Introduction to Quantum Mechanics · Problem 2.23

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · delta_function_properties

Problem

Delta functions live under integral signs, and two expressions (

D_1(x)

and

D_2(x)

) involving delta functions are said to be equal if

\int_{-\infty}^{+\infty} f(x)\,D_1(x)\,dx = \int_{-\infty}^{+\infty} f(x)\,D_2(x)\,dx,

for every (ordinary) function

f(x)

. (a) Show that

\delta(cx) = \frac{1}{|c|}\delta(x),

where

c

is a real constant. (Be sure to check the case where

c

is negative.) (b) Let

\theta(x)

be the step function:

\theta(x) \equiv \begin{cases} 1, & x > 0, \\ 0, & x < 0. \end{cases}

(In the rare case where it actually matters, we define

\theta(0)

to be

1/2

.) Show that

d\theta/dx = \delta(x)