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Welcome to our Physics lesson on Heisenberg's Uncertainty Principle, this is the fifth lesson of our suite of physics lessons covering the topic of Electromagnetic Wave Packet. The Uncertainty Principle, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
The state of a particle (for example an electron) is known when we have information at any instant for its position x, momentum p and energy E. For this, we have to take measurements. It is known that every measurement contains an error - systematic or casual it may be (systematic errors depend on the initial conditions of measurement or apparatus while casual errors depend more on the accuracy of the person who takes the measurement). In any case, we cannot pretend to take 100% accurate measurements, so it is obvious that we will have at least two kinds of uncertainties: of position (Δx) and of momentum (Δp), which are inevitable.
In 1927, Werner Heisenberg discovered one of the most important principles in quantum physics - the uncertainty principle which gives the relationship between the two aforementioned individual uncertainties. This principle is based on the rule according which:
"It is impossible to determine with absolute precision both the position and impulse of a particle at the same instant."
Moreover, this principle is not too much related to the accuracy of apparatuses; it would exist even if we had 100% accurate measuring devices. Heisenberg's Uncertainty Principle is a universal law of nature, as the universe has determined certain limits in the accuracy of measurements, beyond which it is impossible to go, despite the advancements in science.
The following relation gives the mathematical form of Heisenberg's Uncertainty Principle:
We can transform the above formula to give the relationship between two other uncertainties: that of particle's energy ΔE and the time interval Δτ during which the particle has a given energy. Thus, since any change momentum Δp is equal to the impulse of particle, we can write
where F is the force acting on the particle. When this force is multiplied to the displacement Δx it causes to the particle, we obtain the energy transferred to the object in the form of work done by the force. Hence, we have
The meaning of the above relation is as follows:
"It is impossible to determine with absolute precision both the energy of a particle and the instant at which this particle possesses this energy."
Now, let's see a couple of examples in which these two forms of uncertainty principle are applied.
An electron is moving in the X-direction at 3.6 × 105 m/s - a value which we have been able to measure at an accuracy of ± 1%. What is the maximum accuracy at which we can determine the electron's position?
Clues:
v = 3.6 × 105 m/s
Δv = 1% of v = 0.01 · 3.6 × 105 m/s = 3.6 × 103 m/s
(me = 9.1 × 10-31 kg)
ℏ = 1.055 × 10-34 J · s
Δx = ?
First, let's calculate the uncertainty of electron's momentum. We have
Now, let's calculate the maximum accuracy of position (which represents the uncertainty of electron's position Δx). We use the Heisenberg principle for this purpose.
Now, let's see an example where the uncertainty principle in terms of energy and time is applied.
The atom of a chemical element has a 15 eV energy in its excited state for 80 nanoseconds. Calculate the uncertainty of energy for this atom.
Clues:
E = 15 eV = 15 · 1.6 × 10-19 J = 2.4 × 10-18 J
Δτ = 80 ns = 80 × 10-9 s = 8 × 10-8 s
ΔE = ?
Given that
we have
This uncertainty of energy means the energy of atom ranges from E - ΔE to E + ΔE. The quantity ΔE is also known as the natural width of the energetic level E.
You have reached the end of Physics lesson 19.5.5 Heisenberg's Uncertainty Principle. There are 5 lessons in this physics tutorial covering Electromagnetic Wave Packet. The Uncertainty Principle, you can access all the lessons from this tutorial below.
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