Problem 14 - Physics for Scientists and Engineers with Modern Physics

Question

(11) How accurately can the position of a 3.50 -keV electron be measured assuming its energy is known to 1.00\(\% ?\)

Step-by-Step Solution

Answer

The position of the electron can be measured with an uncertainty inversely related to its energy precision but not calculable with direct value in absence of momentum linkage; estimatively an indirect large-scale indication suggests extended value due to undetailed high energy uncertainty involvement.

Step 1: Determine Energy Uncertainty

First, calculate the uncertainty in the energy of the electron. The energy of the electron is given as 3.50 keV, which is equivalent to 3.50 × 10³ eV. The uncertainty in energy ( ΔE ) is 1.00% of 3.50 keV. Therefore, ΔE = 0.01 × 3.50 × 10³ eV = 35 eV.

Step 2: Use Heisenberg's Uncertainty Principle

According to Heisenberg's uncertainty principle, the relationship between the uncertainty in energy ( ΔE ) and the uncertainty in time ( Δt ) is given by ΔE × Δt ≥ ℏ/2, where ℏ is the reduced Planck's constant (1.0545718 × 10⁻³⁴ J·s). We need to convert 35 eV to joules by using the conversion factor 1eV = 1.60218 × 10⁻¹⁹ J. Therefore, ΔE = 35 eV × 1.60218 × 10⁻¹⁹ J/eV = 5.60763 × 10⁻¹⁸ J.

Step 3: Relate Energy Uncertainty to Position Uncertainty

Knowing ΔE, use ΔE × Δt = ℏ/2 to find Δt. Rearranging gives Δt = ℏ/(2ΔE). Now, calculate Δt = (1.0545718 × 10⁻³⁴ J·s)/(2 × 5.60763 × 10⁻¹⁸ J) ≈ 9.40 × 10⁻¹⁸ seconds. Typically, uncertainty in position Δx is indirectly related to Δt via velocity or momentum, but for estimation with minimal details, continue using ΔE × Δt to deduce that faster energy changes imply greater position spread.

Step 4: Calculate Position Uncertainty

For an electron at this energy, considering its velocity and wavelength context through associated equations like λ = h/p and relating Δp to Δx using ΔxΔp = ℏ/2 for approximate position uncertainty. Given the simplistic approach, position can be roughly calculated if complete kinetic parameters were stated, making use of primary estimation trajectory.

Key Concepts

Energy Uncertainty

When we talk about energy uncertainty, we refer to the precision of our knowledge about the energy of a particle, like an electron. In the context of Heisenberg's Uncertainty Principle, energy uncertainty (\(\Delta E\)) is the blur or fuzziness associated with a particle's energy measurement. For example, if we know the energy of an electron is 3.50 keV,
but our knowledge is only accurate to 1%, this means that the uncertainty in this energy (\(\Delta E\)) is 35 eV.
Heisenberg's principle gives us a mathematical relationship:

• \(\Delta E \times \Delta t \geq \hbar/2\)

where \(\hbar\) is the reduced Planck's constant. This formula tells us that the uncertainty in energy and the uncertainty in time are inversely related.
This intrinsic blurriness in measurements forms the foundation of the quantum world, influencing other uncertainties, like position.

Position Uncertainty

Position uncertainty represents how precisely we can know the position of a particle, such as an electron. In quantum mechanics, complete precision is unattainable. Due to the nature of quantum particles, there is always a level of uncertainty.
For instance, the less precise we are about a particle's energy, the harder it becomes to predict its position exactly. This uncertainty in position (\(\Delta x\)) is intertwined with momentum uncertainty, expressed as:

• \(\Delta x \cdot \Delta p \geq \hbar/2\)

where \(\Delta p\) is the uncertainty in momentum and \(\hbar\) is the reduced Planck's constant.
Therefore, if energy changes faster, leading to a greater momentum spread, the position uncertainty increases. Calculating an electron's exact position becomes trickier, and often estimates rely on understanding its kinetic conditions. This highlights the fascinating challenge of particle localization in quantum physics.

Planck's Constant

Planck's constant (\(h\)) is a fundamental quantity in physics that plays a crucial role in quantum mechanics. It defines the scale at which quantum mechanical effects become significant, setting a fundamental limit on the accuracy of simultaneously measuring certain pairs of physical properties, like position and momentum.
The reduced Planck's constant (\(\hbar = h/2\pi\)) often appears in the Heisenberg Uncertainty Principle:

• \(\Delta x \cdot \Delta p \geq \hbar/2\)
• \(\Delta E \times \Delta t \geq \hbar/2\)

These equations show how Planck’s constant ties into limits of measurement precision.
Planck's constant profoundly influences all quantum behaviors, determining the "quantum leap" that particles can make between energy levels. Its introduction revolutionized physics by establishing that energy levels are quantized, such that they can only occur in discrete amounts. This concept sheds light on the wave-particle duality and remains central to understanding atomic and subatomic processes.