- The spin quantum number s characteristic of the type of particle 37 (often called simply its spin), can be written as: s = n 2 , where n may be zero or a natural number (“an integer or half-integer” number). - an integer s (e.g., s = 1 for deuteron, photon. - The magnetic 39 spin quantum number m s quantizes the z component of the spin angular momentum.. - The spin coordinate differs widely from a spatial coordinate, because it takes only 2s + 1 discrete values (Fig. - For electrons, the spin coor- dinate σ takes two values, often called. - “up” and “down”. - 37 Note, the length of the spin vector for an elementary particle is given by Nature once and for all.. - Thus, if there is any relation between the spin and the rotation of the particle about its own axis, it has to be a special relation. - One cannot change the angular momentum of such a rotation.. - A non-zero s value is associated to the magnetic dipole, which in magnetic field acquires 2s + 1 ener- getically non-equivalent positions.. - One of the values is represented by σ. - Diagram of the spin angular momentum vector for a particle with spin quantum number s = 1 2 . - The only measurable quantities are the spin length. - 2 3 h ¯ and the projection of the spin on the quantization axis (chosen as coincident with the vertical axis z), which takes only the values. - Possible positions of the spin angular momentum with respect to the quantization axis z (b) since the x and y components of the spin remain indefinite, one may visualize the same by locating the spin vector (of constant length. - According to the postulate (p. - 25), the square of the spin length is always the same and equal to s(s + 1) h ¯ 2 = 3 4 h ¯ 2 . - The maximum projection of a vector on a chosen axis is equal to 1 2 h, while the length of the vector is larger, equal to. - We conclude that the vector of the spin angular momentum. - We shall now construct operators of the spin angular momentum.. - 0 −1 40 In the general case, the spin of a particle may take the following angles with the quantization axis:. - 41 In the same spirit as wave functions represent vectors: vector components are values of the function for various values of the variable.. - Indeed, after applying S ˆ z to the spin basis functions one obtains:. - Therefore, functions α and β represent the eigenfunctions of the S ˆ z operator with corresponding eigenvalues 1 2 h ¯ and − 1 2 h. - Therefore, both basis functions α and β represent the eigenfunctions of S ˆ 2 and correspond to the same eigen- value. - Wolfgang Pauli re- ceived the Nobel Prize in 1945 “for the discovery of the Exclusion Principle, also called the Pauli Principle”.. - 42 These formulae are easy to memorize, since the sequence of the indices is always “rotational”, i.e.. - and one of the components of S (denoted by S z = N. - Particular values of S (often called simply the spin) and of the spin magnetic number M S depend on the directions of vectors s i . - The ground states of the important nuclei 12 C and 16 O correspond to S = 0, while those of 13 C, 15 N, 19 F have S = 1 2. - 43 Also, note that the mean values of S x and S y are both equal to zero in the α and β state, e.g., for the α state one has. - This means that in an external vector field (of direction z), when the space is no longer isotropic, only the projection of the total angular momentum on the field direction is conserved. - This means that the total electron spin momentum moves on the cone surface making an angle of 5474 ◦ with the external field axis in α state and an angle 180. - 5474 ◦ in the β state. - Whatever the motion, it must satisfy α| ˆ S x α = α| ˆ S y α = 0 and β| ˆ S x β = β| ˆ S y β = 0 No more information is available, but one may imagine the motion as a precession just like that of the top.. - the sodium atom with 23 nucleons (each of spin 1 2 ) in the nucleus and 11 electrons moving around it, also represents a boson.. - When one adds together two electron spin vectors s 1 + s 2 , then the maximum z component of the spin angular momentum will be (in h ¯ units. - This corresponds to the vectors s 1 s 2 , called “parallel” to each other, while the minimum | M S. - If no direction in space is privileged, then all the three states correspond to the same energy (triple degeneracy). - For the singlet state |S| 2 = S(S + 1) h ¯ 2 = 0 hence 1 + cos ω = 0 and ω = 180 ◦ This means the two electronic spins in the singlet state are antiparallel.. - For