Question

(a) According to the Bohr model, an electron in the ground state of a hydrogen atom orbits the nucleus at a specific radius of \(0.53 \AA\). In the quantum mechanical description of the hydrogen atom, the most probable distance of the electron from the nucleus is \(0.53 \AA\). Why are these two statements different? (b) Why is the use of Schrödinger's wave equation to describe the location of a particle very different from the description obtained from classical physics? (c) In the quantum mechanical description of an electron, what is the physical significance of the square of the wave function, \(\psi^{2}\) ? \(6.55\) (a) For \(n=4\), what are the possible values of \(l\) ? (b) For \(l=2\), what are the possible values of \(m_{l}\) ? (c) If \(m_{l}\) is 2 , what are the possible values for \(l\) ?

Short answer

(a) The Bohr model assumes a definite orbit for the electron, while the quantum mechanical description provides a probability of the electron's position. (b) Schrödinger's wave equation gives a probability distribution representing the likelihood of finding a particle in a certain location, which differs from the deterministic description of classical physics. (c) The square of the wave function represents the probability density of finding the electron at any given point in space. (6.55a) For n=4, the possible values of l are \(0, 1, 2, 3\). (6.55b) For l=2, the possible values of ml are \(-2, -1, 0, 1, 2\). (6.55c) The possible values for l when ml=2 are \(l = 2, 3, 4, 5, ...\).

Step-by-step solution

(a) Comparing electron's position in Bohr model and quantum mechanical description

In the Bohr model, the electron orbits the nucleus at specific radii called orbits. The electron orbiting in the ground state in hydrogen atom has a specific radius of \(0.53 \AA\). In contrast, the quantum mechanical description doesn't refer to precise orbits, but instead, it works with probability distributions. In this description, the most probable distance of the electron from the nucleus is also \(0.53 \AA\). The difference between the two statements is that the Bohr model suggests a definite orbit, while the quantum mechanical description provides a probability of the electron's position.

(b) Description of particle location in Schrödinger's wave equation vs. classical physics

In classical physics, the location of a particle is described by a definite position and its velocity at a given time. However, Schrödinger's wave equation gives us a probability distribution representing the likelihood of finding a particle in a certain location. For any point in space, this function gives a probability distribution instead of a precise location, reflecting the principles of quantum mechanics, which states that subatomic particles do not have well-defined positions or velocities. This approach contrasts with the deterministic description of classical physics.

(c) Physical significance of the square of the wave function

The square of the wave function, or \(|\psi|^2\), is a measure of the probability density of finding the electron at any given point in space. The value of \(|\psi|^2\) for a specific point represents the likelihood of the electron being found around that point. By integrating \(|\psi|^2\) over a particular region, one can determine the probability of finding the electron in that region.

(6.55a) Possible values of l for n=4

For an electron in the n=4 energy level, the possible values of the angular momentum quantum number l are given by a range of integers from 0 to n-1. Since n=4, the possible values are \(l = 0,1,2,3\).

(6.55b) Possible values of ml for l=2

The magnetic quantum number, ml, has possible values from -l to +l, including 0. For l=2, the possible values for ml are \(-2, -1, 0, 1, 2\).

(6.55c) Possible values of l for ml=2

Since ml values range from -l to +l, the minimum value of l necessary for ml=2 is l=2. Any energy level n with l=2 or greater will have ml=2 as one of the possible ml values. Therefore, the possible values for l when ml=2 are \(l = 2, 3, 4, 5, ...\).

Key concepts

Bohr Model

The Bohr Model, introduced by Niels Bohr in 1913, revolutionized our understanding of atomic structure. In this model, electrons are depicted as orbiting the nucleus in distinct paths or "orbits" with defined radii. This idea was groundbreaking at the time because it provided a quantized view of atomic physics. Bohr suggested that an electron in the hydrogen atom in its ground state orbits the nucleus at a radius of approximately \(0.53 \text{Å}\), known today as the Bohr radius.
However, the limitation of the Bohr Model lies in its simplistic and classical representation akin to planets orbiting the sun. It does not account for electron behavior in terms of probabilities, which is crucial in explaining phenomena at the atomic scale. As a bridge to more modern quantum mechanics, it is useful principally to initiate visualization of atomic structure.
Understanding the Bohr Model helps in appreciating how scientific models evolve, setting the stage for the complex quantum descriptions that succeeded it.

Schrödinger's Wave Equation

Schrödinger's Wave Equation is a fundamental formula used in quantum mechanics, introduced by Erwin Schrödinger in 1926. This equation is pivotal because it allows for the calculation of wave functions, which are crucial in determining the probabilistic location of particles at the quantum level.
Unlike classical physics, which predicts an exact position and velocity for particles, Schrödinger's equation offers a probability distribution for electron positions. This nature is why quantum mechanics diverges from deterministic descriptions of classical physics, suggesting that particles have a certain probability of being at various locations rather than a set path.
The wave function itself encodes complex information about a system. When solved, this equation provides insights into the electron distribution, energies, and even interactions within atoms, reflecting the uncertainty and probabilistic nature of quantum particles.

Probability Density

In quantum mechanics, probability density gives us insight into the likelihood of finding a particle, such as an electron, in a given space. This is associated with the square of the wave function, denoted as \( \psi^2 \). The value of \( \psi^2 \) at a specific point in space provides the probability density, essentially a map of where an electron is most likely to be observed.
To visualize, imagine the atom like a cloud where the denser parts of the cloud represent where an electron has a higher probability of being found. The concept of probability density refines our approach from seeing electrons as particles on fixed paths to understanding them as having certain, yet uncertain, regions of probable positioning around the nucleus.

• This concept helps explain various atomic behaviors, such as chemical bonding and electron arrangements.
• It is fundamental in predicting chemical properties and reactions, offering a more nuanced view than classical interpretations.

Quantum Numbers

Quantum Numbers are a set of numerical values that describe the unique quantum state of an electron in an atom. Each electron in an atom has its own set of four quantum numbers, and these numbers describe characteristics such as the energy level, shape, and orientation of the orbital.

• **Principal quantum number \(n\):** Indicates the energy level of the electron, taking values \(n = 1, 2, 3, ...\).
• **Angular momentum quantum number \(l\):** Determines the shape of the orbital, with possible ranges from \(0\) to \(n-1\).
• **Magnetic quantum number \(m_l\):** Describes the orientation, taking integer values between \(-l\) and \(+l\).
• **Spin quantum number \(m_s\):** Represents the spin direction of the electron, either \(+1/2\) or \(-1/2\).

Understanding quantum numbers is essential for predicting electron configurations and assessing the chemical behavior of atoms. For instance, for an electron with \(n=4\), permissible \(l\) values range between \(0\) and \(3\), further refining the nuanced picture offered by quantum mechanics.