Introduction to Quantum Mechanics · Problem 1.4

The Wave Function — David J. Griffiths

Physics — Quantum Mechanics · wave_function_normalization

Problem

At time

t = 0

a particle is represented by the wave function

\Psi(x, 0) = \begin{cases} A(x/a), & 0 \leq x \leq a, \\ A(b - x)/(b - a), & a \leq x \leq b, \\ 0, & \text{otherwise}, \end{cases}

where

A

a

, and

b

are (positive) constants. (a) Normalize

\Psi

(that is, find

A

, in terms of

a

and

b

). (b) Sketch

\Psi(x, 0)

, as a function of

x

. (c) Where is the particle most likely to be found, at

t = 0

? (d) What is the probability of finding the particle to the left of

a

? Check your result in the limiting cases

b = a

and

b = 2a

. (e) What is the expectation value of

x