The radius of the proton is about R0=10^-15 m . The probability that the electron is inside the

step 1 We need to compute the probability for the hydrogen ground state. So from problem 65, we have p

The radius of the proton is about R0=10^-15 m . The...

The radius of the proton is about $R_{0}=10^{-15} \mathrm{~m} .$ The probability that the electron is inside the volume occupied by the proton is given by $$ P=\int_{0}^{R_{0}} P(r) d r $$ where $P(r)$ is the radial probability density. Compute $P$ for the hydrogen ground state. (Hint: Show that $e^{-2 r / a_{0}} \approx 1$ for $r \ll a_{0}$ is valid for this calculation.)

The radius of the proton is about $R_{0}=10^{-15} \mathrm{~m} .$ The probability that the electron is inside the volume occupied by the proton is given by
$$
P=\int_{0}^{R_{0}} P(r) d r
$$
where $P(r)$ is the radial probability density. Compute $P$ for the hydrogen ground state. (Hint: Show that $e^{-2 r / a_{0}} \approx 1$ for $r \ll a_{0}$ is valid for this calculation.)

Key Concepts

Hydrogen Atom Ground State

The hydrogen atom ground state refers to the lowest energy solution of the Schrödinger equation for the hydrogen atom. This state is described by an exponential wavefunction that provides the probability amplitude for finding the electron at a given distance from the nucleus. Understanding this state is crucial as it forms the foundation for analyzing electron distributions and energy levels in quantum mechanics.

Radial Probability Density

Radial probability density is derived from the square of the wavefunction and gives the probability per unit radial distance of finding the electron at a specific radius. In spherical coordinates, this density includes a factor from the Jacobian (typically r²) and is integrated over a radial interval to determine the likelihood of locating the electron within that range. This concept is key in determining probabilities in quantum systems like the hydrogen atom.

Small-r Approximations

Small-r approximations involve simplifying expressions under the condition that the radial distance is much smaller than a characteristic length scale (such as the Bohr radius). For example, exponential factors like exp(-2r/a0) can be approximated as 1 when r is very small compared to a0. This approximation simplifies integrals and calculations of probabilities in regions where the variable r is negligibly small relative to the overall scale of the problem.

Quantum Mechanical Probability Integration

Quantum mechanical probability integration is the process of integrating the probability density function over a specified volume or region to obtain the total probability of finding a particle within that region. This method is fundamental in quantum mechanics for connecting the mathematical description of a particle's state with measurable outcomes, such as the likelihood of finding an electron within a particular volume inside an atom.

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