(a) Show that the total number of atomic states (including...

(a) Show that the total number of atomic states (including different spin states) in a shell of principal quantum number $n$ is $2n^2$. $[Hint$: The sum of the first $N$ integers 1 + 2 + 3 + $\cdots$ + $N$ is equal to $N$$(N + 1)$/2.] (b) Which shell has 50 states?

Key Concepts

Principal Quantum Number

This concept refers to the main energy level in which an electron resides in an atom. It is denoted by n and determines the overall size and energy of the orbital, with higher numbers corresponding to orbitals that are larger and have higher energy levels.

Orbital Angular Momentum Quantum Number

This quantum number, denoted by l, defines the shape of the electron’s orbital. For a given principal quantum number n, the orbital angular momentum quantum number can take on integer values from 0 to n-1. Each value of l is associated with a number of magnetic sublevels, which contribute to the orbital degeneracy.

Degeneracy and Magnetic Quantum Number

Degeneracy in this context refers to the number of quantum states that have the same energy. For a given l, the magnetic quantum number m_l ranges from -l to +l, resulting in 2l + 1 possible states. The total degeneracy in a shell is determined by summing these orbital state counts.

Electron Spin

Electron spin is an intrinsic form of angular momentum possessed by electrons, commonly represented as a quantum number with two possible values (+1/2 and -1/2). This property results in a twofold multiplicative factor for the number of states, as each electron orbital can accommodate two distinct spin states.

State Counting in Atomic Shells

Combining the contributions from orbital degeneracy and electron spin, the total number of states in a shell with principal quantum number n is calculated by summing (2l + 1) for each orbital (where l runs from 0 to n-1) and then multiplying by 2 to account for spin. This calculation ultimately leads to the formula 2n^2, representing the total number of atomic states including different spin orientations.

Transcript

00:01 So to show that the total number of atomic states in a shell of a principal quantum number of n is equal to 2n squared, we're going to consider the fact that the angular momentum quantum number l runs from 0 to n minus 1.

00:14 So if n is equal to 3, this is going to be 0, 1, 2.

00:17 Okay? so the maximum number of electrons a shell can hold is the sum of the electrons in its filled sub -shells.

00:23 So this number, n, is equal to the sum over l equal to 0 to n minus 1 because that's our range of l.

00:30 Of 2 times 2l plus 1.

00:34 Sorry, i forgot the plus 1 there.

00:40 Okay.

00:41 So we can expand this, right? and what this is equal to is 2.

00:47 Now, let's just replace it, right? so if l was equal to 0, this would be 1, right? if l was equal to 1, this would be 3.

00:55 If l was equal to 2, this would be 5, and so on, plus dot, dot, dot.

01:07 And then it would be two times n minus 1 plus 1 because the maximum number is when l is equal to n minus 1.

01:17 So this would be 2 times n minus 1 plus 1 close the bracket.

01:26 So 2 times n minus 1 plus 1 is the same as 2n minus 1.

01:30 So this is equal to, this is a bracket here, not a parenthesis, 2 times again we have 1 plus 3 plus dot dot dot and 2 in minus 1 plus 1 plus 1.

01:53 So that's 2n minus 2 plus 1 or 2n plus 1 or excuse me 2n minus 1, right? but the so the quantity in the bracket has in terms whose average value is equal to 1 half times 1 plus 2n minus 1...

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