Heisenberg Uncertainty Principle

A non-trivial result follows from wave packet equation, the product of the finite extent of the wave packet Delta x and the range of momentum Delta k ≡ Delta p/h chosen to construct the wave packet of the said extent is

\(\Delta x\Delta k=4\pi \Rightarrow \Delta x\Delta p=4\pi h\)

The upshot of equation is if we try to get smaller wave train to better describe a localized particle, we have to superpose matter waves of a wider range of Delta k implying imprecise knowledge of the particle’s momentum, while superposing too few matter waves in smaller ?k range will lead to greater uncertainty in the particle position ?x.

The above result is formally summarized in Heisenberg’s Uncertainty Principle:

\(\Delta x\Delta p \geq \frac{h}{2}\)

Heisenberg uncertainty relation imposes a restriction on the accuracy of simultaneous measurement of position and momentum the more precise our measurement of position is, the less accurate will be our momentum measurement and vice-versa. The physical origin of the uncertainty principle is with the quantum system, determination of position by performing measurement on the system disturbs it sufficiently to make the determination of momentum imprecise and vice-versa.

Heisenberg’s γ-ray Microscope

A striking thought experiment illustrating the uncertainty principle is Bohr’s / Heisenberg’s Gamma-ray microscope. To observe a particle, say an electron, we shine it with the light ray of wavelength λ and collect the Compton scattered light in an microscope objective whose diameter subtends an angle θ with the electron as shown in the figure below

The precision with which the electron can be located, Delta x, is defined by the resolving power of the microscope,

\(sin \theta =\frac{\lambda }{\Delta x}\Rightarrow \Delta x=\frac{\lambda }{sin \theta }\)

It appears that by making λ small, that is why we choose γ-ray, and by making sin θ large, Delta x can be made as small as desired. But, according to uncertainty principle, we can do so only at the expense of our knowledge of x-component of electron momentum.

In order to record the Compton scattered photon by the microscope, the photon must stay in the cone of angle θ and hence its x-component of the momentum can vary within ±(h/λ) sin θ. This implies, the magnitude of the recoil momentum of the electron is uncertain by

\(\Delta p_{x}=\frac{2h}{\lambda }sin \theta\)

The product of the uncertainty yields,

\(\Delta x\Delta p_{x}=\frac{\lambda }{sin \theta }\frac{2h}{\lambda }sin \theta =4\pi h\)

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