Built into quantum physics is a remarkable limitation on the information that exists in nature about certain observable quantities that characterize a physical system. The origin of the limitation comes from the fact that in quantum physics, observable quantities are represented by operators, things that perform some mathematical operation on the state function Ψ of the system. If the order in which two operators are applied to the same state makes a difference, the operators are said not to commute, and an uncertainty relation exists for the physical quantities represented by the operators. The operators for position, r, and momentum, p, do not commute, and that fact limits the information that can simultaneously exist in a physical state about the values of those quantities. In one dimension we can write

where ℏ is Planck's constant divided by 2π and the Δ quantities represent uncertainty in the observable. This has nothing to do with experimental error or measurement uncertainty, it involves the actual physical system not possessing a definite value for the observable.

18th and 19th Century optical pioneers used pinholes, not slits.

An electron or photon in a state with well-defined momentum in the direction toward a barrier with a hole in it will have a momentum uncertainty afterward, if it passes through the hole, because the fact that it passed through the hole constitutes a measure of its position. The accuracy with which the size of the hole determines the position will result in a corresponding destruction of the original momentum information. The momentum vector can point in many different possible directions, so if we detect the particles on a screen that glows when they hit it, we will see a broad distribution. The smaller the hole, the broader the distribution.

An analogous relation exists for Fourier transforms of wave number k versus position x. Since k = 2π/λ the relation between the Fourier conjugate variables that leads to Δ k × Δ x ≃ 2π agrees with the momentum-position uncertainty relation, since p = h/λ = ℏk.

Remember the de Broglie relation λ = h/p. Since in general quantum particles do not have a definite momentum, the concept of wavelength is in general useless. Ψ is NOT a "wave function."

A similar relation exists between the energy possessed by a system and the time required for processes occurring within the system. If no processes or changes can happen in the system, then it has a definite energy E. But if there is a process that can occur within time τ, the energy is uncertain by roughly the amount ΔE ≈ ℏ/τ.  This happens because in Fourier analysis, the time and frequency are conjugate variables, related by Δω × Δt ≃ 2π, and E = ℏω. Another way to understand how this relation arises is to consider that the operator for energy is iℏ(∂/∂t) while the operator for time is just the time t. Clearly t and the derivative with respect to t do not commute!

In high energy physics, which studies fundamental particles and simple systems made of them, it is not unusual for a given system to exist for only a very short time, such as 10^-23 seconds. As a result of τ being so small, the mass of the particle is given by a fairly broad distribution. The quoted mass is the value of the peak of the distribution, with the width of the distribution being given roughly by ΔM ≈ ℏ/(c²τ).

A thin barrier which can be penetrated 50% of the time!

A remarkable example of these phenomena is seen in the penetration of a quantum particle through a barrier. Suppose a free particle encounters a region of finite width where the potential energy abruptly changes from 0 to U_o. Since the total energy is E = K + U, if U is greater than K, then since E does not change, K must be negative when the particle is inside the potential region, which is classically impossible. However, quantum states of negative kinetic energy do exist, although the probability amplitude falls off exponentially with distance. Basically inside the barrier the wave number  k looks more like iκ, where κ is real. So inside the barrier e^ikx becomes e^−κx. Thus, if the barrier is thin enough, or E is close enough to U, the probability amplitude converts to a reflected part and a substantial transmitted part, the total probability still summing to the original value.  This is often called quantum tunneling.