Quantization space

An effect of space quantization of orbital angular momentum may be observed if a magnetic field is introduced along what we now identify as the z axis. The orbital angular momentum vector P, of magnitude Pi, may take up only certain orientations such that the component (Pi) along the z axis is given by... 

Figure 1.9 Space quantization of orbital angular momentum for T = 3...

Figure 1.10 Space quantization of electron spin angular momentum...

Just as with other angular momenta there is space quantization of rotational angular momentum so that the z component is given by... 

As we saw in Equation (1.61), space quantization of rolational angular momenlum of a diatomic or linear polyatomic molecule is expressed by... 

Equation (1.48) shows that, for I =, space quantization of nuclear spin angular momentum results in the quantum number Mj taking the values 5 or — 5. The nuclear spin wave function J/ is usually written as a or /i, corresponding to Mj equal to 5 or —5,... 

It can be shown quite easily that, for a filled sub-shell such as 2p or L = 0. Space quantization of the total orbital angular momentum produces 2L - - 1 components with M] = L, L —, —L, analogous to space quantization of f. In a filled sub-shell... 

This expression coincides with (2.24), obtained on the basis of classical concepts. Such a coincidence is understandable from the point of view of the correspondence principle. One must bear in mind that the limit J — 00 means nothing other than that the number of projections of the angular momentum upon the z-axis, permitted by the rule of space quantization and equalling 2J + 1, becomes infinitely large, and the angular momentum becomes classical. 

This) space quantization of the Kepler orbits is without doubt the most surprising result of the quantum theory. The simplicity of the results and their derivation is almost like magic. 

Experiments have demonstrated that the electron behaves rather like a spinning top and so has an intrinsic angular momentum, the value of which is s(s+l) l/21i = ( 3/2)1i, in which s(= 5) is the spin quantum number. Again, there is space quantization, and the components of angular momentum in a direction defined by an internal or applied magnetic field are j/i. 

The results of the Stern-Gerlach experiment were in complete contradiction to the classical interpretation and its predictions. Silver atoms turned out to possess a magnetic moment, but instead of a single, smeared-out distribution, two spots centered around Mup and Mdown were observed. Thus the magnetic moment of a silver atom is space-quantized by an inhomogeneous magnetic field, and this magnetic moment can adopt only two values, hz Hhl ... 

Let us first consider the normal Zeeman effect, which applies to transitions between electronic states with zero total spin magnetic moment, so-called singlet states. Like the projection Ms of S in the Stern-Gerlach experiment, the projection Ml of the spatial angular momentum L is space quantized in the external magnetic field. We shall describe the quantization of the spatial angular momentum by means of quantum mechanical methods in detail later. Suffice it to say that each state with spatial angular momentum quantum number L splits into 2L + 1 components, i.e., a P state (L = 1) splits into three components with... 

This number of levels is really doubled, however, by the electron spin. An electron has an intrinsic permanent magnetism, and associated with it a permanent angular momentum of magnitude %h/2ir. This can bo oriented in either of two opposite directions, giving a component of ih/2tt along a fixed direction. This, as will be seen, is in harmony with the space quantization just described, for the special case Z = For each stationary state of an electron neglecting spin, we can have the two possible orientations of the spin, so that actually an s level has two... 

It is very important and significant that L is equal to h [/ (/ + l)]1 /2, and not equal to h l in the old days this was called space quantization. This means that, even in the absence of external fields, the angular momentum vector L makes an angle of cos 1 (m [l (l + 1)] l/2)2with the z axis (polar axis). 

It is very important to realize that space quantization is a fundamental quantum effect that limits the allowed orientations of or (ie with respect to... 

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