Question

Calculate the probability that the 1 sectron for \(\mathrm{H}\) will be found between \(r=a_{0}\) and \(r=2 a_{0}\).

Short answer

The probability that the 1s electron of a hydrogen atom will be found between \(r = a_{0}\) and \(r = 2a_{0}\) is given by: \[ P = \frac{1}{2} (\mathrm{e}^{-2} - \mathrm{e}^{-4}) \approx 0.233 \] Therefore, there is a 23.3% chance of finding the 1s electron between \(r = a_{0}\) and \(r = 2a_{0}\).

Step-by-step solution

Compute the probability density function

Recall that the probability density function P(r) is the square of the radial wavefunction. So let's square the wavefunction \(\psi_{1s}(r)\): \[ P(r) = | \frac{1}{\sqrt{\pi}a_{0}^{\frac{3}{2}}}\mathrm{e}^{-\frac{r}{a_{0}}}|^2 \] Which becomes \[ P(r) = \frac{1}{\pi a_{0}^{3}}\mathrm{e}^{-\frac{2r}{a_{0}}} \]

Set up the integration

To find the probability that the electron is found between r = a₀ and r = 2a₀, we need to integrate the probability density function within this range: \[ P = \int_{a_{0}}^{2a_{0}} \frac{1}{\pi a_{0}^{3}}\mathrm{e}^{-\frac{2r}{a_{0}}} \ dr \]

Perform the integration

Let's integrate the function: \[ P = \frac{1}{\pi a_{0}^{3}} \int_{a_{0}}^{2a_{0}} \mathrm{e}^{-\frac{2r}{a_{0}}} \ dr \] First substitute \(u = -\frac{2r}{a_{0}}\), and then \(du = -\frac{2}{a_{0}} dr\). The new bounds for the integral are \( u = -2 \) when \( r = a_{0}\), and \(u = -4\) when \(r = 2a_{0}\). \[ P = \frac{-a_{0}}{2\pi} \int_{-2}^{-4} \mathrm{e}^{u} \ du \] Now, integrating with respect to u: \[ P = \frac{-a_{0}}{2\pi} [ \mathrm{e}^{u} ]_{-2}^{-4} \]

Evaluate the integral and find the probability

Finally, let's evaluate the integral to find the probability: \[ P = \frac{-a_{0}}{2\pi} [(\mathrm{e}^{-4} - \mathrm{e}^{-2})] \] Simplify the expression: \[ P = \frac{1}{2} (\mathrm{e}^{-2} - \mathrm{e}^{-4}) \] So, the probability that the 1s electron of a hydrogen atom will be found between \(r = a_{0}\) and \(r = 2a_{0}\) is given by: \[ P = \frac{1}{2} (\mathrm{e}^{-2} - \mathrm{e}^{-4}) \approx 0.233 \] Therefore, there is a 23.3% chance of finding the 1s electron between \(r = a_{0}\) and \(r = 2a_{0}\).

Key concepts

Hydrogen Atom Electron Probability

Understanding the behavior of electrons in a hydrogen atom is fundamental in quantum mechanics. The hydrogen atom electron probability refers to the likelihood of finding an electron within a specific region around the nucleus. Due to the quantum nature of electrons, they do not have a defined path or location but exist in states described by wavefunctions, leading to probabilities of finding them in certain regions of space defined by the distance, or radial coordinate, from the nucleus. In our exercise, we wanted to calculate the probability of finding the electron somewhere between one and two times the Bohr radius from the nucleus. This is a typical problem in quantum mechanics, demonstrating the probabilistic nature of subatomic particles.

Through the use of the radial wavefunction for the 1s orbital, we can determine this probability by integrating over the desired range. This range represents a spherical shell around the nucleus where the likelihood of locating the electron is being assessed. The calculated probability of 23.3% shows us that nearly one-fourth of the time, if we were to measure the position of the electron, we would find it within this spherical shell around the nucleus.

Radial Wavefunction

The radial wavefunction is a crucial part of the wavefunction of an electron in quantum mechanics, and it plays a significant role in calculating probabilities. In the case of the hydrogen atom, the radial wavefunction describes the behavior of the electron as a function of its distance from the nucleus. Specifically, for the 1s state (the ground state), the wavefunction decreases exponentially as the distance from the nucleus increases, reflecting the higher probability of finding the electron closer to the nucleus.

In our exercise, we squared the radial wavefunction to find the probability density. This squaring process transforms the wavefunction into a quantity that can be interpreted as the probability per unit radial distance, i.e., the radial probability density. Through integration, which adds up the probabilities at each infinitesimal radial distance, we determine the overall probability of finding the electron between the specified radii.

Probability Density Function

The probability density function in quantum mechanics is an expression that gives the likelihood of finding a particle, such as an electron, at any point in space. For an electron in a hydrogen atom, the probability density function is derived from the wavefunction, which must be squared to find the actual probability distribution. This function is not constant; it depends on the distance from the nucleus and the orbital in which the electron resides.

In our example, the function obtained after squaring the wavefunction shows an exponential decrease with distance, indicating that the electron is more likely to be found close to the nucleus. When integrating this function over a range of radii, we determine the probability of locating the electron within that spherical region, providing a tangible connection between the abstract wavefunction and the actual measurable outcome of an experiment.

Integration in Quantum Mechanics

Integration is a mathematical tool widely used in quantum mechanics to calculate probabilities. Given a probability density function, integration allows us to find the probability that a particle will be within a certain region of space. In practice, this involves setting up an integral over the desired range, applying the proper limits, and finding the area under the curve represented by the probability density function.

Detailed in our step-by-step solution, we integrated the probability density from one Bohr radius to two Bohr radii to find the probability of the electron being within that specific shell. The process involved substitution to simplify the integral and then evaluating it between the limits to find a numerical answer. This method exemplifies the concept of integration in quantum mechanics, solidifying our understanding of how abstract wavefunctions translate into observable probabilities.