Heisenberg’s γ-ray Microscope

Heisenberg’s γ-ray Microscope is a very high-resolution microscope that uses high-energy gamma rays for illumination. However, this microscope does not exist at present, but Heisenberg imagined it could see and locate the position of an electron with great accuracy. This high-resolution microscope uses high-energy light, giving the electron a bigger kick. According to the uncertainty principle, it is clear that the more precisely we try to determine the position of the electron, the less precise or more uncertain will be its momentum. His theories are of great significance in physics, giving the discipline of quantum physics a new direction. 

What are Gamma Rays?

Gamma rays are the product of the radioactive disintegration of the atomic nuclei and subatomic particles. They have the shortest wavelength (shorter than X-rays) but have the highest energies. When these rays come in contact with cells it can kill them. Therefore, they are extensively used in killing cancer cells via radiation. 

Heisenberg’s γ-ray Microscope

In Heisenberg’s corrected version of the γ-ray microscope, a free electron will sit directly beneath the microscopes’ lens centre, forming a circular cone of angle 2A. Then, we shall illuminate the electron from the left by γ-rays. According to the principle of wave optics, a microscope is capable of resolving objects to size i.e. Δx. The expression of this relation is as follows:

Δx = L / (2sinA)

In quantum mechanics, gamma rays striking the electron move it as they tend to behave like particles. On the diffraction of an electron to the microscope’s lens, the electron deflects to the right. For the observation, gamma rays need to scatter at any angle within the cone of angle 2A. The equation is as follows:

p = h / L,

h = Planck’s constant (value= 6.62607015 × 10−34 joule second)

p’x = (h sinA ) / L’,

L’ =wavelength of the deflected gamma-ray.

In an extreme case, when gamma rays recoil backward on hitting the left edge of the microscope’s lens. The expression for total momentum shall be:

p”x – (h sinA ) / L”.

The final momentum is always equal to the initial momentum according to the law of conservation of momentum. Thus, the expression for the final momentum shall be:

p’x = (h sinA ) / L’ will be equal to the p”x – (h sinA ) / L”

Assume A as small, then we have:

p”x – p’x = Δpx ~ 2h sinA / L [L’ ~ L” ~ L]

As mentioned before, Δx = L / (2 sinA ), we get the reciprocal relationship between the minimum uncertainty and the uncertainty in its momentum along the x-axis. In the direction x as:

Δpx ~ h / Δx or Δpx Δx ~ h.

We can add the greater than sign for the uncertainties more than the minimum.

What Is Heisenberg’s Uncertainty Principle?

Heisenberg, a German Physicist in 1927 put forward the famous uncertainty principle. This principle states that it is impossible to determine the position and momentum of any particle with absolute accuracy or without any uncertainty, at the same time. However, this principle is only applicable for microscopic particles such as electrons, protons or neutrons and not for macroscopic particles such as automobiles. The principle is based on the wave-particle duality of matter that states that matter exists as both particles and waves.

Δ p. Δ x ≥ h/4π.

Δ t ΔE ≥ h/4π

Where h= Planck’s constant

Δ= Uncertainty.

Importance of Heisenberg γ-ray Microscope

The Heisenberg γ-ray Microscope’s significance lies in the determination of the exact position of an electron by using the high-energy gamma rays.

Solved Question

Question: Suppose an electron is moving at a speed of 40m/s with uncertainty in the momentum (Δp) is 10−6 of its momentum. Determine the uncertainty in its position (Δx) using Heisenberg’s uncertainty formula. [Mass of the electron= 9.1×10−31 kg].

Solution:

According to the question:

v = 40m/s

m = 9.1×10−31 kg

 h = 6.626×10−34 Js

 Δp = 10−6

Applying the formula as P = m × v

P = 9.1×10−31×40

P = 364×10−31 kgm/s or

Δp = 364×10−37

According to Heisenberg’s uncertainty formula,

Δ x. Δ p ≥ h/4π

Δ x≥ h/4π Δ p      

Δ x≥1.44 m

Conclusion

Heisenberg’s discoveries have great importance in quantum physics along with Schrodinger’s theories. The uncertainty principle states that it is impossible to accurately determine both the position and momentum of any particle simultaneously. As such, he proposed a γ-ray Microscope, for the determination of the accurate position of an electron by using high-energy gamma rays with a microscope. The rays produce the disintegration and decay of radioactive elements. They have the shortest wavelength (shorter than X-rays) but the highest energies.