Problem 51 - Chemistry: The Central Science in SI Units, Global Edition

Question

Using Heisenberg's uncertainty principle, calculate the uncertainty in the position of (a) a 1.50 -mg mosquito moving at a speed of \(1.40 \mathrm{~m} / \mathrm{s}\) if the speed is known to within \(\pm 0.01 \mathrm{~m} / \mathrm{s} ;\) (b) a proton moving at a speed of \((5.00 \pm 0.01) \times 10^{4} \mathrm{~m} / \mathrm{s}\) (The mass of a proton is given in the table of fundamental constants in the inside cover of the text.)

Step-by-Step Solution

Answer

The uncertainties in position for both the mosquito and the proton are: a) \(\Delta x_{\text{mosquito}}\) is at least approximately \(3.5 \times 10^{-26}\, \text{m}\) b) \(\Delta x_{\text{proton}}\) is at least approximately \(3.2 \times 10^{-13}\, \text{m}\)

Step 1: Data

Mass of mosquito: \(m = 1.50\,\text{mg} = 1.50 \times 10^{-6} \,\text{kg}\) Uncertainty in speed: \(\Delta v = 0.01\,\text{m/s}\)

Step 2: Calculate uncertainty in momentum (mosquito)

Using the formula \(\Delta p = m\Delta v\), we get: \[\Delta p_{\text{mosquito}} = (1.50 \times 10^{-6}\,\text{kg})(0.01\,\text{m/s}) = 1.50 \times 10^{-8}\,\text{kg m/s}\] b) Proton

Step 3: Data

Mass of proton: \(m_{\text{proton}}= 1.67 \times 10^{-27} \,\text{kg}\) Uncertainty in speed: \(\Delta v_{\text{proton}} = 0.01 \times 10^4\,\text{m/s}\)

Step 4: Calculate uncertainty in momentum (proton)

Using the formula \(\Delta p = m\Delta v\), we get: \[\Delta p_{\text{proton}} = (1.67 \times 10^{-27}\,\text{kg})(0.01 \times 10^4\,\text{m/s}) = 1.67 \times 10^{-22}\,\text{kg m/s}\] ##Step 2: Calculate the uncertainty in position## We will now use Heisenberg's uncertainty principle formula \(\Delta x \Delta p \geq \frac{\hbar}{2}\) to find the uncertainties in position.

Step 5: Heisenberg's uncertainty principle formula

Using the uncertainty principle formula and solving for \(\Delta x\), we get: \[\Delta x \geq \frac{\hbar}{2\Delta p}\] a) Mosquito

Step 6: Calculate uncertainty in position (mosquito)

For the mosquito, we have \(\Delta p_{\text{mosquito}} = 1.50 \times 10^{-8}\,\text{kg m/s}\). Using the inequality above, we get: \[\Delta x_{\text{mosquito}} \geq \frac{1.054 \times 10^{-34}\, \text{J s}}{2(1.50 \times 10^{-8}\,\text{kg m/s})} \approx 3.5 \times 10^{-26}\, \text{m}\] b) Proton

Step 7: Calculate uncertainty in position (proton)

For the proton, we have \(\Delta p_{\text{proton}} = 1.67 \times 10^{-22}\,\text{kg m/s}\). Using the inequality above, we get: \[\Delta x_{\text{proton}} \geq \frac{1.054 \times 10^{-34}\, \text{J s}}{2(1.67 \times 10^{-22}\,\text{kg m/s})} \approx 3.2 \times 10^{-13}\, \text{m}\]

Step 8: Final answers

The uncertainties in position for both the mosquito and the proton are: a) \(\Delta x_{\text{mosquito}}\) is at least approximately \(3.5 \times 10^{-26}\, \text{m}\) b) \(\Delta x_{\text{proton}}\) is at least approximately \(3.2 \times 10^{-13}\, \text{m}\)

Key Concepts

Uncertainty in Position

Heisenberg's uncertainty principle tells us that there is a limit to how precisely we can know both the position and momentum of a particle simultaneously. The more accurately we know one, the less accurately we can know the other. This limitation is critical in quantum mechanics and is expressed mathematically as:
\[\Delta x \Delta p \geq \frac{\hbar}{2}\]
Here, \(\Delta x\) is the uncertainty in position, \(\Delta p\) is the uncertainty in momentum, and \(\hbar\) is the reduced Planck constant, approximately \(1.054 \times 10^{-34}\, \text{J s}\). This principle reveals the inherent fuzziness in particle position at the quantum level.
When calculating position uncertainty, like in the case of a mosquito or proton, the uncertainty in their velocity translates to a certain imprecision in knowing precisely where they are. This concept becomes crucial in fields like nanotechnology and quantum computing.

Uncertainty in Momentum

Momentum, in this context, refers to the product of a particle's mass and its velocity. Uncertainty in momentum comes from two factors: uncertainty in the mass (usually negligible in practice) and uncertainty in the velocity. For our calculations, we use the formula:
\[\Delta p = m \Delta v\]
where \(m\) is the mass and \(\Delta v\) is the uncertainty in velocity. In our exercise, for both the mosquito and the proton, the given uncertainty in velocity helps us to determine \(\Delta p\).
For instance, the uncertainty in the mosquito's momentum is calculated and found to be \(1.50 \times 10^{-8}\,\text{kg m/s}\). Similarly, for the proton, it is \(1.67 \times 10^{-22}\,\text{kg m/s}\). These values are crucial for finding the respective uncertainties in their positions.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with particles at atomic and subatomic levels. It challenges classical mechanics' notion of deterministic behavior, introducing probabilities and limitations like the uncertainty principle.
In a quantum world, particles like electrons, protons, and even tiny objects like mosquitoes, demonstrate behaviors that seem paradoxical but are foundational to their understanding. Instead of having precise positions and velocities, they exist in probabilistic states.

• The act of measuring one property more precisely causes the other property to become less certain.
• This notion of uncertainty is not due to flaws in measurement but is a fundamental property of nature.
• Quantum mechanics influences technologies such as lasers, semiconductors, and MRI machines.

Understanding these principles expands the boundaries of modern science and technology across various fields. It's a paradigm shift from the intuitive to the abstract, revealing the universe's fundamental characteristics.