Show that the real and imaginary parts of the complex numbers listed below are as indicated. The real and imaginary parts of the last equation give precisely the Cauchy-Riemann conditions.
2 Calculus of residues
By the way, it has no other singularities in the finite part of the complex plane (remember that sin z is an entire function). The singularity of the corresponding transformed function f(1/w) ≡ φ(w) at the origin w= 0 then determines the nature of the singularity of f(z) at z=∞.
3 Linear response; dispersion relations
Continuous relaxation spectrum: If the number of contributing relaxation modes is infinite, or if there is a continuum of such modes, much more complex dynamical
The point is that the structure of the singularity χ(ω) in the lower half-plane in ω can now be quite complex. Of course, it continues to remain holomorphic in UHP.) In general, a functional form such as that given by the last equation above will have logarithmic branch points at ω = −i/τmax and ω = −i/τmin. These, in turn, are associated with physically interesting types of non-Debye relaxation of the response functions, including "glassy" or "slow dynamics".
4 Analytic continuation and the gamma function
Connection with the Gaussian integral: The gamma function helps us express a number of very useful integrals in closed form
It follows immediately that the value of the gamma function of a half-odd integer can be written explicitly. In the integral from t= 0 to t= 1, expand e−t in powers of t and integrate term by term to obtain the Mittag-Leffler expansion of the gamma function.
The logarithmic derivative of Γ(z) is also known as the digamma function, and is denoted by ψ(z). We have
In other words, the behavior of the gamma function near z = 0 is given by z. As in the case of the gamma function, we can try to extend the region of analyticity to the left in the z-plane by integrating by parts with respect to t.
5 M¨ obius transformations
Furthermore, it is a subgroup of the special linear group SL(2,C), and thus of the general linear group GL(2,C). A type (ii) transformation maps the interior of the unit circle in the z-plane to the exterior in the w-plane, and vice versa. a) the general form of a M¨obius transformation z → w mapping the unit circle to the unit circle is as given above;.
6 Multivalued functions; integral representations
But the contour of integrationC does not pass through the singularities of the integrand at t = 0 and t = 1. The contour can be distorted away from these points without changing the value of the integral. Case (iii) corresponds to case (i), with the roles of the branch points at 1 and −1 reversed.
This is determined by the behavior of the singularities in the t-plane of the integrand φ(t, z). Note the configuration of the contour C1 in the two cases, together with the branch sections of the integrand.
7 Laplace transforms
The iterate of the Laplace transform: Show that, provided the integrals concerned exist,
Note that the transient component of the solution, qtr(t), is characterized by a frequency ωu that depends on the circuit parameters L, C, and R. To check this, consider the general values q(0) and ˙q( 0) = I(0), respectively, of the initial charge on the capacitor and the initial current in the circuit. Referring to the inhomogeneous differential equation forq(t), the complete solution given above represents the particular integral, while the solution of the homogeneous equation represents the complementary function.
If we choose the coefficient of zn in the power series expansion of the exponent, we get the Poisson distribution. Let Pj(t) indicate the probability that the walker will reach sitej in time. The random variable in this problem is j).
Unbiased random walk in d dimensions: Consider an unbiased simple RW on an infinite hypercubic lattice in d dimensions. Each site is labelled by a set of d
Therefore, in an unbiased RW, Pj(t) has a leading asymptotic time dependence ∼t−1/2, which is characteristic of purely diffusive behavior.27 In clear contrast, the leading asymptotic behavior of Pj(t) for a biased random walk is given by Pj(t)∼t−1/2e−λt(1−2√pq). Unbiased random walk in d dimensions: Consider an unbiased simple RWon an infinite hypercubic lattice in d dimensions. Each site is labeled by a set of d. a) Show that the equation satisfied by this generating function is
This specific power-law decay of the probability is characteristic of purely diffusive motion. As in the one-dimensional case, the long-time behavior of P(j, t) is now a decaying exponential in t, rather than a pure power-law decay.
8 Fourier transforms
The Fourier transform of the probability density of a random variable is called its characteristic function. Consider the Fourier transform of the convolution of and g, namely of the function of k given by. We will use this result in the sequel, in the derivation of the Poisson summation formula.
The result above shows that the square of the Fourier transform operator is just 2π times the parity operator. The Fourier transform operator is therefore proportional to a 'fourth root' of the unity operator.
9 QUIZ 1
Solutions
Due to the Kronecker delta, the dependence of f(r) on the angles θ and ϕ also vanishes, so f(r) =f(r). The Coulomb potential in d-dimensional space is (apart from a proportionality constant) the fundamental Green function of the Laplacian operator in d dimensions. That is, the Coulomb potential in d dimensions, i.e., the fundamental Laplace Green's function, is essentially the inverse Fourier transform of -1/k2.
It is clear from the last equation that the Green's function is only a function of the magnitude R of the vector R. The basic Green's function of the Laplacian (i.e., the Coulomb potential) in 2-dimensional Euclidean space is proportional to the logarithm of R.
