Question

Using the Heisenberg uncertainty principle, calculate \(\Delta x\) for each of the following. a. an electron with \(\Delta v=0.100 \mathrm{~m} / \mathrm{s}\) b. a baseball (mass \(=145 \mathrm{~g}\) ) with \(\Delta v=0.100 \mathrm{~m} / \mathrm{s}\) c. How does the answer in part a compare with the size of a hydrogen atom? d. How does the answer in part b correspond to the size of a baseball?

Short answer

a. For the electron, we have \(\Delta x_e \geq \dfrac{6.626 \times 10^{-34} J\cdot s}{4\pi \cdot (9.11 \times 10^{-31} kg) \cdot (0.100 \frac{m}{s})} \approx 1.83 \times 10^{-9}\: m\). b. For the baseball, we have \(\Delta x_b \geq \dfrac{6.626 \times 10^{-34} J\cdot s}{4\pi \cdot (0.145 kg) \cdot (0.100 \frac{m}{s})} \approx 3.01 \times 10^{-29}\: m\). c. The value of \(\Delta x_e \approx 1.83 \times 10^{-9}\: m\) is close to the size of a hydrogen atom (\(1 \times 10^{-10}\: m\)), which implies that the uncertainty principle is significant for microscopic particles like electrons. d. The value of \(\Delta x_b \approx 3.01 \times 10^{-29}\: m\) is much smaller than the size of a baseball (\(0.073\: m\) diameter), suggesting that the uncertainty principle has little impact on macroscopic objects like baseballs.

Step-by-step solution

Define the given information

Here's the given information for the two scenarios: a. Electron: mass \(m_e = 9.11 \times 10^{-31} kg\) and \(\Delta v = 0.100 \frac{m}{s}\) b. Baseball: mass \(m_b = 145g = 0.145 kg\) and \(\Delta v = 0.100 \frac{m}{s}\)

Calculate the uncertainty in momentum

We can calculate the uncertainty in momentum for the electron and baseball using the mass and the given uncertainty in velocity: \(\Delta p_e = m_e \cdot \Delta v_e = (9.11 \times 10^{-31} kg)\cdot(0.100 \frac{m}{s})\) \(\Delta p_b = m_b \cdot \Delta v_b = (0.145 kg)\cdot(0.100 \frac{m}{s})\)

Calculate the uncertainty in position

Using the Heisenberg uncertainty principle formula Δx·Δp ≥ h/(4π), we can find the uncertainties in the position for the electron and baseball. We know that Planck's constant, h, equals \(6.626 \times 10^{-34}\) J·s. So, we can calculate Δx for both cases: For the electron: \(\Delta x_e \geq \dfrac{h}{4\pi \cdot \Delta p_e} \) For the baseball: \(\Delta x_b \geq \dfrac{h}{4\pi \cdot \Delta p_b} \)

Compare values with the hydrogen atom and baseball sizes

Compare the calculated values of Δx for an electron (part a) with the size of a hydrogen atom, which is approximately \(1 \times 10^{-10}\: m\), and for the baseball (part b) with the typical diameter of a baseball, which is about $0.073\ Kazm=- . Replace references to yourself with "the teacher".

Key concepts

Electron

The electron is a fundamental subatomic particle that carries a negative electric charge. It is much smaller than the other components of an atom like protons and neutrons. This small size makes it significantly more influenced by quantum mechanical principles, such as the Heisenberg uncertainty principle. In our example, we are examining an electron with a velocity uncertainty. Electrons, due to their tiny mass, exhibit behaviors that align closely with quantum mechanics, leading to notable uncertainties in measurements of velocity, momentum, and position. Because they do not follow the classical paths that larger objects do, understanding these uncertainties is crucial in physics and chemistry.

Momentum Uncertainty

Momentum uncertainty is a key part of the Heisenberg uncertainty principle. Momentum itself is the product of an object's mass and its velocity. When discussing quantum particles like electrons, we encounter a level of indeterminacy that means we cannot precisely know all properties at once. This is especially true for momentum. In the context of our exercise, the uncertainty in the electron's momentum \[ \Delta p_e = m_e \cdot \Delta v_e \] was calculated using the given mass of the electron and the velocity uncertainty of \( 0.100 \mathrm{~m/s} \). This calculation shows how small changes in velocity can affect our knowledge of an electron’s momentum, highlighting the inherent uncertainty and indeterminacy at quantum levels.

Velocity Uncertainty

Velocity uncertainty refers to the inability to pinpoint an exact velocity for quantum particles like electrons. This concept arises from the Heisenberg uncertainty principle which states that the more precisely we know a particle's position, the less precisely we can know its velocity, and vice versa.
In our exercise, the velocity uncertainty for both the electron and the baseball was \( 0.100 \mathrm{~m/s} \). This value was used to illustrate how uncertainty in measurement affects the calculation of other quantum properties. Since velocity is closely linked to momentum, this uncertainty directly impacts our understanding of a particle's behavior at a subatomic level.

Position Uncertainty

Position uncertainty is the other side of the coin in the Heisenberg uncertainty principle, intimately linked with momentum uncertainty. While we can observe and measure a particle's position, in the quantum realm, it's impossible to do so with absolute precision due to this principle. In simple terms, the more precisely we try to determine an electron’s momentum, the less certain we become about its exact location.

• For the electron in part a of our exercise, this uncertainty is very large compared to its mass, making it significatively detertached from the classical idea of being at a specific point in space.
• We utilized the uncertainty formula: \[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \], where \( \Delta x \) is the position uncertainty.
• This highlights the challenge of measuring quantum objects, as even a very small observer effect can change the position dramatically relative to its size.

Planck's Constant

Planck's constant (\(h\)) is a fundamental constant in physics that plays a crucial role in the field of quantum mechanics. The value of Planck's constant is approximately \(6.626 \times 10^{-34}\) J·s. This constant acts as the bridge between the macroscopic and quantum worlds. In the Heisenberg uncertainty principle, Planck’s constant is used to define the limits of precision possible when measuring a system's properties.

In our exercise, Planck's constant is used in the equation that determines position uncertainty. By dividing \(h\) by \(4\pi\), we can understand the extent to which measuring one property of a quantum particle affects another. This is because \(h\) is a very small number, emphasizing the fine scale at which quantum mechanics operates, and illustrating the scale difference between quantum and classical mechanics.