A quantum particle confined to one-dimensional potential well of width 1nm is in its ground state. What is the probability of finding the particle over an interval of 0.1 nm at the center of the box?
Answers
Answer:
Calculate the wavelength associated with an electron with energy 2000 eV.
Sol: E = 2000 eV = 2000 × 1.6 × 10–19 J
images
2. Calculate the velocity and kinetic energy of an electron of wavelength 1.66 × 10 –10 m.
I assume you know the basics of quantum mechanics and schrodinger equation. Please refer to your university textbook if necessary.
Given,
n=1 (Ground state)
a= 1nm
Interval: 0.45 -0.55
To find the probability of finding the electron in the interval (0.45,0.55) you have to integrate mod(psi^2)ex where psi = root(2/a) * sin((n*π*x)/a) from 0.45 to 0.55.
I have attached the final answer as an image which is a verified answer.
Please reply if you want me to elaborate more since it takes some time to write the steps. Also don't forget to upvote if it's helpful!
The probability of finding the electron is 0.198363 or 19.8%
Note: When you are calculating sin values set your calculator to radians only or you will encounter an error /garbage value. Ex if you take the angle in degrees, your answer will come out to be 0.097 -0.098 which is totally WRONG! (It's not possible for n =1 plot a graph and you'll see!)
You could check for errors through the following ways:
1) If summation of the probabilities of all intervals = 1 then you're good!
2) Keep the angles in radian! It's CALCULUS!
3)By plotting a reference value of mod psi square for the given quantum number and interpreting your answer based on the reference diagram.
4) Think intuitively and use Common Sense!
Hope it helps!
If you want the answer to the other question then please reply. Don't forget to upvote!
It takes time to reply and edit so I might take some time to reply)
