\u03b1 \u03b2 invariant braceleftBig R T \u03b1 R \u03b2 R \u03b1 bracerightBig \u03b2 R T \u03b1 R \u03b2 \u03b2 R T \u03b1 \u03b2 \u03b2

# Α β invariant braceleftbig r t α r β r α

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α ) β invariant: braceleftBig R T ( α ) R ( β ) R ( α ) bracerightBig β = R T ( α ) R ( β ) β = R T ( α ) β = β . (4 . 66) Hence β is the axis of this rotation. Therefore R T ( α ) R ( β ) R ( α ) = R ( β ) = R ( R ( α ) β ) . (4 . 67) In Box 4.3 we showed that when | α | is infinitesimal, R ( α ) β β α × β , so when β is also infinitesimal, equation (4.67) can be written in terms of the classical generators (4.63) as (1 + i α · J ) (1 i β · J ) (1 i α · J ) 1 i( β α × β ) · J . (4 . 68) The zeroth order terms (‘1’) and those involving only α or β cancel, but the terms involving both α and β cancel only if α i β j [ J i , J j ] = i α i β j summationdisplay k ǫ ijk J k . (4 . 69) This equation can hold for all directions α and β only if the J i satisfy [ J i , J j ] = i summationdisplay k ǫ ijk J k , (4 . 70) which is identical to the ‘quantum’ commutation relation (4.33). Our red- erivation of these commutation relations from entirely classical considerations is possible because the relations reflect the fact that the order in which you rotate an object around two different axes matters (Problem 4.6). This is a statement about the geometry of space that has to be recognised by both quantum and classical mechanics. In Appendix D it is shown that in classical statistical mechanics, each component of position, x i , and momentum, p i , is associated with a Hermi- tian operator ˆ x i or ˆ p i that acts on functions on phase space. The operator ˆ p i generates translations along x i , while ˆ x i generates translations along p i (boosts). The operators ˆ L i associated with angular momentum satisfy the commutation relation [ ˆ L x , ˆ L y ] = i H ˆ L z , where H is a number with the same dimensions as ¯ h and a magnitude that depends on how ˆ x i and ˆ p i are nor- malised. If the form of the commutation relations is not special to quantum me- chanics, what is? In quantum mechanics, complete information about any system is contained in its ket | ψ ) . There is nothing else. From | ψ ) we can evaluate amplitudes such as ( x | ψ ) for the system to be found at x with orientation μ . If we do not care about μ , the total probability for | ψ ) to be found at x is Prob(at x | ψ ) = summationdisplay μ vextendsingle vextendsingle ( x | ψ ) vextendsingle vextendsingle 2 . (4 . 71) Eigenstates of the x operator with eigenvalue x 0 are states in which the system is definitely at x 0 , while eigenstates of the p operator with eigenvalue ¯ h k are states in which the system definitely has momentum ¯ h k . By contrast, in classical statistical mechanics we declare at the outset that a well defined state is one that has definite values for all measurable quantities, so it has a definite position, momentum, orientation etc. The eigenfunctions of ˆ p or ˆ L do not represent states of definite momentum or angular momentum, because we have already defined what such states are.

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• Physics, mechanics, The Land, David Skinner, probability amplitudes, James Binney, Physics of Quantum Mechanics