A particle can be in two different states given by orthonormal wavefunctions ψ1 and ψ2. If the probability of being in state ψ1 is 1/3, find out normalized wave function for the particle.
Answers
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
