Basic quantum mechanics by J. M. Cassels FRS (auth.)

By J. M. Cassels FRS (auth.)

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1. l 2 , the chance of l being found to have the value l•. The equation ic 1 12 + ic2 12 + ic3 l2 = 1 corresponds to the usual relation for the sum of the squares of direction cosines. In general the vector representing I{! changes its direction as time goes on. One exception occurs if I{! represents a stationary state of energy E"' and f does not mention t explicitly; then the vector representing I{! does not move because the projection of I{! on each u. t and is therefore of MATHEMATICAL AND PHYSICAL DEVELOPMENT 25 constant length.

The commutator of two operators f and 1ft is defined to be fm- mf and written as [f, m]. 26) Typical commutation relations for the three-dimensional position and momentum of a particle are (pxx)l{! - ihl{! } ax ax (pxy)l{! 28) where the suffixes i, j take the values 1, 2, 3 for the x, y, z components of r and p respectively. If l does not commute with m it may be shown that fm is not a Hermitian operator, and therefore does not represent an observable. 29) The form of this equation shows that the symmetrical product fm + mf is always a Hermitian operator.

But this must be understood to mean (t{11 I kt{12 ) and not (kt{11 I t/Jz). r r Orthogonality of eigenfunctions If u 1 , u2 are two eigenfunctions of l belonging to the eigenvalues lt. 13) and so (1 2 -1 1)(u 1 I u2 ) = 0. 14) BASIC QUANTUM MECHANICS 22 and uh u2are said to be orthogonal. t Generally, if u,. and two eigenfunctions of f,

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