1. Mathematical Tools

Transcription

1 1. Mathematical Tools 1.1 Coordinate Systems Suppose u 1, u 2, and u 3 are the coordinates of a general coordinate coordinate system in which the (ê 1, ê 2, ê 3 ) unit or basis vectors specify the directions of the orthogonal coordinate axis. Then the vector locating any point in space with respect to the origin of this coordinate system is and the usual geometrical components are: Line element: Surface element: Volume element: The h 1, h 2, and h 3 are scale factors which must be found for each transformation from Cartesian coordinates to other coordinate systems: We will restrict ourselves to working in Cartesian, cylindrical and spherical polar coordinates, since that is what we need for the remainder of the course. Note, however, that other coordinate systems exists, such as elliptical, prolate spherical and others, which have various advantages under specific circumstances.

2 The so called Jacobian of a coordinate transformation is given by Cartesian Coordinates We use this as our base system, with respect to which all others are determined. Therefore, Where we have defined:

3 Cylindrical Coordinates So for the transformation from a Cartesian basis to a cylindrical basis, we have:

4 We then also have: The Jacobian of the transformation, to be included in the integrals and such, is:

5 Spherical Coordinates

6 So one finds the transformation matrix from Cartesian to spherical polar coordinates to be of the form:

7 The appropriate Jacobian of the transformation to spherical coordinates is:

8 1.2 Vector Operators Cartesian Cylindrical

9 Cylindrical Cont. Spherical

10 1.3 Integral Theorems The integral theorems are mostly used in scattering theory and in the description of radiative excitation or decay of atoms and nuclei. In other words, mostly where QM and E&M come into play at the same time. We start with Gauss theorem: Note that in terms of our earlier definitions: In the context of electromagnetism the vector A may represent some vector quantity, say the electric field E. Then E represents the charge density and Gauss theorem (also called the divergence theorem) represents the equality of counting the charge contained within some volume τ and measuring the flux of E through the surface S. In QM this relation is used both in calculations involving the radiation flux from a decaying quantum system (atom or nucleus) and in the theory of scattering where one wants to use the integral form of the Schrödinger equation for general (non-spherical) potentials and higher energies where partial wave expansion techniques become too cumbersome.

11 Stokes theorem: Note that, for any closed surface S the integral since the line C bounding such as surface is point. Thus, Here, A could be a vector potential, associated with many tiny current loops. Then A is the magnetic induction B due to the loops and Stokes theorem may be interpreted as the flux of B through any non-closed surface bounded by the line C, being equal to the sum of all A components along the line elements dl. In QM and more advanced studies, such as QED (quantum electro dynamics) the sort of current distributions described here (the loops) and the associated vector potential are associated with moving charged particles in a quantum system and radiated photons respectively.

12 1.4 The Dirac Delta (δ) Function The Dirac delta function is the integral (or continuous variable) analogue to the Kronecker delta for sum: Both of them are important when dealing with orthogonal basis vectors and functions, in particular with respect to the very important statement of completeness in QM, which we will get to later on. Basic properties: The first property follows from the second with λ=0. If you want to be fussy about things, then you d have to face the fact that the Dirac delta is not a function at all, since its integral is not defined (0 =?). So the proper definition is as an integral over the delta function which, by its definition, produces the desired result:

13 Which is OK, because all we will ever use it for is when evaluating functions under integrals. So, for an arbitrary function f (x), which is continuous and finite around x = 0, we can write δ(x)f(x) = δ(x)f(0), since both sides are zero for x 0, and they agree at x = 0 (from Brian Serot s notes). Then we can write: Listed here are the most important additional properties of the Dirac delta, for our purposes:

14

15 1.5 The Fourier Transform In a general n-dimensional space, the Fourier and inverse Fourier transforms are: So that for one dimension: If f(x) and f(k) are odd functions, then the transforms are given by If they are even functions, then the sin(kx) is replaced by cos(kx).

16 Fourier transforms of a delta function Using the definition of the delta function from before, we have The Plancherel Theorem The inner product of two functions g(x) and f(x) is defined by Where the star on the f(x) indicates complex conjugation. For now, you should regard the left hand side of this last statement as nothing more than a convenient notation, but the larger meaning of this will become clear later on. When you substitute in the Fourier transforms of the functions, you get:

17 So that we have the relation: This is the Plancherel theorem. It says that the inner product of two functions in position space is equal to their inner product in momentum space and the change from one space to the other happens via a Fourier transform. Descriptions of physical phenomena in position and momentum space are therefore equivalent. Don t worry if the significance of the previous two statements doesn t come across yet. Hopefully, by the end of the course this will become more clear. Derivatives of Fourier Transforms

18 So if we are transforming to momentum space, then we can write (using the QM momentum operator which we will be deriving later on hopefully you ve seen this one before though)

19 1.6 The Fourier Series For any periodic function f(x) with period 2l, one may use an expansion in terms of functions with the same period to recast f(x). That is,

20 Dirichlet Conditions If f(x) is a periodic of period 2l and if, between l and l, it is single valued, has a finite number of local minima and maxima, and a finite number of discontinuities, and if is finite, then the Fourier series in (1.6-1a) with the coefficients defined by (1.6-1b) and (1.6-1c) converges to f(x) at all points where f(x) is continuous and it converges to the midpoint of the jump where f(x) is discontinuous. Even and Odd f(x) If f(x) is either even or odd, then the coefficients are given by: If you need more help with any of the material discussed in this lecture, please come and see me in my office. You can also review this material using one of the usual mathematical methods books, such as the one by Mary Boas.