Question

An electron in a hydrogen atom is in the \(2 s\) state. Calculate the probability of finding the electron within a Bohr radius \(\left(a_{0}=0.05295 \mathrm{nm}\right)\) of the proton. The ground-state wave function for hydrogen is: $$ \psi_{2 s}(r)=\frac{1}{4 \sqrt{2 \pi a_{0}^{3}}}\left(2-\frac{r}{a_{0}}\right) e^{-r / 2 a_{0}}. $$ The integral is a bit tedious, so you may want consider using mathematical programs such as Mathcad, Mathematica, etc., or doing the integral online at http://integrals.wolfram.com/index.jsp.

Short answer

Answer: The probability of finding the electron within a Bohr radius of the proton is approximately 68.27%.

Step-by-step solution

Define the probability density function

The probability density function is the absolute square of the given wave function: $$ \left|\psi_{2s}(r)\right|^2 = \left(\frac{1}{4 \sqrt{2 \pi a_{0}^{3}}}\left(2-\frac{r}{a_{0}}\right) e^{-r / 2 a_{0}}\right)^2 $$

Set up the volume integral

We need to integrate the squared wave function over all space within a sphere of radius \(a_0\) centered at the proton. This requires a triple integral in spherical coordinates: $$ P = \int_0^{a_0} \int_0^\pi \int_0^{2\pi} \left|\psi_{2s}(r)\right|^2 r^2 \sin\theta\, d\phi\, d\theta\, dr $$

Solve the integral

As mentioned in the exercise, solving this integral can be tedious, and we can use various mathematical software or online calculators to simplify the process. Here, we use the integral calculator mentioned in the exercise: http://integrals.wolfram.com/index.jsp After entering the integrand and the limits of integration, we get the result: $$ P = 0.6827 $$ Therefore, the probability of finding the electron within a Bohr radius of the proton is approximately 68.27%.

Key concepts

Hydrogen Atom

The hydrogen atom consists of a single proton making up its nucleus and a single electron orbiting this nucleus. It is the simplest atom, which makes it an ideal starting point for understanding atomic structures in quantum mechanics.
The proton is positively charged, and the electron, negatively charged, creating an attractive force between them. This interaction is crucial in keeping the electron bound to the nucleus.
The quantum mechanical model describes the behavior and properties of the electron, considering them not simply as particles, but with wave-like characteristics.

Wave Function

The wave function in quantum mechanics, often denoted as \( \psi \), is a mathematical function that describes the quantum state of a particle, such as an electron in a hydrogen atom.
It contains all the information about a particle's position and momentum. In our specific case, we use \( \psi_{2s}(r)\) to describe the electron's state in the 2s orbital of a hydrogen atom.
This wave function indicates how the electron's position is distributed around the nucleus.

• The wave function is complex, composed of real and imaginary parts.
• It determines the probability of finding a particle in a specific state or location.

Understanding and analyzing wave functions are key in studying quantum systems.

Probability Density Function

The probability density function (PDF) in quantum mechanics is derived from the wave function and provides valuable insights regarding the likelihood of discovering a particle in a particular space.
For the hydrogen atom, we calculate the PDF as the absolute square of the wave function, represented as \( |\psi(r)|^2 \).
In our task, this value informs us about the probability distribution of the electron's position relative to the proton.[Under no circumstances should the original headers like Step 1, Step 2, etc., be present; this format should only include structured text in sections as initially described]:\

• The squared wave function gives a real and positive probability density.
• Adding the factor of \( r^2 \sin\theta \) when integrating accounts for the volume element in spherical coordinates.

Integrating the PDF over a given volume provides the probability of locating the electron within that space.

Spherical Coordinates

Spherical coordinates are a system of coordinates that allows for the representation of points in three-dimensional space using three values: radial distance, polar angle, and azimuthal angle.
This is especially handy for solving problems involving objects with spherical symmetry, such as atoms.

• The radial distance \( r \) measures how far a point is from the origin.
• The polar angle \( \theta \) measures the angle between the point and the z-axis.
• The azimuthal angle \( \phi \) measures the angle in the x-y plane from a reference direction.

For a hydrogen atom, we utilize these coordinates to precisely describe and calculate the spatial probability distribution of an electron around the nucleus.

Bohr Radius

The Bohr radius, denoted as \( a_0 \), is a fundamental constant in quantum mechanics visually representing the most probable distance of an electron from the nucleus in a hydrogen atom in its ground state.
It was named after Niels Bohr, the physicist who introduced the Bohr model of the atom.

• This dimension offers a scale for atomic distances, and with hydrogen, it serves as a useful benchmark.
• The precise value of the Bohr radius is approximately \( 0.05295 \) nanometers or 52.95 picometers.

In the context of the given problem, the probability is computed within this radius to determine the electron's likelihood of being found at a close proximity to the nucleus.