Angular momentum quantum numbers

In summary, for angular momentum quantum numbers j, l, and s, it is not generally true that m_s and m_l must be less than m_j. There are cases where the magnitude of m_j can be smaller than either m_s or m_l, due to the z components of spin and orbital angular momenta being in opposite directions. Additionally, when dealing with an electron, there may be ambiguity in determining the values of m_l and m_s for a given m_j. It is important to consider the Clebsch-Gordan coefficients in these cases.
  • #1
BeauGeste
49
0
For angular momentum quantum numbers j, l, and s must it be true that [tex] m_s, m_l < m_j [/tex]?
It would seem that it is true because I assume that [tex] m_s +m_l = m_j [/tex], but I have not actually seen that written down anywhere and am curious.
Thanks.
 
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  • #2
No. It is not generally true. Remember that m tells you the component of angular momentum along some certain direction. This means that it can be either positive or negative. Consider, then, the case where [tex]m_s = -\frac{1}{2}[/tex]. It is hopefully clear that [tex]m_l = m_j + \frac{1}{2}[/tex].
 
  • #3
ok, I think I meant the magnitude of the m's. i.e.
[tex] |m_s|, |m_l| \leq |m_j| [/tex].
For example, take the situation where [tex]m_j = -1/2[/tex], j=3/2, s=1/2. The orbital and spin angular momenta magnetic numbers can add to this for two cases:
1. [tex] m_s = -1/2, m_l = 0 [/tex].
2. [tex] m_s = 1/2, m_l = -1 [/tex].
I would argue from above that case 2 is not a viable option because [tex]|m_l| \nleq |m_j| [/tex].
What do you think?
 
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  • #4
If you're using magnitudes, you can't have negative numbers. However, even if you want to compare the magnitudes of [tex]m_s[/tex], [tex]m_l[/tex], and [tex]m_j[/tex], you'll find that there are states where [tex]|m_j|[/tex] is smaller than either [tex]|m_s|[/tex] or [tex]|m_l|[/tex]. All this requires is that the spin and orbital angular momenta have their z components in opposite directions. This is something we should expect from normal vector analysis (i.e. it has nothing to do with quantum mechanics specifically). If I add two vectors which have projections in opposite directions along the z-axis, I should expect that the magnitude of the z component of the resultant vector must be smaller than at least that of one of the two vectors I added; and, it may be smaller than both.
 
  • #5
darn, I wrote that wrong. Hopefully this makes my question more clear:

if j=3/2 then possible [tex] m_j [/tex] values are -3/2 to 3/2 by ones. if j=3/2 and we're dealing with an electron then l = 1 and s = 1/2. [tex] m_l [/tex] values are 1,0, and -1.
If [tex] m_j = - 3/2 (+3/2) [/tex] then of course [tex] m_l = -1 (+1) [/tex] and [tex] m_s = -1/2 (+1/2) [/tex] respectively. Those are the only possibilities to form [tex] m_j [/tex] for the maximum values of [tex] m_j [/tex].
Now when [tex] m_j = \pm 1/2[/tex] it seems ambiguous as to what [tex] m_{l,s} [/tex] are. For instance:
1. [tex] m_s = -1/2, m_l = 0 [/tex]
2. [tex] m_s = 1/2, m_l = -1 [/tex]
both give [tex] m_j = -1/2[/tex].

Is that ok or is one of them not correct?
 
  • #6
BeauGeste said:
1. [tex] m_s = -1/2, m_l = 0 [/tex]
2. [tex] m_s = 1/2, m_l = -1 [/tex]
both give [tex] m_j = -1/2[/tex].

The state with [itex]m_j = -1/2[/itex] is a linear combination of the two states with [itex](m_s = -1/2, m_l = 0)[/itex] and [itex](m_s = +1/2, m_l = -1)[/itex]. The coefficients of the linear combination are called Clebsch-Gordan coefficients:

http://farside.ph.utexas.edu/teaching/qm/lectures/node47.html

To relate the notation on that page to your example, substitute [itex]l[/itex] for its [itex]j_1[/itex], [itex]s[/itex] for its [itex]j_2[/itex], [itex]m_l[/itex] for its [itex]m_1[/itex], [itex]m_s[/itex] for its [itex]m_2[/itex], and finally [itex]m_j[/itex] for its [itex]m[/itex].
 
  • #7
BeauGeste said:
darn, I wrote that wrong. Hopefully this makes my question more clear:

if j=3/2 then possible [tex] m_j [/tex] values are -3/2 to 3/2 by ones. if j=3/2 and we're dealing with an electron then l = 1 and s = 1/2. [tex] m_l [/tex] values are 1,0, and -1.
If [tex] m_j = - 3/2 (+3/2) [/tex] then of course [tex] m_l = -1 (+1) [/tex] and [tex] m_s = -1/2 (+1/2) [/tex] respectively. Those are the only possibilities to form [tex] m_j [/tex] for the maximum values of [tex] m_j [/tex].
Now when [tex] m_j = \pm 1/2[/tex] it seems ambiguous as to what [tex] m_{l,s} [/tex] are. For instance:
1. [tex] m_s = -1/2, m_l = 0 [/tex]
2. [tex] m_s = 1/2, m_l = -1 [/tex]
both give [tex] m_j = -1/2[/tex].

Is that ok or is one of them not correct?

Adding to what jtbell said, you should also be aware that there are [tex]j=3/2[/tex] states arising from [tex]l=2,\ s=1/2[/tex]. Here, there are 10 possible [tex]|m_l, m_s\!\!>[/tex] states; but, only 8 of them give allowed values of [tex]m_j[/tex].
 
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1. What is the significance of the angular momentum quantum number in quantum mechanics?

The angular momentum quantum number, denoted as ℓ, describes the magnitude of angular momentum of an electron in an atom. It is a fundamental property of an electron and plays a crucial role in determining the energy levels and orbitals of an atom.

2. How is the value of the angular momentum quantum number determined for an electron?

The value of the angular momentum quantum number depends on the principal quantum number (n) and can have values ranging from 0 to n-1. It is determined by the shape of the orbital in which the electron is located.

3. What is the relation between the angular momentum quantum number and the orbital shape?

The angular momentum quantum number determines the shape of the orbital in which the electron is located. For example, an ℓ value of 0 corresponds to an s orbital, while an ℓ value of 1 corresponds to a p orbital. The higher the ℓ value, the more complex the orbital shape becomes.

4. Can two electrons in the same atom have the same angular momentum quantum number?

No, according to the Pauli exclusion principle, no two electrons in an atom can have the same set of quantum numbers, including the angular momentum quantum number. This means that if one electron has an ℓ value of 2, the other electron in the same atom must have a different ℓ value.

5. How does the angular momentum quantum number affect the energy of an electron in an atom?

The angular momentum quantum number plays a crucial role in determining the energy levels of an atom. It affects the energy through its relationship with the principal quantum number (n) and the magnetic quantum number (mℓ), which together determine the specific energy level and sublevel of an electron.

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