Questions on Quantum Numbers (introductory physics)

In summary, quantum numbers are numerical values used to describe the energy levels and properties of an electron in an atom. There are four quantum numbers - principal, angular momentum, magnetic, and spin - which determine the electron's location, energy, and spin within an atom. The principal quantum number (n) represents the energy level and size of an electron's orbital, while the angular momentum quantum number (l) describes the shape of the orbital. The magnetic quantum number (m<sub>l</sub>) can have both positive and negative values and determines the orbital's orientation in space.
  • #1
nonequilibrium
1,439
2
Hello, I'm taking a course in introductory modern physics, and I have a few questions on the nature of quantum numbers:

  1. Why is [tex]\Delta l = \pm 1[/tex]? My book simply states this is due the angular momentum of photons, but we have that [tex]L_{photon} = \pm 1 \hbar[/tex] and [tex]L_{electron} = \sqrt{l(l+1)} \hbar[/tex]. The fact that l must change with one doesn't seem compatible...?
  2. (possibly vague question, skip if confusing) What is the meaning of [tex]m_l[/tex] (measure for the projection of [tex]\vec L[/tex] on the z-axis) if we don't define a z-axis? Or in other words: sure we can always define a z-axis, but this can be totally arbitrary (assuming no external B-field or something), suggesting that due to symmetry it's true for every axis, but that is not true, so what is up with that?
  3. Why doesn't spin influence the spectral lines of certain atoms? I've read somewhere it's because sometimes the net spin of the atom is zero, but I don't see the relevance of that. For example: say we have an electron with spin up (and there's an external magnetic field) and it drops down from [tex]l=1,m_l=1[/tex] to [tex]l=0,m_l=0[/tex]. Now wouldn't we always expect a different energy jump (and thus some other frequency of the emitted light) if the electron had spin down? (due to the external magnetic field)
  4. Trying to understand the Stern & Gerlach experiment: looking up silver, we see that its configuration is [tex](Kr) 4d^{10} 5s^{1}[/tex], so if I understand correctly the reason [tex]m_l[/tex] does not play a role in the experiment is because in the s-orbit, [tex]m_l = 0[/tex], and in the other orbits for every [tex]m_l[/tex] there's another one with the opposing sign? But there is a net spin, resulting in the famous result of the experiment. Okay, that sounds understandable. This, however, means that quantummechanically, without the knowledge of spin, we would expect one line, i.e. no deviation (because net magnetic orbital number is zero). However, the experiment was to measure the quantization of [tex]L[/tex]: how were they planning to measure that? Or in other words: I don't understand what they (Stern & Gerlach) were expecting to see (which indicates I might be misunderstanding the whole experiment).

Thank you!
 
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  • #2
nonequilibrium said:
Why is [tex]\Delta l = \pm 1[/tex]? My book simply states this is due the angular momentum of photons, but we have that [tex]L_{photon} = \pm 1 \hbar[/tex] and [tex]L_{electron} = \sqrt{l(l+1)} \hbar[/tex]. The fact that l must change with one doesn't seem compatible...?
One has to be careful not to mix up a particle's angular momentum and the projection of that angular momentum. A photon is a spin-1 particle with projections ##\pm \hbar##, but the magnitude of the spin is ##\sqrt{s ( s+1)} = \sqrt{2} \hbar##, so there is no difference with how we consider angular momentum for the electron.

nonequilibrium said:
What is the meaning of [tex]m_l[/tex] (measure for the projection of [tex]\vec L[/tex] on the z-axis) if we don't define a z-axis? Or in other words: sure we can always define a z-axis, but this can be totally arbitrary (assuming no external B-field or something), suggesting that due to symmetry it's true for every axis, but that is not true, so what is up with that?
There is no ##m## quantum number without a defined axis, as one must specify which operator one is considering. If an atom is unpolarized (no preferred quantization axis), then it would be found in a superposition of ##m## states, whichever axis is chosen.

nonequilibrium said:
Why doesn't spin influence the spectral lines of certain atoms? I've read somewhere it's because sometimes the net spin of the atom is zero, but I don't see the relevance of that. For example: say we have an electron with spin up (and there's an external magnetic field) and it drops down from [tex]l=1,m_l=1[/tex] to [tex]l=0,m_l=0[/tex]. Now wouldn't we always expect a different energy jump (and thus some other frequency of the emitted light) if the electron had spin down? (due to the external magnetic field)
At the level of theory used, the electromagnetic field responsible for the transition between levels does not couple with the spin, therefore transitions involving a change in total spin can be ignored. However, because of spin-orbit coupling, some transitions will be possible or not depending on whether the spin of the electrons can help conserve angular momentum.

nonequilibrium said:
Trying to understand the Stern & Gerlach experiment: looking up silver, we see that its configuration is [tex](Kr) 4d^{10} 5s^{1}[/tex], so if I understand correctly the reason [tex]m_l[/tex] does not play a role in the experiment is because in the s-orbit, [tex]m_l = 0[/tex], and in the other orbits for every [tex]m_l[/tex] there's another one with the opposing sign? But there is a net spin, resulting in the famous result of the experiment. Okay, that sounds understandable. This, however, means that quantummechanically, without the knowledge of spin, we would expect one line, i.e. no deviation (because net magnetic orbital number is zero). However, the experiment was to measure the quantization of [tex]L[/tex]: how were they planning to measure that? Or in other words: I don't understand what they (Stern & Gerlach) were expecting to see (which indicates I might be misunderstanding the whole experiment).
In a classical world, one would still find that electrons have spin (spin is actually a consequence of special relativity), but any orientation would be possible. The SG experiment would thus have resulted in an elongated splotch. The fact that two spots appeared meant that spin was quantized (Stern and Gerlach could also deduce the value of that spin and found it was related to Planck's constant).
 

1. What are quantum numbers?

Quantum numbers are numerical values that are used to describe the energy levels and properties of an electron in an atom. They are used to determine the electron's location, energy, and spin within an atom.

2. How many quantum numbers are there?

There are four quantum numbers: the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (ml), and the spin quantum number (ms).

3. What is the significance of the principal quantum number?

The principal quantum number (n) represents the energy level of an electron in an atom. It can have any positive integer value and determines the size and energy of an electron's orbital.

4. How does the angular momentum quantum number affect an electron's behavior?

The angular momentum quantum number (l) describes the shape of an electron's orbital. It can have values ranging from 0 to n-1 and determines the subshell an electron occupies.

5. Can the magnetic quantum number have negative values?

Yes, the magnetic quantum number (ml) can have both positive and negative values, including 0. It determines the orientation of an electron's orbital in space.

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