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, \(\psi^{2}\) ?

Short answer

(a) In the Bohr model, a specific radius of 0.53 Å represents the actual orbit of the electron around the nucleus. In the quantum mechanical description, 0.53 Å is the most probable distance of the electron from the nucleus, considering the electron as a probability distribution rather than being in a fixed orbit. (b) Schrödinger's wave equation, unlike classical physics, considers the inherent uncertainty at the atomic level, thus providing a probabilistic view of particles' locations and behaviors instead of deterministic trajectories. (c) The square of the wave function \(\psi^2\) signifies the probability density of finding the particle in a certain region of space, thereby detailing how likely it is to pinpoint the particle at a specific point in space.

Step-by-step solution

(a) Comparing Bohr model and Quantum Mechanical description distances

The Bohr model assumes that the electron orbits the nucleus in circular paths of specific radii, whereas the quantum mechanical description treats the electron as a probability distribution. When both descriptions talk about 0.53 Å, the Bohr model refers to the radius of the electron's orbit, while the quantum mechanics description refers to the most probable distance at which the electron is likely to be found from the nucleus.

(b) Difference between Schrödinger's wave equation and Classical Physics

The classical description of a particle (such as an electron) relies on well-defined trajectories (like orbits in the Bohr model) and deterministic laws of motion. On the other hand, Schrödinger's wave equation belongs to quantum mechanics, which describes particles in terms of their wave functions. This approach takes into account the inherent uncertainty in the position and momentum of particles at the atomic level, thus providing a probabilistic view of their location and behavior, rather than definitive trajectories.

(c) Significance of the square of the wave function

The wave function, represented by \(\psi\), describes the probability amplitude of a particle's position, energy, and other characteristics. The square of the wave function, \(\psi^2\), has physical significance because it gives us the probability density of finding the particle in a certain region of space. In other words, it tells us how likely it is to find the particle at a specific point in space. The probabilities must be normalized to 1, meaning that the sum of all probabilities (∫\(\psi^2\) dV) within the entire space must equal 1, since the particle has to be somewhere in that space.

Key concepts

Bohr model

The Bohr model revolutionized the understanding of atomic structure in the early 20th century. Proposed by Niels Bohr in 1913, this model describes the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus, much like planets revolving around the Sun. According to this model, these electrons can only occupy certain allowed orbits, with each orbit corresponding to a specific energy level.

One of the key postulates of the Bohr model is that an electron in an atom can only have certain discrete energy values. When an electron transitions between these orbits, it must absorb or emit energy equal to the difference between the energy levels. This is observed in the discrete spectral lines of elements. Bohr's theory explains why the ground state, the state of lowest energy for the electron in a hydrogen atom, has a specific radius of 0.53 Ångström (Å). This distance represents the radius of the purported orbit in a classical sense, which was a significant step forward in atomic physics at the time.

Schrödinger's wave equation

When Erwin Schrödinger proposed his famous wave equation in 1926, he laid the groundwork for a fully developed quantum mechanical portrayal of subatomic particles. Schrödinger's wave equation allows us to calculate the wave function, \( \psi \), associated with a quantum particle such as an electron in an atom. This wave function is a mathematical expression that contains all the information about the system's state.

The equation itself is a linear partial differential equation that describes how the wave function changes over space and time. More importantly, it is central to quantum mechanics because it treats particles as waves with a certain amplitude and phase, rather than as fixed-point particles. This implies that rather than moving along a single, predictable path, particles like electrons occupy a cloud of probability, indicating where they are likely to be found at any given time. Solutions to this equation yield eigenfunctions corresponding to specific energy levels, similar to the allowed orbits in the Bohr model, but with a richer set of possibilities.

Probability amplitude

In the realm of quantum mechanics, the probability amplitude is a core concept that represents the complex probability of a particular quantum event, like an electron being at a certain location. This concept is described by the wave function, \( \psi \), which we have from Schrödinger's wave equation. However, \( \psi \) itself doesn't give a direct probability. Instead, its square magnitude, or absolute square \( |\psi|^2 = \psi^*\psi \), where \( \psi^* \) is the complex conjugate of \( \psi \), provides the probability density.

The probability of finding an electron within a specific volume of space around the nucleus can then be calculated by integrating the probability density over that volume. This provides the likelihood of the electron's presence in various regions around the nucleus and forms the basis for the electron cloud model which depicts areas where electrons are most likely to be observed.

Quantum mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is notably different from classical mechanics, which can describe the motion of everyday-sized objects. Quantum mechanics incorporates the key principles of uncertainty and probability, making it inherently probabilistic.

The heart of quantum mechanics is the quantum mechanical model, where subatomic particles are represented by wave functions, as introduced in Schrödinger's wave equation. This model implies that we can only predict the likelihood of finding a particle in a certain state, rather than determining its exact properties at any given moment. Quantum mechanics also accounts for phenomena like superposition, where particles may exist in multiple states simultaneously, and entanglement, where quantum particles can be interconnected in such a way that the state of one instantaneously influences the state of another, regardless of the distance separating them.