Example 29.2 calculates the most probable value and the...

Example 29.2 calculates the most probable value and the average value for the radial coordinate $r$ of the electron in the ground state of a hydrogen atom. For comparison with these modal and mean values, find the median value of $r$ Proceed as follows. (a) Derive an expression for the probability, as a function of $r,$ that the electron in the ground state of hydrogen will be found outside a sphere of radius $r$ centered on the nucleus. (b) Make a graph of the probability as a function of $r / a_{0} .$ Choose values of $r / a_{0}$ ranging from 0 to 4.00 in steps of $0.250 .$ (c) Find the value of $r$ for which the probability of finding the electron outside a sphere of radius $r$ is equal to the probability of finding the electron inside this sphere. You must solve a transcendental equation numerically, and your graph is a good starting point.

Key Concepts

Radial Probability Distribution

This concept pertains to the probability density function that describes how the likelihood of finding an electron at a certain distance from the nucleus varies with the radial coordinate. It results from combining the square of the wave function with the volume element in spherical coordinates, and it underpins many calculations in quantum mechanics related to electron distributions.

Cumulative Distribution and Median

In the context of probability distributions, the cumulative distribution function (CDF) quantifies the probability that the electron is found within a given radius. The median is a specific value of the radial coordinate at which the cumulative probability is 0.5, meaning there is an equal chance of finding the electron inside or outside the sphere defined by this radius.

Transcendental Equations and Numerical Solutions

When equalizing the probability of finding the electron inside and outside a sphere, the resulting equation is transcendental, meaning it cannot be solved with standard algebraic methods. Instead, numerical methods or graphical techniques are used to find an approximate solution to this equation.

Hydrogen Atom Ground State and Normalization

The calculation relies on the normalized ground state wave function of the hydrogen atom, which describes the electron's spatial distribution. Normalization ensures that the total probability of finding the electron anywhere in space is one, allowing for meaningful application of probability concepts such as the mean, mode, and median.

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