Example 41.3 calculates the most probable value and the...

Example 41.3 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

The radial probability distribution quantifies the likelihood of finding a particle, such as an electron, at a particular distance from the nucleus in a quantum system. It is obtained by squaring the radial part of the wavefunction and including the appropriate volume element, which often incorporates a factor of r² in spherical coordinates. This concept is central to understanding the spatial distribution of electrons in atoms.

Ground State of the Hydrogen Atom

The ground state of the hydrogen atom represents the lowest energy configuration, characterized by a specific wavefunction that solves the Schrödinger equation. This state exhibits a unique radial probability distribution that is used as a fundamental example for illustrating quantum mechanical concepts, such as quantization, probability density, and the behavior of electrons in potential wells.

Measures of Central Tendency (Mode, Mean, Median)

In probability and statistics, different measures of central tendency—namely the mode (most probable value), mean (average value), and median—provide distinct insights into the distribution of a variable. In quantum mechanics, comparing these measures for the radial coordinate helps illustrate the asymmetry and shape of the electron’s spatial distribution around the nucleus.

Integration of Probability Density

Integrating the probability density function over a specific region, such as from a given radius to infinity, yields the probability of finding a particle within that region. This method is essential for determining cumulative probabilities and for analyzing regions of space where the electron is most likely to be found.

Transcendental Equations in Quantum Mechanics

Transcendental equations are non-algebraic equations that often emerge from equating probability expressions derived from wavefunctions. Since these equations cannot be solved exactly with standard algebraic methods, numerical techniques or graphical analysis are employed to approximate their solutions, which is a common scenario in advanced quantum mechanics.

Numerical and Graphical Methods

Numerical and graphical methods are essential tools for solving equations that do not have closed-form solutions, such as transcendental equations. In the context of finding the median of a probability distribution, graphing the cumulative probability function and using iterative approximation methods can effectively determine the point where the probability of finding a particle inside equals that outside.

Transcript

00:01 For the median probability, or the radius for the median probability for the 1st state of hydrogen, where the probability density function is a thin spherical shell in three dimensions, and you're finding probability of the electron being found somewhere in that shell at that radius.

00:24 So i've taken the liberty of pre -plotting that probably density function, where over units of four radius from zero to four.

00:42 So the ordinary probability density is the square of the wave function because it's real value functions.

00:50 There's no complex conjugate.

00:52 Probability density function is 4 pi are squared times that because it's the spherical shell, which works out to what this equation here for the 1s state.

01:06 Previously we an example of 41 .3 was found that the mode of the probability that's the function is the bore radius and the probability of being found outside further out from the bore radius is actually almost 68%.

01:39 So you're more likely to be found further away from bore radius than then within it.

01:46 And then the average or the mean position was actually three halves times a board radius, so 1 .5 bore radius.

01:57 So the mode is a not.

02:12 Mean is at 1 .5 a0.

02:15 So the question is where is the median? and the median is going to be somewhere around here by inspection.

02:26 But we to actually calculate it.

02:35 So let's see.

02:37 The first thing we have to do is find an expression for the probability of the electron being found outside some radius r.

02:48 So that's probability ranging from r to infinity.

02:55 So that's out further away than r.

02:56 This is, those are the limits of integration.

03:00 Probability for the 1s state.

03:09 Let's see, and you just pull that constant out.

03:21 Cute time.

03:32 So as a function of r, that's what's looking for.

03:37 I am not going to leave it that way.

03:39 I'm going to follow what the textbook does to put this in terms of units of bore radius.

03:47 So i'm going to let z equals 2 times r, a ,0.

03:56 So then p equals 4a, a, cubed.

04:43 We're switching to z terms.

05:32 I think they did that backwards...

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