1.3 Simple Applications of Schrödinger Equation
The Schrödinger equation has been expressed as (1.48). The equation is a secondorder linear differential equation (SOLDE). In particular, our major interest lies in
solving an eigenvalue problem of (1.55). Eigenvalues consist of points in a complex
plane. Those points sometimes form a continuous domain, but we focus on the
eigenvalues that comprise discrete points in the complex plane. Therefore, in our
studies the eigenvalues are countable and numbered as, e.g., λ n (n ¼ 1, 2, 3, Á Á Á). An
example is depicted in Fig. 1.2. Having this common belief as a background, let us
first think of a simple form of SOLDE.
Example 1.1 Let us think of a following differential equation:
d
2 y x
ð Þ
dx 2 þ λy x
ð Þ ¼ 0,
ð1:61Þ
where x is a real variable; y may be a complex function of x with λ possibly being a
complex constant as well. Suppose that y(x) is defined within a domain [ÀL, L]
(L > 0). We set boundary conditions (BCs) for (1.61) such that
y L
ð Þ ¼ 0 and y ÀL
ð Þ ¼ 0 L > 0
ð
Þ:
ð1:62Þ
The BCs of (1.62) are called Dirichlet conditions. We define the following differential operator D described as
D À
d
2
dx 2 :
ð1:63Þ
Then rewriting (1.61), we have
z
1
i
0
Fig. 1.2 Eigenvalues
λ n (n ¼ 1, 2, 3, Á Á Á) on a
complex plane
14
1 Schrödinger Equation and Its Application