Complex-conjugate multipliers

In this section we continue our study of the bifurcation of a pair of complex-conjugate multipliers of the periodic orbit over and beyond the unit circle... 

It is the net intensity, not the electric field, which concerns us. We previously used the fact that intensity is proportional to E to evaluate i. Using complex numbers to represent E requires one slight modification of this procedure. In the present case we must multiply E by its complex conjugate -obtained by replacing / 1 by to evaluate intensity ... 

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... 

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 product of a function and its complex conjugate is always real and is positive everywhere. Accordingly, the wave function itself may be a real or a complex function. At any point x or at any time t, the wave function may be positive or negative. In order that F(x, t)p represents a unique probability density for every point in space and at all times, the wave function must be continuous, single-valued, and finite. Since F(x, /) satisfies a differential equation that is second-order in x, its first derivative is also continuous. The wave function may be multiplied by a phase factor e , where a is real, without changing its physical significance since... 

Each of the integrands in equations (2.18), (2.19), and (2.20) is the complex conjugate of the wave function multiplied by an operator acting on the wave function. Thus, in the coordinate-space calculation of the expectation value of the momentum p or the nth power of the momentum, we associate with p the operator (h/f) d/dx). We generalize this association to apply to the expectation value of any function f p) of the momentum, so that... 

The coefficients a, are evaluated by multiplying (3.27) by the complex conjugate 0 of one of the eigenfunctions, integrating over the range of the variables, and noting that the 0,s are orthonormal... 

The bracket (bra-c-ket) in

) provides the names for the component vectors. This notation was introduced in Section 3.2 as a shorthand for the scalar product integral. The scalar product of a ket tp) with its corresponding bra (-01 gives a real, positive number and is the analog of multiplying a complex number by its complex conjugate. The scalar product of a bra tpj and the ket Aj>i) is expressed in Dirac notation as (0yjA 0,) or as J A i). These scalar products are also known as the matrix elements of A and are sometimes denoted by Ay. 

To find the first-order correction to the eigenvalue En, we multiply equation (9.22) by the complex conjugate of and integrate over all space to obtain... 

Because the expression obtained contains i, it is still a complex function. Suppose, however, that instead of squaring (a + ib) we multiply by its complex conjugate, (a — ib) ... 

If the roots and S2 are both real numbers, Bq. (6.116) shows that Uq and a I are certainly both real. If the roots and. S are complex, the coefficients and di must still be real and must also satisfy Eq. (6.116). Complex conjugates are the only complex numbers that give real numbers when they are multiplied together and when added together. To illustrate this, let z be a complex number z = X + iy. Let z be the complex conjugate of z z = Jt — iy. Now... 

Multiplying numerator and denominator by the complex conjugate of the denominator gives... 

The modulus on the right-hand side is obtained by multiplying by the complex conjugate, giving... 

If we now multiply both sides of the equation by, the complex conjugate of ipmt and integrate over all space, we obtain the following equation as a result of the orthonormality properties of the functions ftn ... 

We denote the set of complex numbers by C. Readers should be familiar with complex numbers and how to add and multiply them, as described in many standard calculus texts. We use i to denote the square root of —1 and an asterisk to denote complex conjugation if x and y are real numbers, then (x + iy) = X — iy. Later in the text, we will use the asterisk to denote the conjugate transpose of a matrix with complex entries. This is perfectly consistent if one thinks of a complex number x + iy as a one-by-one complex matrix ( x + iy ). See also Exercise 1.6. The absolute value of a complex number, also known as the modulus, is denoted... 

This fact will be at the heart of the proof of our main result in Section 6.5. Proof. First, we show that V satisfies the hypotheses of the Stone-Weierstrass theorem. We know that V is a complex vector space under the usual addition and scalar multiplication of functions adding two polynomials or multiplying a polynomial by a constant yields a polynomial. The product of two polynomials is a polynomial. To see that V is closed under complex conjugation, note that for any x e [—1, 1] and any constant complex numbers flo, , a sN[Pg.102]

The use of the symbol ay is to indicate that this represents the absorption of a photon by an atom. Further, the matrix Uy t) = e mtTJy 0), and when placed substituted into Eq. (77), we arrive at an expression for ay t). Now, when ay(t) is multiplied by its complex conjugate, we have... 

Multiplying Eq. (11) by tjr (x,t) from the left and then subtracting from the result its complex conjugate, we are left with... 

Notice that these equations explicitly include derivatives and the complex conjugate f of the wavefunction. The expression for the momentum even includes i = f—W Complex numbers are not just a mathematical convenience in quantum mechanics they are central to the treatment. Equation 6.6 illustrates this point directly. Any measurement of the momentum (for example, by measuring velocity and mass) will of course always give a real number. But if the wavefunction is purely real, the integral on the right-hand side of Equation 6.6 is a real number, so the momentum is a real number multiplied by ih. The only way that product can be real is if the integral vanishes. Thus any real wavefunction corresponds to motion with no net momentum. Any particle with net momentum must have a complex wavefunction. 

There is no restriction as to the phase of the coefficients they can take positive or negative, real or imaginary values. Computing oscillator strengths f or transition probabilities W, one first has to perform the summation in Eq. [227] before multiplying pel by its complex conjugate. 

First, we scalar-multiply the Maxwell s equations by the complex conjugate modal fields... 

Multiplying by the complex conjugate and rearranging into the real and imaginary components, we obtain... 

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