

Slit | Gauss line shape

Now we can object that we can get to know the impulse and location of photon because in experiment with the slit Δx we can divide it into small arbitrarily pieces and look at the screen where the particle hits the screen. The deviation of the particle from its initial direction shows the impulse change.

Nevertheless, I know only the coordinate x (relevant to one moment) and impulse p (relevant to another, more late moment) in this case. The uncertainty principle states that it's not possible to determine the coordinate and impulse simultaneously with accuracy better than h/2. But it gives the opportunity to determine the coordinate and impulse beforehand.

In general this principle isn't the expression of experimental insufficiency but the numerical expression of measurability. The uncertainty principle is based on the following: each measurement causes disturbances in the system principally and these disturbances are not determined in exact way. The uncertainty principle states that it's not possible to determine the coordinate and impulse simultaneously with accuracy better than h/2.

There are not only the relationship between p and x, i.e. the impulse and coordinate, but there is also the uncertainty principle for other quantities, such as energy and time:

ΔE · Δt ³ h/4π = h/2

where E = hν

Dn · Δt ≥ 1/4π

So, it states the lower border for frequency sharpness of light pulses.

One can see that the uncertainty principle shows there is no principal possibility to predict whether an electron (on the lower figure) ignite or neutralize the H-bomb !