Each quantum state of the hydrogen atom is labeled by a set of four quantum numbers: {n, ℓ, mℓ,

Each quantum state of the hydrogen atom is labeled by a set...

Key Concepts

Quantum Numbers

Quantum numbers are a set of numerical values used to uniquely describe the state of an electron in an atom. They arise from the solutions of the Schrödinger equation and include values such as the principal quantum number (n), which primarily determines the energy of the orbital, the orbital angular momentum quantum number (l), the magnetic quantum number (m?) that describes the orientation of the orbital, and the spin quantum number (m?) that specifies the electron's spin direction.

Energy Level Degeneracy

Degeneracy in quantum mechanics refers to the phenomenon where multiple distinct quantum states share the same energy level. In the context of the hydrogen atom, degeneracy occurs because for a given principal quantum number n, different combinations of l, m?, and m? yield the same energy. This leads to a total degeneracy of 2n² for each energy level, reflecting the contributions from both orbital and spin degrees of freedom.

Hydrogen Atom Quantum Model

The hydrogen atom model is a fundamental system in quantum mechanics that consists of a single electron orbiting a proton. The simplicity of this system allows for analytical solutions of the Schrödinger equation, yielding quantized energy levels and well-defined quantum numbers, which provide deep insight into atomic structure and electron behavior.

Principal Quantum Number

The principal quantum number (n) is one of the key quantum numbers that characterizes the energy and size of an electron's orbital in an atom. In the hydrogen atom, the energy of the electron depends solely on n, and it determines the radial distribution of the electron's probability density. Each increase in n corresponds to a higher energy level and a larger orbital size.

Transcript

00:01 Here we're going to look at the degeneracy in each level in the hydrogen atom.

00:06 And what we mean by degeneracy is states with identical energies.

00:13 So recall that n is the principal quantum number that delineates the energy state.

00:19 And we are going to use an induction approach to prove that degeneracy is 2n squared.

00:30 So we'll start and show that this is true for the first three levels, and then we will carry it forward to any n.

00:40 There are four principal quantum numbers in the hydrogen atom.

00:45 The first one is the primary principal energy level number called n, and it goes with the energy.

00:58 The second one is l, which goes with the angular momentum.

01:03 Of an individual state.

01:08 The next number is m sub -l, which is sometimes called the magnetic quantum number or the z component of the angular momentum.

01:21 It can only take on discrete values.

01:24 And the last one is the spin of the electron.

01:30 And we'll start with n equals 1.

01:32 That's the easiest one.

01:34 But all of these are going to have a degeneracy of because the electron can have a spin plus or minus one half.

01:44 The l, angular momentum, starts with zero and goes to one less than the n.

01:53 So there's only one possible l for n equals one.

01:57 And then m sub l goes from minus l to positive l in integer steps.

02:06 And that can only be zero.

02:07 So for n equals 1, let's make a little list of the degeneracy is 2.

02:15 And so far so good, our formula is working.

02:20 Let's do 2n squared is 2.

02:24 So yeah, that formula looks good for the first energy level.

02:28 For n equals 2, we have l equals 0 and 1.

02:35 With 0, it's the same thing as what we had for the l equals n equals one state.

02:45 And the next state goes from plus one all the way down to minus one.

02:54 So there are three states, each of which has two spin states.

03:01 So that's six.

03:03 And our degeneracy total is eight.

03:09 2n squared is 8.

03:13 So that's looking good.

03:16 We'll keep on going.

03:20 N equals 3.

03:21 L can be 0, 1, and 2.

03:24 And we'll keep everything we had before, but add one more l state.

03:33 Whoops, that should be minus 1.

03:38 And the l equals 2 goes all the way from plus 2 to minus 2 in steps of 1...

Please give Ace some feedback