bei48482_FM

Quantum Theory of the Hydrogen Atom 211

h

h

h

–2

0

2

Lz

ml = 2 l = 2

ml = 1

ml = 0

ml = –1

ml = –2

h h

h

L = l(l + 1)

= 6

Figure 6.4Space quantization of orbital angular momentum. Here the orbital quantum number is

l2 and there are accordingly 2l 1 5 possible values of the magnetic quantum number ml, with

each value corresponding to a different orientation relative to the zaxis.

L = l(l + 1)h

(b)

z

L

∆z

θ

ml

(a)

z

L

∆z = 0

h

Figure 6.5The uncertainty prin-

ciple prohibits the angular mo-

mentum vector Lfrom having a

definite direction in space.

The space quantization of the orbital angular momentum of the hydrogen atom is

show in Fig. 6.4. An atom with a certain value of mlwill assume the corresponding

orientation of its angular momentum Lrelative to an external magnetic field if it finds

itself in such a field. We note that Lcan never be aligned exactly parallel or antiparallel

to Bbecause Lzis always smaller than the magnitude l(l 1 ) of the total angular

momentum.

In the absence of an external magnetic field, the direction of the zaxis is arbitrary.

What must be true is that the component of Lin anydirection we choose is ml. What

an external magnetic field does is to provide an experimentally meaningful reference

direction. A magnetic field is not the only such reference direction possible. For

example, the line between the two H atoms in the hydrogen molecule H 2 is just as

experimentally meaningful as the direction of a magnetic field, and along this line the

components of the angular momenta of the H atoms are determined by their mlvalues.

The Uncertainty Principle and Space Quantization

Why is only one component of Lquantized? The answer is related to the fact that L

can never point in any specific direction but instead is somewhere on a cone in space

such that its projection Lzis ml. Were this not so, the uncertainty principle would be

violated. If Lwere fixed in space, so that Lxand Lyas well as Lzhad definite values,

the electron would be confined to a definite plane. For instance, if Lwere in the

zdirection, the electron would have to be in the xyplane at all times (Fig. 6.5a). This

can occur only if the electron’s momentum component pzin the zdirection is infinitely

uncertain, which of course is impossible if it is to be part of a hydrogen atom.

However, since in reality only onecomponent Lzof Ltogether with its magnitude

Lhave definite values and LLz, the electron is not limited to a single plane

(Fig.6.5b). Thus there is a built-in uncertainty in the electron’s zcoordinate. The

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