Understanding Hydrogen Atom Orbitals and Wave Functions

In quantum mechanics, the behavior of electrons in atoms, like the Hydrogen atom, is described by wave functions, denoted by $\psi$. For a Hydrogen atom or any one-electron species, the wave function $\psi$ depends on the quantum numbers: the principal quantum number ($n$), the angular momentum quantum number ($l$), and the magnetic quantum number ($m_l$). The wave function can be separated into two parts:

• Radial Wave Function ($R_{nl}(r)$): This part depends on the distance of the electron from the nucleus ($r$) and the quantum numbers $n$ and $l$. It determines the probability of finding the electron at a certain distance from the nucleus.
• Angular Wave Function ($Y_{lm_l}(\theta, \phi)$): This part depends on the angles $\theta$ and $\phi$ (in spherical coordinates) and the quantum numbers $l$ and $m_l$. It describes the shape of the orbital and its orientation in space.

The complete wave function is the product of these two parts: $\psi_{nlm_l}(r, \theta, \phi) = R_{nl}(r) \cdot Y_{lm_l}(\theta, \phi)$.

The Specific Case of 's' Orbitals

The question specifically asks about the 's' orbitals in the Hydrogen atom. For 's' orbitals, the angular momentum quantum number is $l=0$.

• When $l=0$, the angular wave function $Y_{00}(\theta, \phi)$ becomes a constant value, independent of the angles $\theta$ and $\phi$. This means that 's' orbitals do not have directional properties; they are spherically symmetrical.
• The radial wave function $R_{n0}(r)$, however, still depends on $r$, $n$, and $l=0$. This function dictates the probability distribution of the electron as a function of distance from the nucleus.

Therefore, the 's' orbital's characteristics (specifically its probability distribution) depend solely on the radial wave function and are independent of the angular wave function because the angular part is just a constant for $l=0$.

Analysis of Options

Let's analyze each statement based on this understanding:

1. Statement 1:
   For H-atom, s orbital is dependent on the radial wave function (r) only and independent of the angular wave function ($\theta$ and $\phi$).

   This statement aligns perfectly with our understanding. Since $l=0$ for s orbitals, the angular part $Y_{00}$ is constant, making the orbital dependent only on the radial part $R_{n0}(r)$.

2. Statement 2:
   For H-atom, s orbital is dependent on both the radial wave function (r) and the angular wave function ($\theta$ and $\phi$).

   This is incorrect. While 's' orbitals have radial dependence, they are independent of the angular part ($\theta$ and $\phi$) because $l=0$. Angular dependence arises for orbitals with $l \geq 1$ (like p, d, f orbitals).

3. Statement 3:
   For H-atom, there is no such dependence observed for s orbital; rather, the dependence starts with involvement of p orbital.

   This is incorrect. 's' orbitals have a clear dependence on the radial wave function, which determines their size and energy levels. Dependence on angular functions starts with p orbitals ($l=1$), not that it *only* starts there.

4. Statement 4:
   For H-atom, s orbital is dependent on the angular wave function ($\theta$ and $\phi$) only and independent of the radial wave function (r).

   This is the opposite of the actual situation. 's' orbitals are independent of the angular functions but are dependent on the radial function.

Based on the analysis, the first statement accurately describes the characteristics of the 's' orbital in the Hydrogen atom.