The wave function of a particle is

ψx=b1+x2-1mmx<0mmc1+x20mmx1mm

and zero elsewhere

Short Answer

Expert verified

a) Suppose that the wave function is a continuous function

b2=c2

0+-10b21+x22dx+01c21+x4dx+0=1

There fore the values of b and c are 0.382, 0.382

the probability that the particle will be found to the right of the origin is 90%

Step by step solution

01

Step 1:The wave function of a particle is 

ψx=b1+x2-1mmx<0mmc1+x20mmx1mm

and zero elsewhere

02

Sub part a) step 2: Suppose that the wave function is a continuous function

limx02b21+x22=limx02c21+x4b21+02=c21+02b2=c2

03

Sub part 2 (b) step 3:The graphs of the wave function, and the probability density function over the interval

-2mmx2mm

04

Step 4:

05

Step 5:

06

Sub part (c) step 6:The probability that the particle will be found to the right of the origin, means that we have to

calculate the value of the probability of finding the particle in the interval 0mmx1mm

First of all, we have to calculate the values of b and c For the probability interpretation of ψxto make sense, the wave function must satisfy the condition

-+ψx2dx=1--1ψx2dx+-10ψx2dx+01ψx2dx+1+ψx2dx=10+-10b21+x22dx+01c21+x4dx+0=1

07

Step 7:

Since b=c then

-10b21+x22dx+b2011+x4dx=1b2-1011+x2dx+b2011+x4dx=1

08

Step 8:

b212xx2+1+tan-1x-10+b2011+4x3+6x2+4x+x4dx=1b220+0-12-tan-11+b21+1+2+2+15-0=1b2212+π4+b26.2=1b2212+π4+6.2b2=1b26.45+0.3925=1b2=16.8425=0.146b=0.382

There fore the values of b and c are 0.382, 0.382

09

Step 9:

Now, the probability that the particle will be found to the right of the origin is

p0mmx1mm=01ψx2dx=01c1+x22dx=01c1+x4dx=c2011+4x+6x2+4x3+x4dx=0.146011+4x+6x2+4x3+x4dx=0.146x+4x22+6x33+4x44+x5501=0.1466.2x100%=90.52%=91%

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Most popular questions from this chapter

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