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
Step by step solution
Understanding the problem
Determine the radius of the n=1 Bohr orbit
Use Heisenberg's Uncertainty Principle
Calculate the minimum uncertainty in momentum
Calculate the momentum of the electron in n=1 orbit
Compare the momentum uncertainties
Discuss the results
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Bohr Model
- 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
- 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
- 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 \(\Delta p \approx 9.935 \times 10^{-25} \/ \text{kg·m/s}\), which, despite being somewhat small numerically, is significant relative to the electron's actual momentum.
Hydrogen Atom
- 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.