Question

A particle limited to the x- axis has the wave function ѱ= ax between x=0 and x=
1 and ѱ = 0 elsewhere. Find the probability that the particle can be found
between x = 0.45 and x = 0.55.

Answer 1

Given:

A particle limited to the x- axis has the wave function,

ѱ= ax between x=0 and x=  1 and

ѱ = 0 elsewhere.

To Find:

The probability that the particle can be found  between x = 0.45 and x = 0.55.

Solution:

From the wave equation we can infer that the probability distribution is given by,

• PD = |ѱ|²
• PD = |ax|² = a²x²

Since the particle is limited to x axis,

• Total probability of finding a particle between x0 and x1 is :
• Probability =  $\int\limits^{x1}_{x0} {|\psi|}^{2} \, dx$

We need to find Probability for finding the particle between x = 0.45 and x  =0.55

• P = $\int\limits^{0.55}_{0.455} {a^{2}x^{2} } \, dx$  =  $a^{2} \int\limits^{0.55}_{0.45} {x^{2} } \, dx$  = a² (0.55)³ - (0.45)³ /3
• P = a² x (0.166375 - 0.091125 )/3 = 0.0251a²

Therefore the probability of finding a particle between x =0.45 and x=0.55 is 0.0251a².