The hydrogen atom is the only chemical species which admits a closed-form solution to the Schrodinger wave equation. In general, the wave functions are expressed in spherical coordinates but fortunately they are spherically symmetric for all the \(s\)-orbitals. Consider the wave function for the \(1 s\) orbital \(\left(n=1, l=0, m_{I}=0\right)\). $$ \psi(r, \theta, \phi)=\frac{1}{\sqrt{\pi}}\left(\frac{1}{a_{0}}\right)^{3 / 2} e^{-r / \omega_{0}} . $$ where \(a_{a}\) is the Bohr radius, given by \(a_{0}=\varepsilon_{0} h^{2} / \pi \mu e^{2}\). a. Demonstrate that this wave function is properly normalized. b. Show that the probability density function for the radial position of the \(1 s\) electron is given by $$ f(r)=\frac{4 r^{2}}{a^{3}} e^{-2 r / a} . $$ c. Determine the average radial position, \(r\), for the ls orbital. d. Calculate the uncertainty in radial position by determining the root-mean- square deviation, \(\Delta\) r. e. Evaluate \(\Delta r / r\). Comment on the physical implications of your result.

Step by step solution

01

Check Normalization

To check if the wave function \(\psi(r, \theta, \phi)\) is normalized, the integral of \(|\psi(r, \theta, \phi)|^2\) over all space must equal 1. The normalization condition in three dimensions with spherical symmetry is \[ \int_0^{\infty} \int_0^{\pi} \int_0^{2\pi} |\psi(r, \theta, \phi)|^2 r^2 \sin(\theta)\, d\phi\, d\theta\, dr = 1. \]Because \(\psi\) is spherically symmetric, it doesn't depend on \(\theta\) and \(\phi\), and the angular integrals simplify to a factor of \(4\pi\). Then, the equation simplifies to\[ \int_0^{\infty} |\psi(r)|^2 4\pi r^2\, dr. \]

02

Perform the Radial Integration

Substituting the given wave function \(\psi(r)\) into the normalization integral and performing the integration, we find that\[ \int_0^{\infty} \left( \frac{1}{\sqrt{\pi}} \left(\frac{1}{a_{0}}\right)^{\frac{3}{2}} e^{-r / a_{0}} \right)^2 4\pi r^2\, dr = \int_0^{\infty} \frac{4 r^2}{\pi a_{0}^3} e^{-2r / a_{0}}\, dr. \]Evaluating this integral (which becomes a gamma function after the substitution \(u = 2r/a_0\)) will confirm that it equals 1, demonstrating that the wave function is properly normalized.

03

Determine the Probability Density Function

The probability density function for any wave function \(\psi\) is given by \(|\psi|^2\). Thus, for the radial probability density function \(f(r)\), we have that\[ f(r) = 4\pi r^2 |\psi(r)|^2. \]By substituting the wave function into this expression and factoring out the constants, we can derive that\[ f(r) = \frac{4 r^2}{a_{0}^3} e^{-2r / a_{0}}. \]

04

Calculate the Average Radial Position

The average radial position \(\langle r \rangle\) is given by\[ \langle r \rangle = \int_0^{\infty} rf(r)\, dr. \]Inserting the expression for \(f(r)\) obtained in the previous step, we have:\[ \langle r \rangle = \int_0^{\infty} r \times \frac{4 r^2}{a_{0}^3} e^{-2r / a_{0}}\, dr. \]This integral will yield a value that represents the expected radial distance of the electron in the 1s orbital.

05

Calculate the Uncertainty in Radial Position

The uncertainty in radial position, \(\Delta r\), is calculated from the root-mean-square deviation, which in this case is\[ (\Delta r)^2 = \langle r^2 \rangle - \langle r \rangle^2. \]We already have \(\langle r \rangle\) from the previous step, so we need to calculate \(\langle r^2 \rangle\), which is given by\[ \langle r^2 \rangle = \int_0^{\infty} r^2 f(r)\, dr. \]After performing this integration, we take the square root of \((\Delta r)^2\) to find \(\Delta r\).

