By Umezawa H.

This paintings starts off by way of distinguishing the adaptation among quantum mechanics and quantum box concept. It then makes an attempt to increase box concept through including a thermal measure of freedom to phenomena taking place inside of a vacuum. The ensuing quantum box conception is named Thermo box Dynamics (TFD).

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R A ) = EΨ0 (r 1 , r 2 , . . 6) can be separated by using the ansatz wave function Ψ0 (r 1 , r 2 , . . , r A ) = φα1 (r 1 )φα2 (r 2 ) · · · φαA (r A ) . 8) with the separation constants εαi satisfying the condition A E= εαi . 9) i=1 The solution of the many-nucleon Schr¨ odinger equation is thus a product of single-particle wave functions obtained by solving a one-nucleon Schr¨ odinger equation for an external potential well. In this way the mean-ﬁeld concept has turned the complicated many-nucleon problem into a simple one-nucleon one.

To begin with, let J 1 , J 2 , J 3 and J 4 be four commuting angular momentum vectors, with J = J 1 + J 2 + J 3 + J 4 their vector sum. We have now several possible ways to couple the four angular momenta to the total angular momentum J . Let us look more closely at the following two ways: J = J 12 + J 34 , J = J 13 + J 24 , J 12 ≡ J 1 + J 2 , J 13 ≡ J 1 + J 3 , J 34 ≡ J 3 + J 4 , J 24 ≡ J 2 + J 4 . e. |j1 j2 (j12 ) j3 j4 (j34 ) ; j m = |j1 j3 (j13 ) j2 j4 (j24 ) ; j m . 80) the number of linearly independent states is the same and the two sets both form complete sets of states.

The two separate reduced matrix elements are combined by a 9j symbol to produce the desired ﬁnal matrix element. In this way the original, usually rather complicated reduced matrix element can be expressed as a product of two, usually much simpler, matrix elements. g. to two diﬀerent particles or to the spatial and spin variables of one particle. 55) × (j1 T L j1 )(j2 S L j2 ) . 54), is applied to a single particle with spin. The spin–orbit-coupled wave functions are of the 34 2 Tensor Operators and the Wigner–Eckart Theorem form |n l 12 j m .