Chapter 7, Problem 7.62QE

Interpretation Introduction

Interpretation:

The sketch of contour surface for an electron that has quantum numbers $n$ equals to 3, $l$ equals to 0, $m_l$ equals to 0, and $m_s$ equals to $-\text{1/2}$ has to be drawn.

Concept Introduction:

Erwin Schrödinger gave a model to describe the behavior of an electron in atoms through a wave function. The wave function is defined as a mathematical function for an electron wave that gives the amplitude of the wave at any point in space. It is represented by a Greek letter $\Psi$.

The wave function of an electron has no physical significance but when the square of its absolute magnitude takes then it gives a probability of finding electron at any particular point of time.

The expression of the probability of finding electrons is as follows:

  $\text{Probability of finding electrons}=\left|{\Psi }^2\right|$        (1)

The acceptable wave function is characterized by a set of four quantum numbers. These The acceptable wave function is characterized by a set of four quantum numbers. These quantum numbers are related to the shape, size of the electron wave and the location of the electron in space.

1. These quantum numbers are represented by $n$, $l$, $m_l$, and $m_s$. The quantum number $n$ is known as principal quantum number and it allowed values are all positive whole numbers. The quantum number $l$ is known as angular momentum quantum number and its allowed values are all positive integer from 0 to $n-1$.

2. The notation of subshell is used to determine by a combination of both principal quantum number and angular quantum number. In notation, numerical values of $n$ is used while value of $l$ denotes the respective subshell. The table for the $l$ values is as follows:

$\begin{array}{cccccccc}l& 0& 1& 2& 3& 4& 5& 6\\ \text{Subshell}& s& p& d& f& g& h& i\end{array}$

3. The quantum number $m_l$ is known as the magnetic quantum number. It tells about the orbitals present in the subshell and its allowed values lie between $-l$ to $+l$ include 0.

4. The quantum number $m_s$ is known as the electron spin quantum number. It represents the spin of the electron and its allowed values are $+\text{1/2}$ and $-\text{1/2}$.