Section 4-6
Section 4-6 Angular Momentum and Magnetic Moment
development and is contained in Appendix 4. (It is recommended that Chapter 6 be completed before reading Appendix 4.) We give only the results. They are ˆLzfl,m = m¯hfl,m, m = −l,−l + 1,... ,l − 1,l
(4-60)
ˆL2fl,m = l (l + 1) ¯h2fl,m
(4-61)
These look like the results already given in Eqs. (4-55) and (4-57). There is a difference, however. Here there is no indication that m is an integer, whereas in Eq. (4-55) m must be an integer, as indicated by the presence of zero in its value list. There are two ways in which we can have a sequence of the form −l, −l + 1, . . . , l − 1, l. One way is to have an integer series, for example, −2, −1, 0, +1, +2, which must contain zero.
The other way is to have a half-integer series, for example, − 3 , − 1 , + 1 , + 3 , which
2
2
2
2
skips zero. If we work only with the properties of the operators, we find that either possibility is allowed. But if we assume that the as-yet-unspecified eigenfunctions fl,m are separable into θ - and φ-dependent parts, we find ourselves restricted to the integer series. For orbital angular momentum (due to motion of the electron in the atomic orbital), the z component must be (in atomic units) an integer, for we have seen that the state functions ψ contain the spherical harmonics Yl,m, which are indeed separable.
Spin angular momentum for an electron (to be discussed in more detail in the next chapter), has half-integer z components of angular momentum, and the eigenfunctions corresponding to spin cannot be expressed with spherical harmonics.
4-6 Angular Momentum and Magnetic Moment
If a charged particle is accelerated, a magnetic field is produced. Since circular motion of constant velocity requires acceleration (classically) it follows that a charged particle
having angular momentum will also have a magnetic moment. The magnetic moment is proportional to the angular momentum, colinear with it, and oriented in the same direction if the charge is positive. For an electron, the magnetic moment is given by
µ = −βeL
(4-62)
where βe, the Bohr magneton, has a value of 9.274078 × 10−24 J T−1 (equal to 1 a.u.),
2
where T is magnetic field strength in Tesla. (βe is defined to contain the ¯h that belongs
√
to L, so it is only the l(l + 1) part of L that is used in the calculation.)
EXAMPLE 4-6 What is the magnitude of the orbital magnetic moment for an elec tron in a 3d state of a hydrogen atom? In a 4d state of He+?
√
√
SOLUTION For any d state, l = 2, so, in a.u., |µ| = βeL = βe l(l + 1) = 6βe = 2.27 × 10−23JT−1. (Since we want magnitude, we can ignore the minus sign.) The value does not depend on the quantum number n nor on atomic number Z, so it is the same for He+.
Applying a magnetic field of strength B defines a z-direction about which the mag netic moment vector precesses. The z-component, µz, of the precessing vector interacts with the applied field B. The interaction energy is E = −µzB = βeLzB = βemB
(4-63)
Chapter 4 The Hydrogenlike Ion, Angular Momentum, and the Rigid Rotor
This means that some of the degeneracies among the energy levels of the hydrogenlike ion will be removed by imposing an external magnetic field. For instance, the 2p+1 and 2p−1 energy levels will be raised and lowered in energy, respectively, while 2s and 2p0 will be unaffected (see Fig. 4-14). This, in turn, will affect the atomic spectrum for absorption or emission. The splitting of spectral lines due to the imposition of an external magnetic field is known as Zeeman splitting. Because the splitting of levels depicted in Fig. 4-14 is proportional to the z component of orbital angular momentum, given by m ¯h, it is conventional to refer to m as the magnetic quantum number.
In the absence of external fields, eigenfunctions having the same n but different l and m are degenerate. We have seen that this allows us to take linear combinations of eigenfunctions, thereby arriving at completely real eigenfunctions like 2px and 2py, instead of 2p+1 and 2p−1. When a magnetic field is imposed, the degeneracy no longer exists, and we are unable to perform such mixing. Under these conditions, 2px, 2py, 3dxy, etc. are not eigenfunctions, and we are restricted to the pure m = 0, ±1, ±2 . . .
type solutions.
Thus far we have indicated that the stationary state functions for the hydrogenlike ions are eigenfunctions for ˆ L2 and ˆ Lz, and we have compared this to the fact that |L| and Lz are constants of motion for a frictionless gyroscope precessing about an external field axis. But how about atoms with several electrons? And how about molecules? Are their stationary state functions also eigenfunctions for ˆ L2 and ˆ Lz? A general approach to this kind of question is discussed in Chapter 6. For now we simply note that the spherical harmonics are eigenfunctions of ˆ L2 and ˆ Lz [Eqs. (4-55) and (4-57)] and that any state function of the form ψ(r, θ, φ) = R(r)Yl,m(θ, φ) will necessarily be an eigenfunction of these operators. But the spherical harmonics are solutions associated with spherically symmetric potentials. Therefore, it turns out that eigenfunctions of the time-independent hamiltonian operator are also eigenfunctions for ˆ L2 and ˆ
Lz onlyif the potential is spherically symmetric. In the more restricted case in which ψ has
Figure 4-14 Energy levels of a hydrogenlike ion in the absence and presence of a z-directed magnetic field.
development and is contained in Appendix 4. (It is recommended that Chapter 6 be completed before reading Appendix 4.) We give only the results. They are ˆLzfl,m = m¯hfl,m, m = −l,−l + 1,... ,l − 1,l
(4-60)
ˆL2fl,m = l (l + 1) ¯h2fl,m
(4-61)
These look like the results already given in Eqs. (4-55) and (4-57). There is a difference, however. Here there is no indication that m is an integer, whereas in Eq. (4-55) m must be an integer, as indicated by the presence of zero in its value list. There are two ways in which we can have a sequence of the form −l, −l + 1, . . . , l − 1, l. One way is to have an integer series, for example, −2, −1, 0, +1, +2, which must contain zero.
