Heisenberg Uncertainty Principle - Online Tutor, Practice Problems & Exam Prep

The Heisenberg uncertainty principle, formulated by physicist Werner Heisenberg, posits that the velocity and position of an electron cannot be precisely determined simultaneously. This principle arises from the dual wave-particle nature of electrons, a concept known as complementarity. The uncertainty in an electron's momentum (ΔP), which is its mass in motion, is expressed in kilograms times meters per second. The uncertainty principle formula is:

$\Delta X\cdot \Delta P\ge \frac{h}{4\pi }$

where ΔX is the uncertainty in position and h is Planck's constant (6.626 x 10^-34 Joule seconds), quantifies the limits of precision when measuring an electron's properties. This relationship underscores a fundamental limit to our knowledge of subatomic particles, impacting various fields of physics and chemistry.

Heisenberg Uncertainty Principle

Now, the physicist Werner Heisenberg theorized that the velocity and position of an electron cannot be calculated simultaneously, meaning that you might know the velocity or speed of an electron as it's traveling, but you won't know its position, and conversely, you might know where it's located, but you wouldn't know how fast it's moving.

Now, related to an electron behaving both as a wave and a particle is a reason for this issue. We're going to say here that the velocity or speed of an electron is related to its wave nature. Remember, light energy can move in the form of a wave, and we're going to say that the position of an electron is related to its particle nature.

Again, some say that light energy can be seen as just a cluster of particles known as photons. Now this relationship we call it complementarity, where electrons can be seen as either particles or waves, but not both simultaneously. And that's the reason why you can't know both the velocity and position of an electron at the same exact time.

Heisenberg Uncertainty Principle

With the inability to determine both the velocity and position of an electron comes the Heisenberg uncertainty principle. Now here we're going to say that it can be broken down in terms of uncertainty and momentum, and then the uncertainty principle formula itself. Now we're going to say momentum can be described as mass in motion. Here we're going to say that ΔP equals our uncertainty and momentum, and here it's in units of kilograms times meters over seconds.

We're going to say here that our uncertainty in position ΔP = m × ΔV here is just the mass in kilograms of our electron. And then V here represents the uncertainty and it will be ΔV the uncertainty in our velocity. Now with this we go on to the uncertainty principle formula itself. Now here it's going to say it's used when given the uncertainties in position and momentum. So here it's going to be that ΔX which is our uncertainty in position in terms of meters times ΔP which we just said is the uncertainty in our momentum.

Now remember this can be further expanded by substituting in mass times uncertainty and velocity. And then H is Planck's constant which is 6.626 × 10 - 34 J × s. Now one additional thing. Remember that one Joule is equal to kilograms times meter squared over second squared. That would mean that joules times seconds equals KG times meters squared over second squared times S, so one second. We'll cancel out with one second here. So this is equivalent to also saying kilograms times meters squared over seconds. So keep that in mind when it comes to Planck's constant. The units can either be in joules times seconds or kilograms times meters squared over seconds.

Heisenberg Uncertainty Principle Example 1

In this example question, it states calculate the uncertainty and velocity of a neutron if the uncertainty in its position is 712 picometers. Here we're told the mass of the neutron is 1.67510 * 10^-27 kilograms. All right, so we're going to be utilizing the Heisenberg uncertainty principle to answer this question. Now remember that it's going to be that the uncertainty in our position, which is delta X times the mass of our object, in this case the neutron, times the uncertainty in its velocity, will be greater than or equal to Planck's constant divided by 4 * π.

All we have to do here is fill in the information we know. Let's see. So we know what the uncertainty in the position is at 712 picometers. But remember, we need the units of length to be in meters, not picometers. So we're going to do a conversion here. We have 712 picometers. Remember 1 pico is 10^-12, so this is equal to 7.112 * 10^-10 meters. So this will be our delta X. Next we're going to say that M represents the mass of our neutron, which were given as 1.67510 * 10^-27 kilograms. And then that's going to be times our uncertainty and velocity, which we don't know this would be greater than or equal to.

