A hydrogen atom in a 3p state is placed in a uniform...

A hydrogen atom in a 3$p$ state is placed in a uniform external magnetic field $\overrightarrow{\boldsymbol{B}} .$ Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment. (a) What field magnitude $B$ is required to spit the 3$p$ state into multiple levels with an energy difference of $2.71 \times 10^{-5} \mathrm{eV}$ between adjacent levels? (b) How many levels will there be?

Key Concepts

Zeeman Effect

The Zeeman effect is the splitting of atomic energy levels into multiple components when an external magnetic field is applied. This occurs because the magnetic field interacts with the magnetic dipole moments associated with the angular momentum of electrons, causing shifts in the energy levels that depend on the magnetic quantum number. It is fundamental for explaining how a uniform magnetic field can cause distinct splittings in the spectrum of an atom.

Magnetic Quantum Number

The magnetic quantum number, often denoted as m, specifies the component of the orbital angular momentum of an electron along a chosen axis (typically the direction of the external magnetic field). For a given orbital angular momentum quantum number l, m can take on values ranging from -l to +l in integer steps. This quantum number determines the number of distinct energy levels that appear in the presence of a magnetic field, as each value of m corresponds to a different energy shift due to the Zeeman effect.

Bohr Magneton

The Bohr magneton is a physical constant that represents the natural unit for expressing the magnetic dipole moment of an electron due to its orbital (or spin) angular momentum. It is defined in terms of fundamental constants and is crucial for quantifying the interaction energy between an electron’s magnetic moment and an external magnetic field. In the context of the Zeeman effect, the Bohr magneton provides the scaling factor for the energy splitting between different m levels.

Orbital Magnetic Dipole Moment

The orbital magnetic dipole moment arises from the motion of an electron in its orbital path, which creates a magnetic moment analogous to a current loop. This magnetic moment is proportional to the orbital angular momentum and is given by a factor involving the electron charge and mass. Its interaction with an external magnetic field leads to energy shifts in the atomic levels, a primary cause of the Zeeman splitting of spectral lines.

Transcript

00:01 In this problem, we're told that we have a hydrogen atom in a 3p state.

00:08 And we're asked two things about this situation, namely we want to know what kind of magnetic field, or what is the value of a magnetic field that would give splitting between levels equal to a delta e of 2 .71 times 10 to the negative fifth, electron volts.

00:34 And the second question is if we have the scenario, then how many levels will there be due to this introduction of this magnetic field or due to what is otherwise called the zeman splitting? so in the first case, it's actually easier to figure out how many levels there are before we figure out the value of b.

00:55 So if we think about 3p, what this actually tells us is that we have two quantum numbers to consider.

01:05 The first one is that n is equal to three, our principal quantum number.

01:09 And then p, if we think about the chemistry notation or spectrographic notation, we have s p, d, and f, where s corresponds is zero, p is one, d is two, f is three.

01:24 So if we have p, this means that our l value is one.

01:28 So that is one.

01:31 And this also tells us that ml is bounded by l here, because we know that ml is going to be in the range from negative l to positive l.

01:48 Okay? so that means that m sub l is going to be between negative 1 and 1 in this case.

01:55 So that means right away we know that we have three possible energy levels.

02:00 When m sable is negative one, zero, and one.

02:04 So then the next thing to look at is if we know this, and we start to consider the energy splitting between these levels, we know that there's an expression which actually will use you because that's how it appears in the book, which goes like you is equal to negative mu sub z, the magnetic moment in z direction, times magnetic field, where we know that mu sub z is this value here.

02:32 So let's rewrite it...

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