the triplet state | S | 2 = S(S + 1) h ¯ 2 = 2 h ¯ 2 and hence 3 2 (1 + cos ω) h ¯ 2 = 2 h ¯ 2 , i.e. - Despite forming the angle ω = 7052 ◦ the two spins in the triplet state are said to be “parallel”.. - However, in the same hydrogen molecule we have two protons, whose spins may also be “parallel” (orthohydrogen) or antiparallel. - Spin angular momentum for a system with two electrons (in general, particles with s = 1 2. - Then, the spin vectors of individual electrons (see Fig. - the other is triply degenerate (triplet state), and the three components of the state have S = 1 and S z = 1 0 −1 in h ¯ units. - In the singlet state (a) the vectors s 1 and s 2 remain on the cones of different orientation, and have the op- posite (“antiparallel”) orientations, so that s 1 + s 2 = 0. - The three triplet components (b,c,d) differ by the direction of the total spin angular momentum (of constant length. - In each of the three cases the angle between the two spins equals ω = 7052 ◦ (although in textbooks – including this one – they are said to be “parallel”. - Unlike classical mechanics, quantum mechanics is radical: it requires that two particles of the same kind (two electrons, two protons, etc.) should play the same role in the system, and therefore in its description enshrined in the wave. - 45 Quantum mechanics guarantees that the roles played in the Hamil- tonian by two identical particles are identical. - Within this philosophy, exchange of the labels of two identical particles (i.e. - In short, 1 ↔ 2) leads, at most, to a change of the phase φ of the wave function: ψ(2 1. - However, when we exchange the two labels once more, we have to return to the initial situation: ψ(1 2. - spin integer par- ticles) 1 2 3 N has to be symmetric with respect to the exchange of coordinates x i y i z i σ i and x j y j z j σ j , i.e. - Let us see the probability density that two identical fermions occupy the same position in space and, additionally, that they have the same spin coordinate (x 1 y 1 z 1 σ 1. - Conclusion: two electrons of the same spin coordinate (we will sometimes say: “of the same spin”) avoid each other. - 47 The reason for the hole is the antisymmetry of the electronic wave. - Thus, the probability density of finding two identical fermions in the same position and with the same spin coordinate is equal to zero. - They can be at the same point in space.. - It requires that all pairs of the identical particles follow the same rule.. - 47 Besides that any two electrons avoid each other because of the same charge (Coulombic hole). - 48 The Pauli exclusion principle is sometimes formulated in another way: two electrons cannot be in the same state (including spin). - The wave function evolves in a deterministic way according to the time- dependent Schrödinger equation (1.10).. - From axioms of quantum mechanics one can prove that a product of errors (in the sense of standard deviation) of measurements of two mechanical quantities is greater than or equal to 1 2 [ ˆ A B], where ˆ [ ˆ A B] ˆ is the mean value of the commutator [ ˆ A B ˆ. - Wieman and Wolfgang Ketterle (Nobel Prize 2001 “for discovering a new state of matter. - In the Bose condensate the bosons (alkali metal atoms) are in the same place in a peculiar sense. - The total wave function for the bosons was, to a first approximation, a product of identical nodeless wave functions for the particular bosons (this assures proper symmetry).. - Each of the wave functions extends considerably in space (the Bose condensate is as large as a fraction of a millimetre), but all have been centred in the same point in space.. - Heisenberg received the Nobel Prize in 1932 “for the cre- ation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen”.. - However, just after re- turning from his honeymoon, the rector of the university called him, saying that there was a problem. - In the SS weekly an article by Prof. - Johannes Stark (a Nobel Prize winner and faithful Nazi) was about to appear claim- ing that Professor Heisenberg is not such a good patriot as he pretends, because he so- cialized in the past with Jewish physicists. - The interrogation took place in the basement.. - One of the questioners was a Ph.D. - In the post scriptum there was a precisely tailored phrase:. - Recall the definition of the variance, or the square of the standard deviation (A) 2 of measurements of the quantity A:
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