Remember Planck's constant here, 6.626 * 10^-34? Remember the units here are Joules times seconds. Joules itself is equal to kilograms times meter squared over second squared, and that's going to be multiplying by seconds. So one of these seconds cancels out with this second. So the units here will really be kilograms times meter squared over seconds. That's going to be divided by 4 * π. Now what we're going to do here is we're going to try to simplify all of this for ourselves. We're going to multiply these two together. When we multiply them together, that's going to give me 1.1926712 * 10^-36. Here it's going to be kilograms times meters, and the only reason I'm doing it in that order is because Planck's constant puts kilograms of four meters squared, and that's going to be times our unknown or uncertainty in velocity, which will be greater than or equal to.

So divide now Planck's constant by 4 Pi. When we do that, that's going to give me 5.2728 * 10^-35 kilograms times meter squared over seconds. Finally, we want to isolate the uncertainty in our velocity. So divide both sides now by this 1.1926712 * 10^-36 kilograms times meters on both sides. So when we do that here, this cancels out, and then over here kilograms cancel out with kilograms. This meter cancels out with one of these meters, which at the end gives us what? Meters or seconds? Which makes sense because those are the customary units for velocity. So we're going to say here, when we punch that into our calculator, the uncertainty in our velocity will be greater than or equal to 44.2 meters per second. This would be our final answer to this example question.

To what uncertainty (in m) can the position of a baseball traveling at 51.0 m/s be measured if the uncertainty of its speed is 0.12%? The mass of the baseball is 150 g.

• A. 4.2 x 10^-33 m
• B. 5.7 x 10^-33 m
• C. 5.9 x 10^-33 m
• D. 4.8 x 10^-33 m

An electron with a mass of 9.11 x 10^-31 kg has an uncertainty in its position of 630 pm. What is the uncertainty in its velocity?

• A. 9.0 x 10³ m/s
• B. 5.9 x 10³ m/s
• C. 9.2 x 10⁴ m/s
• D. 7.8 x 10⁴ m/s

A proton with a mass of 1.67 x 10^-27 kg traveling at 4.7 x 10⁵ m/s has an uncertainty in its velocity of 1.77 x 10⁵ m/s. Determine its uncertainty in position.

• A. 1.78 x 10 ^-13 m
• B. 3.56 x 10^-12 m
• C. 2.50 x 10^-13 m
• D. 4.24 x 10^-12 m

Here’s what students ask on this topic:

What is the Heisenberg uncertainty principle?

The Heisenberg Uncertainty Principle is a fundamental concept in quantum mechanics, introduced by Werner Heisenberg in 1927. It states that there is a limit to how precisely we can simultaneously know certain pairs of properties of a particle. The most commonly discussed pair of properties are position and momentum (momentum being mass times velocity). According to the principle, the more accurately we know a particle's position, the less accurately we can know its momentum, and vice versa.

This is not due to limitations in our measurement tools, but rather a fundamental property of the quantum world. The principle arises because particles at the quantum level exhibit both wave-like and particle-like properties. When you try to measure one property precisely, you inherently disturb the other, hence the uncertainty.

Mathematically, the principle is expressed as:

$\Delta x\cdot \Delta p\ge \frac{\hslash }{2}$

where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ℏ (h-bar) is the reduced Planck constant. This relationship shows that the product of the uncertainties is never smaller than the reduced Planck constant divided by two, indicating there's always some level of inherent uncertainty in our measurements at the quantum scale.

Which of the following is a consequence of the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle, a fundamental theory in quantum mechanics, states that it is impossible to simultaneously know both the exact position and exact momentum of a particle. One consequence of this principle is that the more precisely we determine the position of a particle, the less precisely we can know its momentum, and vice versa. This is not due to limitations in measurement or technology, but rather a fundamental property of the quantum world.

This inherent uncertainty has profound implications for our understanding of the micro-world. It means that at a subatomic level, particles do not have definite positions or velocities until they are measured. This principle also leads to the concept of quantum "fuzziness," where particles exist in a state of probability rather than certainty, and it underpins the behavior of atoms and subatomic particles, influencing their interactions and the structure of molecules.