Heisenberg’s Uncertainty Principle

The measurement of a particle’s variable has inherent uncertainty, according to Heisenberg’s uncertainty principle. The principle states that the more precisely the position is known, the more uncertain the momentum is, and vice versa. It is commonly applied to a particle’s position and momentum. This is in contrast to classical Newtonian physics, which states that with sufficient equipment, all particle variables can be measured to an arbitrary uncertainty.

What is Heisenberg’s Uncertainty Principle?

In quantum mechanics, the Heisenberg Uncertainty Principle explains why we cannot simultaneously measure multiple quantum variables. Until the advent of quantum theory, it was assumed that all factors of an object could be recognised with absolute precision at the same time. Newtonian physics sets no limits on how better methods and techniques might reduce measurement uncertainty, so it was theoretically possible to define all information with sufficient care and accuracy. Heisenberg boldly proposed that this precision has a lower limit, making our understanding of a particle highly unpredictable.

More specifically, knowing the precise momentum of a particle makes knowing the precise position impossible, and vice versa. This relationship also holds for energy and time in which a system’s precise energy cannot be measured in a finite amount of time. Heisenberg defined uncertainty in the products of conjugate pairs – momentum/position and energy/time, as having a minimum value equal to Planck’s constant, divided by 4. To put it another way:

Δp Δx ≥ h/4π (1)

Δt ΔE ≥ h/4π (2)

Where h is Planck’s constant and refers to the uncertainty in that variable.

Method

We must first determine the uncertainty in momentum p=mu and then invert equation 7.15, to find the uncertainty in position x=/(2p), given the uncertainty in speed u=1.0103m/s.

Solution

For an electron: 

Δ𝑝Δ𝑥=𝑚Δ𝑢=(9.1×10−31kg)(1.0×10−3m/s)=9.1×10−34kg·m/s, ℏ2Δ𝑝=5.8cm.

For a bowling ball:

Δ𝑝Δ𝑥=𝑚Δ𝑢=(6.0kg)(1.0×10−3m/s)=6.0×10−3kg·m/s, ℏ2Δ𝑝=8.8×10−33m.

Significance

The position uncertainty for the bowling ball is immeasurably small compared to that of the electron. Because Planck’s constant is so small, the uncertainty principle’s limitations are barely noticeable in macroscopic systems like a bowling ball.

What does this imply?

It’s difficult to imagine not understanding precisely where an atom is at any given time. Although it may appear intuitive that when a particle exists in space, we can point to where it is, Heisenberg’s uncertainty principle shows that this is not the case. This is due to the particle’s wave-like nature. A particle is dispersed throughout space, so it does not occupy a single precise location, but a range of roles. Similarly, because a particle is made up of a carton of waves, each with its own momentum, the momentum of a particle can only be described as a range of momentum at best.

Consider whether quantum variables could be precisely measured. According to de Broglie’s equation, a wave with a precisely measurable position is collapsed into a single point with an indefinite wavelength, and thus an indefinite momentum. A wave with a perfectly measurable momentum, on the other hand, has a wavelength that oscillates indefinitely overall space and thus has an indefinite position.

The same thought experiment could be done with energy and time. It would take ages to precisely measure a wave’s energy, whereas measuring a wave’s exact instance in space would require a single moment with infinite energy.

Conclusion

The Heisenberg Principle does have a significant impact on how scientific tests are designed and practised. Consider determining a particle’s momentum or position. To make a measurement, you must interact with the particle in some way that changes its other variables. A collision between an electron and another particle, such as a photon, is required to measure the position of an electron, for example. This will transfer some of the momentum of the second particle to the electron being measured, causing it to change. A particle with a relatively small wavelength and thus more energy would be required for a more precise assessment of the electron’s position. However, this would alter the dynamics even more during a collision.