The other way is to have a half-integer series, for example, − 3 , − 1 , + 1 , + 3 , which
2
2
2
2
skips zero. If we work only with the properties of the operators, we find that either possibility is allowed. But if we assume that the as-yet-unspecified eigenfunctions fl,m are separable into θ - and φ-dependent parts, we find ourselves restricted to the integer series. For orbital angular momentum (due to motion of the electron in the atomic orbital), the z component must be (in atomic units) an integer, for we have seen that the state functions ψ contain the spherical harmonics Yl,m, which are indeed separable.
Spin angular momentum for an electron (to be discussed in more detail in the next chapter), has half-integer z components of angular momentum, and the eigenfunctions corresponding to spin cannot be expressed with spherical harmonics.
4-6 Angular Momentum and Magnetic Moment
If a charged particle is accelerated, a magnetic field is produced. Since circular motion of constant velocity requires acceleration (classically) it follows that a charged particle
having angular momentum will also have a magnetic moment. The magnetic moment is proportional to the angular momentum, colinear with it, and oriented in the same direction if the charge is positive. For an electron, the magnetic moment is given by
µ = −βeL
(4-62)
where βe, the Bohr magneton, has a value of 9.274078 × 10−24 J T−1 (equal to 1 a.u.),
2
where T is magnetic field strength in Tesla. (βe is defined to contain the ¯h that belongs
√
to L, so it is only the l(l + 1) part of L that is used in the calculation.)
EXAMPLE 4-6 What is the magnitude of the orbital magnetic moment for an elec tron in a 3d state of a hydrogen atom? In a 4d state of He+?
√
√
SOLUTION For any d state, l = 2, so, in a.u., |µ| = βeL = βe l(l + 1) = 6βe = 2.27 × 10−23JT−1. (Since we want magnitude, we can ignore the minus sign.) The value does not depend on the quantum number n nor on atomic number Z, so it is the same for He+.
Applying a magnetic field of strength B defines a z-direction about which the mag netic moment vector precesses. The z-component, µz, of the precessing vector interacts with the applied field B. The interaction energy is E = −µzB = βeLzB = βemB
(4-63)
Chapter 4 The Hydrogenlike Ion, Angular Momentum, and the Rigid Rotor
This means that some of the degeneracies among the energy levels of the hydrogenlike ion will be removed by imposing an external magnetic field. For instance, the 2p+1 and 2p−1 energy levels will be raised and lowered in energy, respectively, while 2s and 2p0 will be unaffected (see Fig. 4-14). This, in turn, will affect the atomic spectrum for absorption or emission. The splitting of spectral lines due to the imposition of an external magnetic field is known as Zeeman splitting. Because the splitting of levels depicted in Fig. 4-14 is proportional to the z component of orbital angular momentum, given by m ¯h, it is conventional to refer to m as the magnetic quantum number.
In the absence of external fields, eigenfunctions having the same n but different l and m are degenerate. We have seen that this allows us to take linear combinations of eigenfunctions, thereby arriving at completely real eigenfunctions like 2px and 2py, instead of 2p+1 and 2p−1. When a magnetic field is imposed, the degeneracy no longer exists, and we are unable to perform such mixing. Under these conditions, 2px, 2py, 3dxy, etc. are not eigenfunctions, and we are restricted to the pure m = 0, ±1, ±2 . . .
type solutions.
Thus far we have indicated that the stationary state functions for the hydrogenlike ions are eigenfunctions for ˆ L2 and ˆ Lz, and we have compared this to the fact that |L| and Lz are constants of motion for a frictionless gyroscope precessing about an external field axis. But how about atoms with several electrons? And how about molecules? Are their stationary state functions also eigenfunctions for ˆ L2 and ˆ Lz? A general approach to this kind of question is discussed in Chapter 6. For now we simply note that the spherical harmonics are eigenfunctions of ˆ L2 and ˆ Lz [Eqs. (4-55) and (4-57)] and that any state function of the form ψ(r, θ, φ) = R(r)Yl,m(θ, φ) will necessarily be an eigenfunction of these operators. But the spherical harmonics are solutions associated with spherically symmetric potentials. Therefore, it turns out that eigenfunctions of the time-independent hamiltonian operator are also eigenfunctions for ˆ L2 and ˆ
Lz onlyif the potential is spherically symmetric. In the more restricted case in which ψ has
Figure 4-14 Energy levels of a hydrogenlike ion in the absence and presence of a z-directed magnetic field.