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Magazine Chapter 39: Problem 70
Suppose that the uncertainty of position of an electron is equal to the radius of the \(n\) = 1 Bohr orbit for hydrogen. Calculate the simultaneous minimum uncertainty of the corresponding momentum component, and compare this with the magnitude of the momentum of the electron in the \(n\) = 1 Bohr orbit. Discuss your results.
Short Answer
Expert verified
Minimum momentum uncertainty is about half of the actual momentum, showing significant uncertainty in quantum systems.
Step by step solution
01
Understanding the problem
The problem asks us to calculate the minimum uncertainty in momentum for an electron in a hydrogen atom, given that the uncertainty in position is the radius of the electron's orbit in the hydrogen atom's ground state. We'll also compare this to the electron's actual momentum in the same state.
02
Determine the radius of the n=1 Bohr orbit
The radius of the electron in the n=1 Bohr orbit of hydrogen, denoted as \( r \), can be calculated using the formula: \( r = a_0 \), where \( a_0 = 0.529 \times 10^{-10} \) meters is the Bohr radius.
03
Use Heisenberg's Uncertainty Principle
The Heisenberg Uncertainty Principle states: \( \Delta x \Delta p \geq \frac{\hbar}{2} \), where \( \Delta x \) is the uncertainty in position and \( \Delta p \) is the uncertainty in momentum. Substitute \( \Delta x = a_0 \) to find \( \Delta p \).
04
Calculate the minimum uncertainty in momentum
Plug the values into the uncertainty relation: \( \Delta p \geq \frac{\hbar}{2 a_0} \). Substituting \( \hbar = 1.0545718 \times 10^{-34} \) J路s and \( a_0 = 0.529 \times 10^{-10} \) m, we find \( \Delta p \geq 9.935 \times 10^{-25} \text{ kg路m/s} \).
05
Calculate the momentum of the electron in n=1 orbit
The momentum \( p \) of the electron in the n=1 Bohr orbit is given by \( p = \frac{\hbar}{a_0} \). Using \( \hbar = 1.0545718 \times 10^{-34} \) J路s, calculate \( p = 1.875 \times 10^{-24} \text{ kg路m/s} \).
06
Compare the momentum uncertainties
Compare the calculated minimum momentum uncertainty \( \Delta p \approx 9.935 \times 10^{-25} \text{ kg路m/s} \) with the electron's momentum \( p \approx 1.875 \times 10^{-24} \text{ kg路m/s} \). The minimum uncertainty in momentum is about half of the actual momentum, indicating a significant uncertainty.
07
Discuss the results
The calculated minimum uncertainty in momentum is substantial compared to the actual momentum, consistent with the high level of uncertainty inherent in quantum systems. This result supports the limits imposed by quantum mechanics on simultaneous measurements of position and momentum.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Bohr Model
The Bohr Model is an early theoretical model of atomic structure that introduced the idea of quantized electron orbits around the nucleus. This model was developed by Niels Bohr in 1913 and was pivotal as it laid the foundation for future quantum theories. According to the Bohr Model, electrons move in circular orbits at specific energy levels and do not emit radiation while in these orbits, thus preventing the electrons from spiraling into the nucleus.
- Bohr hypothesized that the orbits are stable and that the angular momentum of electrons is an integral multiple of \(\hbar \/ 2\pi\), where \(\hbar\) is Planck's reduced constant.
- This idea directly leads to quantization, explaining why only certain energy levels are allowed.
- For hydrogen, the electron's orbits are determined using a series of calculations that account for these energy states. The radius for the simplest ground state, \(n=1\), is known as the Bohr radius (\(a_0 = 0.529 \/ 10^{-10} \/ \text{m}\)).
Quantum Mechanics
Quantum Mechanics is the fundamental theory that describes the behavior of particles at atomic and subatomic scales, where classical mechanics no longer applies. It introduces principles such as wave-particle duality, quantization, and uncertainty that fundamentally redefine our understanding of microscopic phenomena.
- Wave-Particle Duality: Particles like electrons exhibit properties of both waves and particles. This dual nature explains phenomena such as interference and diffraction.
- Quantization: Only discrete energy levels are accessible to particles. This is essential to the understanding of atomic structure and electron configuration.
- Uncertainty Principle: A key aspect of quantum mechanics articulates that certain properties, such as position and momentum, cannot be simultaneously measured with arbitrary precision.
Electron Momentum Uncertainty
Electron Momentum Uncertainty is a concept arising from Heisenberg's Uncertainty Principle, a fundamental aspect of quantum mechanics. This principle essentially limits the precision with which certain pairs of physical properties, like position and momentum, can be known simultaneously. If you try to precisely measure one, you'll increase uncertainty in the other.
- The uncertainty formula is given by: \(\Delta x \Delta p \geq \frac{\hbar}{2}\), where \(\Delta x\) is the position uncertainty, and \(\Delta p\) is the momentum uncertainty.
- In the hydrogen atom at the \(n=1\) orbit, setting \(\Delta x\) to the Bohr radius implies a corresponding minimum momentum uncertainty calculated via the formula.
- Our calculations showed \(\Delta p \approx 9.935 \times 10^{-25} \/ \text{kg路m/s}\), which, despite being somewhat small numerically, is significant relative to the electron's actual momentum.
Hydrogen Atom
The Hydrogen Atom is the simplest and most fundamental atom, consisting of a single proton and an electron. Its simplicity makes it an ideal subject for theoretical atomic models and quantum mechanical analysis.
- The hydrogen atom serves as a stepping stone for understanding more complex atoms and was the basis for Niels Bohr's initial modeling of atomic structure.
- In the ground state, the electron occupies the lowest energy level or the first Bohr orbit.
- The behavior of electrons in hydrogen and other atoms is primarily dictated by the laws of quantum mechanics, particularly the quantization of energy levels and the uncertainty principle.
