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What is the probability that an electron in the ground state of hydrogen will be found inside the nucleus? (a) First calculate the exact answer, assuming the wave function (Equation 4.80) is correct all the way down to \(r=0 .\) Let \(b\) be the radius of the nucleus. (b) Expand your result as a power series in the small number \(\epsilon \equiv 2 b / a\), and show that the lowest-order term is the cubic: \(P \approx(4 / 3)(b / a)^{3}\). This should be a suitable approximation, provided that \(b \ll a\) (which it \(i s\) ). (c) Alternatively, we might assume that \(\psi(r)\) is essentially constant over the (tiny) volume of the nucleus, so that \(P \approx(4 / 3) \pi b^{3}|\psi(0)|^{2}\). Check that you get the same answer this way. (d) Use \(b \approx 10^{-15} \mathrm{m}\) and \(a \approx 0.5 \times 10^{-10} \mathrm{m}\) to get a numerical estimate for \(P .\) Roughly speaking, this represents the "fraction of its time that the electron spends inside the nucleus."

Step by step solution

01

Use the wave function for the hydrogen atom

The wave function of a hydrogen atom's electron in its ground state is given by \( \psi(r) = \frac{1}{\sqrt{\pi}} \left( \frac{1}{a} \right)^{3/2} e^{-r/a} \), where \(a\) is the Bohr radius. This will be utilized to find the probability of the electron being inside the nucleus.

02

Calculate the exact probability

To find the probability \(P\) that the electron is inside the nucleus, integrate the probability density \(\psi^2(r)\) from \(r=0\) to \(r=b\): \[ P = \int_0^b \left( \frac{1}{\pi a^3} e^{-2r/a} \right) 4\pi r^2 \, dr \]. Evaluating, \( P = 1 - e^{-2b/a} \left(1 + \frac{2b}{a} + \left(\frac{2b}{a}\right)^2 \right) \).

03

Expand and approximate probability

Assume \( \epsilon = 2b/a\) is small to expand the exponential function for a power series. We have \[ P \approx \frac{4}{3} \left( \frac{b}{a} \right)^3 \] as the lowest order term in \( \epsilon \) for small \(b/a\).

04

Alternative approximation for constant wave function

Assume that \(\psi(r)\) is constant over the nucleus: \(\psi(r) \approx \psi(0)\). Then the probability is \( P \approx \int_0^b |\psi(0)|^2 \, 4\pi r^2 \, dr \). Simplifying, it gives \( P \approx \frac{4}{3} \pi b^3 |\psi(0)|^2 \), matching the form from Step 3.

05

Calculate numerical estimate for P

Using given values, \(b \approx 10^{-15}\,\text{m}\) and \(a \approx 0.5 \times 10^{-10}\,\text{m}\), calculate the numerical probability: \[ P \approx \frac{4}{3} \left( \frac{10^{-15}}{0.5 \times 10^{-10}} \right)^3 \approx 2.688 \times 10^{-15} \]. This indicates the fraction of time an electron spends inside the nucleus.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hydrogen Atom

The hydrogen atom is one of the simplest and most extensively studied atomic systems in quantum mechanics. It consists of a single proton as its nucleus and one electron orbiting it. This simplicity makes it an ideal model for introducing and understanding foundational concepts in quantum mechanics.

In quantum terms, the hydrogen atom is often used to explore the idea of quantized energy levels. The electron can only occupy specific energy levels or "orbits," which are determined by the solutions to the Schrödinger equation. These solutions predict the electron's behavior in terms of wave functions rather than classical paths.

• The atom serves as an excellent model for radial probability distributions, where the radial component describes how electron probability varies with distance from the nucleus.
• This model helps explain why electrons do not spiral into the nucleus, resolving a major inconsistency in classical physics.

The study of the hydrogen atom paves the way for understanding more complex atoms and the details of atomic structure.

Wave Function

In quantum mechanics, the wave function, denoted as \( \psi \), is a mathematical description of the quantum state of a particle. For the hydrogen atom, the wave function describes the behavior and distribution of the electron around the nucleus.

For the ground state (lowest energy level) of a hydrogen atom, the wave function is spherically symmetric and is given by:\[ \psi(r) = \frac{1}{\sqrt{\pi}} \left( \frac{1}{a} \right)^{3/2} e^{-r/a} \]Here, \( r \) represents the radial distance from the nucleus, and \( a \) is the Bohr radius.

• Wave functions provide information about the likelihood of finding an electron at a particular point in space by yielding the probability density when squared.
• They encapsulate both the particle and wave nature of an electron, reflecting the duality inherent in quantum particles.

The concept of a wave function is central in quantum mechanics, as it provides the most complete description available of a quantum system.

Probability Density

The probability density \( \psi^2 \) is derived from the wave function and provides insight into where an electron is most likely to be found at any given moment. In our example of the hydrogen atom, the probability density for the ground state is:\[ \psi^2(r) = \left( \frac{1}{\pi a^3} \right) e^{-2r/a} \]

This function indicates that the electron is more likely to be found closer to the nucleus and less likely as the distance increases.

• Probability density helps in visualizing the cloud-like electron distribution around the nucleus, often referred to as an "electron cloud."
• Integrating the probability density over a certain range allows us to compute the actual probability of finding an electron within that region.

In essence, the probability density provides a practical way to map the extent and intensity of electron presence in different regions around the nucleus.

Bohr Radius

The Bohr radius, denoted as \( a \), is a fundamental constant in quantum mechanics. It represents the most probable distance from the nucleus at which the electron in a hydrogen atom's ground state is likely to be found. The Bohr radius serves as a unit of measurement within the atomic scale.

Quantitatively, it is approximately 0.529 Ångströms or \( 0.529 \times 10^{-10} \) meters. This physical constant is crucial for various calculations in quantum mechanics.

• It provides a scale for understanding the size of orbitals in hydrogen-like atoms.
• The Bohr radius is utilized for scaling wave functions and probability densities, ensuring that all calculations are consistent with observed physical phenomena.

Recognizing the significance of the Bohr radius helps in appreciating its role in atomic models and its fundamental importance in quantum theory.