Mathematical Minimum (for Aspiring Physicists)

Linear Differential Equations

In the more general case when the function depends on more than one variable, the differentiation would have to be performed against one variable at a time while keeping the others fixed. For example, the partial derivative of x for f(x,y) is defined by:

---------- (6)

Many differential equations in physics involve multiple variables. Such kind of equations are referred as partial differential equations. The wave equation in three dimensional space is an example:

---------- (7)

where x, y, z are the spatial coordinates, t is the time, v is the propagation velocity, u(x,y,z,t) is the displacement of the wave, and the second derivative is defined as d²u/dx² = d(du/dx)/dx. For wave motion in one dimension, e.g., along the x axis, the 2nd and 3rd terms in Eq.(7) vanish; the solution for u at any point x and any time t is given by the cosine (or sine) function:

	u(x) = A cos(kx - t) ---------- (8) where = 2, k = 2/, and A is the amplitude. The frequency is related to the wavelength and wave velocity by the formula: = v. Figure 03 shows the cosine wave (for A = 1) at two instants of t = 0, and t = /.
Figure 03 Cosine Wave [view large image]	Note: see footnote for derviation of the differentiation for the sine and cosine functions.

Eqs.(1) and (7) are linear differential equations because they involve only linear terms in either y(t) or u(x,y,z,t). Examples can be found in many physical systems such as the harmonic motion in "Classical Mechanics", and the Schrodinger equation in "Quantum Mechanics". Linear equation (such as Eq.(1) or Eq.(7)) has the special property that if y₁ and y₂ are the solutions of k equals to k₁ and k₂ respectively, the combined solution y = A₁y₁ + A₂y₂ is again a solution of the same differential equation, where A₁ and A₂ are arbitrary constants. This is known as superposition in wave theory such as the one described by Eq.(7). Superposition forms the base in explaining interference and the production of wave packets. The mathematical formula for a wave packet is just the generalization of the superposition with the wave solution in the form of Eq.(8) (at t = 0 for example):

---------- (9)

where the coefficients A_n's are replaced by the weighting function f(k - k_o), and the sum becomes integration over a continuous range of k from -

to +

(see Figure 04a). At x = 0 the cosine function is equal to 1, hence all contributions to the integral comes from the weighting function, and the result is large. As x increases, the cosine function becomes a rapidly oscillating function of k, and its integral tends to cancel out resulting in the form of a pulse (wave packet) as shown in Figure 04a. In the limiting case where the weighting function is expressed in term of a delta function - A

(k - k_o), the wave packet is reduced to the monochromatic form such as shown in Figaure 03 or Eq.(8) with k = k_o; the wave is now extending from -

to +

. In general, there is a certain spreads

x in x, and

k in k. In quantum theory, the uncertainty principle postulates that

p >

=1.054x10^-27erg-sec,

Figure 04a Wave Packet
[view large image]

where the momentum p =

k comes from the de Broglie formula linking p and

. Note:
It can be shown that the inverse of Eq.(9) is given by f(k - k_o) = (1/

)y(x) coskx dx,

which together with Eq.(9) are known as Fourier transforms. Similar linear combination with the exponential function in Eq.(4) is called Laplace transforms. The Laplace transformation offers a direct method of solving linear ordinary and partial differential equations (the mathematical detail will not be pursued here).

By virtue of the orthogonality relation:

the cosine function in Eq.(9) acts like an unit vector¹ in the 3-dimensional space (Figure 04b), such that i_n

i_m =

_nm where i₁ = i, i₂ = j, and i₃ = k, and

_nm = 1 for n = m; = 0 otherwise. Thus, in quantum mechanics we can consider the wave function in force free space (like the wave packet) as an expansion in an infinite dimensional space with cos(nx) as its bases (unit vectors). Similar expansion can be expressed with wave function in bound states (where n and m become integers) for the cases of non-vanishing potential V(x). The idea has been expanded even further in quantum field theory to the more abstract Fock or Hilbert space, where it is just a mathematical symbol such as |n_k

, where n_k represents the number of particles in the state k with certain energy-momentum, spin, ... etc. The corresponding orthogonality relation is now expressed as

Figure 04b Unit Vectors (The vector v can be expressed in terms of the unit vectors)

n_k|n_k'

_{n_kn_k'}. In quantum theory, the coefficient or component (such as the f(k-k₀) in Eq.(9)) for each of such basis has been identified as the probability amplitude with the state k.

¹A vector is a mathematical entity with two or more space dimensions such as the velocity of an object v in Figure 04b. Loosely speaking, a (second rank) tensor is the product of two vector spaces (un-prime and prime) with ii', ij', ..., kk' as its bases.

A practical example of the linear differential equation is given in the next section.