The Complex Conjugate

Multiplying both sides by J/, the complex conjugate of p obtained Ifom it by replacing all i (equal to — 1) by —i, and integrating over all space gives... 

Wavefunctions are often complex quantities, and we have to be careful to distinguish a wavefunction vp from its complex conjugate P. For most of this text, wavefunctions will be real quantities and so we can drop the complex conjugate sign without lack of mathematical rigour. 

If we multiply through from the left by the complex conjugate of t/cq we have... 

The time derivative of the coefficient c ft) for the particular state 4/ (t) is found by multiplying either side of the equation by the complex conjugate of 4/°(t) and integrating. After a little manipulation we find... 

The variation of either S4> ot (60 should make SL = 0. The first two terms in (3.38) should thus be zero, and the last two terms should be zero. Taking the complex conjugate of the last two terms and subtracting them from the first two gives... 

A linear coordinate transformation may be illustrated by a simple two-dimensional example. The new coordinate system is defined in term of the old by means of a rotation matrix, U. In the general case the U matrix is unitary (complex elements), although for most applications it may be chosen to be orthogonal (real elements). This means that the matrix inverse is given by transposing the complex conjugate, or in the... 

We note that Yf = Yfm, where the asterisk denotes the complex conjugate. 

The notation 3(f) will be used to denote the complex conjugate of 3(f)-... 

This space Jf is the set of all functions f(x) satisfying Eq. (8-2), and is in fact self-dual, because the complex conjugate of any function that satisfies Eq. (8-2), itself also satisfies Eq. (8-2), and so is in Jf. It is to be emphasized that the symbol /> represents the function f(x) with its entire range of values, not just the numerical value of the function at some arbitrary point. The variable x does not appear in the symbol /> for the element of... 

The operation of complex conjugation will be denoted by an overscore a denotes the complex conjugate of a. For a matrix A, with matrix elements al the hermitian conjugate matrix with elements afl will be denoted by A ... 

Finally upon taking the complex conjugate of Eq. (9-261), we infer that CC = cl, from which one deduces that c2 = c2 = c 2 so that c is real and with the normalization det (7=1... 

When defined in terms of a real basis such as the vectors e °( ) and e 2)(i), the complex conjugate vector is... 

The time reversal operator may now be determined.6 The simplest anti-unitary operation is the transition to the complex conjugate. This operator K is clearly antilinear and anti-unitary. 

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