The Schrodinger wave equation for hydrogen atom is $$ \Psi(\text { radial })=\frac{1}{16 \sqrt{4}}\left(\frac{z}{a_{0}}\right)^{3 / 2}\left[(\sigma-1)\left(\sigma^{2}-8 \sigma+12\right)\right] e^{-\sigma / 2} $$ where \(a_{0}\) and \(Z\) are the constant in which answer can be expressed and \(\sigma=\frac{2 Z r}{a_{0}}\) minimum and maximum position of radial nodes from nucleus are ......respectively. \(\therefore .\) 1 \(\begin{array}{llll}\text { (a) } \frac{a_{0}}{Z}, \frac{3 a_{0}}{Z} & \text { (b) } \frac{a_{0}}{2 Z}, \frac{a_{0}}{Z} & \text { (c) } \frac{a_{0}}{2 Z}, \frac{3 a_{0}}{Z} & \text { (d) } \frac{a_{0}}{2 Z}, \frac{4 a_{0}}{Z}\end{array}\)

Understanding the concept of radial nodes is a critical stepping stone in quantum mechanics, specifically when examining the Schrödinger wave equation for the hydrogen atom. Radial nodes are particular points within an atom's electron cloud where the probability of finding an electron drops to zero. These points occur because the wave function, which predicts the electron's behavior, is equal to zero at these positions.

In the exercise, the radial nodes are determined by setting the radial wave function to zero and solving for the variable \(\sigma\). The term within the square brackets of the wave function equation takes the form of a cubic equation: \[ (\sigma - 1)(\sigma^2 - 8\sigma + 12) = 0 \.\] Once the values of \(\sigma\) that nullify this equation are found, these are then translated into distances from the nucleus, directly providing the positions of the radial nodes.

The field of quantum mechanics is the underlying theory that governs the behavior of particles at the smallest scales, such as electrons in atoms. In quantum mechanics, particles like electrons are described not by definite positions or velocities but by wave functions that provide the probabilities of finding the particles in different places at different times.

The wave equation provided in the exercise is a direct application of quantum mechanics, specifically formulated by Erwin Schrödinger to describe the behavior of electrons in a hydrogen atom. Solving this equation allows us to predict where an electron is likely to be found and underlies how we know about electron configurations and energy levels within an atom.

The wave function, symbolized as \( \Psi \) in quantum mechanics, is a mathematical function that describes the quantum state of a particle or a system of particles. In essence, the wave function encapsulates all possibilities of a particle’s characteristics, such as position and momentum. However, it directly computes the probability amplitude—when squared, this gives the probability density of finding a particle in a given space.

The wave function provided in the exercise for the hydrogen atom, \( \Psi(\text{radial}) \), holds all the information needed to determine the electron's behavior around the nucleus. This specific wave function is complex, as it arises from solving the Schrödinger equation for the hydrogen atom, which takes into account both the radial distance from the nucleus and the angular components of the electron's position.

The concept of probability in quantum mechanics is a fundamental departure from classical physics. It expresses the likelihood of an event occurring, such as the location of an electron at any given moment. Instead of deterministic predictions, quantum mechanics provides probabilistic outcomes, heavily reliant on the wave function.

For the hydrogen atom's electron, the square of the wave function's magnitude, \( |\Psi|^2 \), gives us the probability density function. In practice, it signifies where we are most likely to find the electron. The exercise challenges the student to find the radial nodes where this probability becomes zero, emphasizing the quantum mechanical nature of particles exhibiting wave-like properties that can lead to such phenomena as nodes where the existence of the electron is improbable.