Answer:

. The wavefunction of a particle confined to the x axis is ψ = e

−x

for x > 0 and ψ = e

+x

for x < 0. Normalize this wavefunction and calculate the probability of finding the

particle between x = −1 and x = 1.

Answer: Normalization refers to the requirement that

Z ∞

−in f ty

ψ

∗ψ dx = 1.

If ψ does not satisfy this property, because it is a SOLUTION to the Schrodinger ¨

equation (SE), which is a linear equation, Nψ, where N is a constant, is also a

solution to the SE.

Ignoring the acceptability of the given function to be a wavefunction, we proceed to

normalize the function. We look for a N, such that R

Nψ

∗Nψ dx = 1. That is

N

2

Z ∞

0

e

−x

e

−x dx +

Z −∞

0

e

x

e

x dx

= 1

Each of the INTEGRALS is a 1

2

so N2 = 1 or the wavefunction is normalized.

The probability of finding the particle between x = −1 and x = 1 is obtained by

integrating ψ

∗ψ over this domain.

P =

R 1

−1

ψ

∗ψ dx

R ∞

−∞

ψ∗ψ dx

In this instance, the denominator in the above expression is unity. The probability

of finding the particle in the DESIRED region is

P =

Z 1

0

e

−x

e

−x dx +

Z −1

0

e

x

e

x dx

=

1

2

e

2 − e

−2