Question

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

Minimum momentum uncertainty is about half of the actual momentum, showing significant uncertainty in quantum systems.

Step-by-step solution

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.

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.

Use Heisenberg's Uncertainty Principle

The Heisenberg Uncertainty Principle states: $\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\hslash }{2}$, where $\mathrm{\Delta }x$ is the uncertainty in position and $\mathrm{\Delta }p$ is the uncertainty in momentum. Substitute $\mathrm{\Delta }x=a_0$ to find $\mathrm{\Delta }p$.

Calculate the minimum uncertainty in momentum

Plug the values into the uncertainty relation: $\mathrm{\Delta }p\ge \frac{\hslash }{2a_0}$. Substituting $\hslash =1.0545718\times {10}^{-34}$ J·s and $a_0=0.529\times {10}^{-10}$ m, we find $\mathrm{\Delta }p\ge 9.935\times {10}^{-25}\text{ kg·m/s}$.

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 $p=\frac{\hslash }{a_0}$. Using $\hslash =1.0545718\times {10}^{-34}$ J·s, calculate $p=1.875\times {10}^{-24}\text{ kg·m/s}$.

Compare the momentum uncertainties

Compare the calculated minimum momentum uncertainty $\mathrm{\Delta }p\approx 9.935\times {10}^{-25}\text{ kg·m/s}$ with the electron's momentum $p\approx 1.875\times {10}^{-24}\text{ kg·m/s}$. The minimum uncertainty in momentum is about half of the actual momentum, indicating a significant uncertainty.

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.

Key concepts

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 $\hslash \text{/}2\pi$, where $\hslash$ 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 ($a_0=0.529\text{/}{10}^{-10}\text{/}\text{m}$).

The Bohr Model, while groundbreaking, had limitations as it could only accurately predict behavior for hydrogen-like atoms and failed to address interactions in multi-electron systems.

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.

Quantum mechanics provided a comprehensive framework to understand phenomena that the Bohr Model could not explain, including the complexity of atoms with many electrons and molecular bonding.

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: $\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\hslash }{2}$, where $\mathrm{\Delta }x$ is the position uncertainty, and $\mathrm{\Delta }p$ is the momentum uncertainty.
• In the hydrogen atom at the $n=1$ orbit, setting $\mathrm{\Delta }x$ to the Bohr radius implies a corresponding minimum momentum uncertainty calculated via the formula.
• Our calculations showed $\mathrm{\Delta }p\approx 9.935\times {10}^{-25}\text{/}\text{kg·m/s}$, which, despite being somewhat small numerically, is significant relative to the electron's actual momentum.

This illustrates the fundamental fuzziness at the quantum level, helping to highlight why quantum mechanics relies on probabilities rather than certainties.

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.

Understanding the hydrogen atom extends beyond its simple structure, revealing the quantum behavior that governs not only hydrogen itself but all matter at the microscopic scale. The hydrogen atom thus becomes a crucial element for teaching the principles of quantum mechanics, electronic transitions, and spectral lines.