06

Evaluate the Ratio \(\Delta r / \langle r \rangle\)

Once we have both \(\Delta r\) and \(\langle r \rangle\), we can find their ratio:\[ \frac{\Delta r}{\langle r \rangle} = \frac{\text{root-mean-square deviation}}{\text{average radial position}}. \]This ratio gives insight into the relative uncertainty: a higher value indicates greater uncertainty in the electron's position relative to its average radial distance.

07

Comment on the Physical Implications

After evaluating \(\Delta r / \langle r \rangle\), we consider its implications in the context of quantum mechanics. A small ratio implies that there's a relatively precise knowledge of the electron's radial position within the atom, whereas a larger ratio indicates a less precise knowledge, which is consistent with the uncertainty principle in quantum mechanics.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Function Normalization

Normalization of wave functions is a crucial process in quantum mechanics, ensuring that the probability of finding a particle within the entire space is equal to one. The Schrödinger wave equation describes how quantum states evolve over time, yet it does not demand the wave functions to be normalized explicitly. It is the responsibility of the physicist to ensure normalization is achieved, as it is a physical requirement for the consistency of the theory.

In the specific case of the hydrogen atom's 1s orbital, normalization involves integrating the absolute square of the wave function over the entire space. When evaluated, this integral should render the result of unity. If not, the wave function must be adjusted by multiplying it by a normalization constant. Such a step is critical before delving into further calculations or interpretations of a quantum system.

Probability Density Function

The probability density function in quantum mechanics is more than a mathematical abstraction; it is directly related to the likelihood of finding an electron at a particular location within an atom. For spherically symmetric states, like the 1s orbital, the function takes the form of a radial probability density that depends only on the distance from the nucleus.

To form this function, we square the magnitude of the wave function and multiply by the surface area of a sphere at radius r, which is expressed as \(4\text{\textpi} r^2\). The resulting function \(f(r)\) provides critical insights into the spatial distribution of an electron. Peaks within this distribution represent regions with a higher probability of finding the electron, offering a vivid picture of the atomic 'cloud' a physicist works with in quantum chemistry.

Quantum Mechanics

Quantum mechanics opens a door to understanding the behavior of particles at the atomic and subatomic levels, diverging significantly from classical mechanics. This branch of physics describes particles in terms of probabilities rather than deterministic paths. The fundamental premise is that particles are described by wave functions, which encapsulate the probabilities of finding a particle in various locations.

Quantum mechanics is characterized by principles such as wave-particle duality, which recognizes particles as having both wave and particle properties, and quantization, which implies that certain properties, like energy, can only take discrete values. These principles defy our everyday experiences and intuition, yet they have been experimentally verified and serve as the basis for modern technologies, including semiconductors and lasers.

Average Radial Position

The average radial position is a quantum mechanical average that represents the expected distance of an electron from the nucleus. It is denoted as \(\text{\textlangle} r \text{\textrangle}\) and is found by integrating the product of the radial position and the probability density function over all possible radial distances.

In other words, it is a form of spatial 'average' that allows physicists to gather information about the electron's most likely range from the nucleus in its ground state. However, unlike a regular arithmetic mean, this average is weighted by the probability distribution of the electron's position. Unlike classical orbits, which define a clear path around the nucleus, quantum mechanics describes this spread of likely positions with the average radial position being a measurable quantity that provides insight into the electron cloud's extent.

Uncertainty Principle

One of the cornerstones of quantum mechanics is the uncertainty principle, which was first formulated by Werner Heisenberg. This principle states that certain pairs of physical properties, like position and momentum, cannot be simultaneously known to arbitrary precision. Specifically, the more accurately we know one value, the less accurately we can know the other.

The exercise involving the hydrogen atom's 1s orbital showcases this principle. The calculation of the uncertainty in the radial position and the average radial position are related to the uncertainty principle. The ratio \(\text{\textDelta}r/\text{\textlangle}r\text{\textrangle}\) is a measure of the relative spread in the probable radial positions of the electron. A larger ratio suggests a more substantial uncertainty and reflects a fundamental property of quantum systems – no matter how precise our instruments, the universe sets a limit on the precision of our measurements.