11 The diffusion equation
In the context of the diffusion equation for the concentrationρ(r, t), such an initial condition represents a point source of unit concentration at the origin. The solution thus obtained is the fundamental solution (or Green's function) of the diffusion equation. Moreover, this is independent of the dimensionality d of the space in which the scattering takes place.
Since this phenomenon involves a medium with a boundary, we will next turn to the solution of the diffusion equation in the presence of finite boundaries. My purpose in using the method of images in the present context of the diffusion equation is twofold.
Finite boundaries: Solution by separation of variables: You are undoubt- edly familiar with an elementary method for the solution of partial differential equa-
6. Now you may ask: can we write each of these solutions as a fundamental Gaussian solution on an infinite line, plus an 'extra' piece arising from the presence of finite boundaries. It is important to remember, however, that the physical region in which these solutions are valid is limited to the line interval [−b, b]. Interestingly, the only difference between the solutions for the reflection and absorption boundary conditions is the additional factor (−1)n in the summation in the latter case.
But this is sufficient to completely change the long-term behavior of the PDF p(x, t), as you will see below. Survival probability: The representations of p(x, t) just found are useful in reading the long-term behavior of the PDF because they are superpositions of .
Survival probability: The representations of p(x, t) just found are useful in reading off the long-time behavior of the PDF, because they are superpositions of
Let Q(t,±b|0)dt be the probability that a particle starting from x= 0 at t = 0 reaches one bor−b for the first time in the time interval (t, t+dt), without e never hit one of the endpoints at any previous time. Therefore, a first transition to one or the other of the endpoints is a safe event, as already stated. Connection to the Schrodinger equation for a free particle: Let us return to the fundamental solution of the diffusion equation in d-dimensional space, for an arbitrary initial PDF.
Now we can write the formal solution of the Schr¨odinger equation by analogy with the solution of the diffusion equation. Therefore, it is an explicit form of the free particle propagator for a non-relativistic particle moving in d-dimensional Euclidean space.
Spreading of a quantum mechanical wave packet: As you know from ele- mentary quantum mechanics, the wave packet representing a free particle undergoes
The state of the particle is therefore no longer a minimum uncertainty state for any t >0. Incidentally, note that the energy of the particle (i.e., the expectation value of its Hamiltonian) in the state under consideration is not p20/(2m), but instead. As you know, the normal modes of vibration of a region R are given by the solutions of the Helmholtz equation (∇2 +k2)u(r, t) = 0, i.e. the eigenvalue equation.
The scattering amplitude; differential and total cross-sections: The time-independent Schr¨odinger equation for the position-space wavefunction of the particle is given by. Integral equation for scattering: To find the scattering amplitude, we need the asymptotic form of the wavefunction solution.
Green function for the Helmholtz operator: The Green function satisfies the equation
We need to derive the asymptotic behavior of the wave function from this integral equation and identify the amplitude of the distribution. Comparing this result with the asymptotic form of the wave function, it immediately follows that the distribution amplitude is given by the formula. Substituting the above solution into the formula forf(k, θ) gives an expression for the scattering amplitude as a power series in λ, called the Born series.
The scattering amplitude in the Born approximation is thus, to a constant factor, the (three-dimensional) Fourier transform of the potential, with momentum. The scattering amplitude in the Born approximation is essentially the matrix element of the potential energy operator between the initial and finite eigenstates of the free particle momentum.
The Yukawa potential: According to quantum field theory, the forces between elementary particles arise from the exchange of other particles which are the quanta
Now, eik·r = eip·r/ =r|pis the position space wave function corresponding to the particular state of initial momentum |p of the particle. Show that, in the Born approximation, the scattering amplitude for the Yukawa potential is given by . where Q is the magnitude of the momentum transfer vector, as usual. For very large values of the drop energy E, the total cross section drops as 1/E.
The Coulomb potential corresponds to the ξ→ ∞ limit of the Yukawa potential.43 It follows that the scattering amplitude in the Born approximation is now fB(k, θ). All these "little miracles" and drawbacks are related to the long-term nature of the Coulomb potential.
13 The wave equation
Therefore, provided that all singularities of the integrand are in the lower half-plane, all our requirements are satisfied. Furthermore, the region of integration (that is, the entire k-space) is invariant under rotations of the coordinate axes. Since sin (cτ k) is the imaginary part of eicτ k, and J0(kR) is real for real values of the argument kR, we have.
Formally, if the limit c→ ∞ is taken in the wave equation, the wave operator reduces to the negative of. This is how the extended nature of the Green function arises in the case of wave propagation in equal-dimensional spaces, leading to the previously mentioned aftereffects.
14 The rotation group and all that
A rotation by an angle ψ of the coordinate axes about the origin in the xy plane gives the new coordinates x =x cos ψ+y sin ψandy =−x sin ψ+y cos ψ. So if the components of the direction vector are given by (n1, n2, n3), we are guaranteed that. This is a convenient place to determine the general forms of the elements of the unit group U(2) and the special unit group SU(2).
In the case of the rotation group SO(3), the parameter space (or group manifold) is doubly connected. 54 When we write π1(G), where G is a Lie group, we mean the fundamental group of the parameter space in G.
15 QUIZ 2
Solutions
