What is the total probability of finding a particle in a one-dimensional box

[bamboy192]

Hi,
This is my first equation that I need to solve and both my textbook and professor have not shown me how to do it. I'm in chemistry and there isn't a prerequisite for calculus so I was wondering if I could be shown a way to solve this problem without doing calculus but if there isn't a way, the answer will still be beneficial.

Thanks

What is the total probability of finding a particle in a one-dimensional box in level n = 3 between x = 0 and x = L/6?

Homework Equations

Psi = sqrt(2/L)sin(n pi x /L)

 

[vela]

How is the probability of finding a particle in a region related to the wavefunction of the particle?

 

[epenguin]

You will probably get some funny looks here about a QM course without calculus. You are not actually forbidden to learn any. You'll have to be smarter to do QM without it than with - but you can do this problem.

You can't without knowing what a sine is and what its graph looks like, what pi and radians are. Maybe that counts as pre-calculus.

Useful to realize until you are more used to it, in a problem like this you are allowed to set L = 1 if you find that simplifies for you. Because L is just a length which is measured in some units and I am going to measure it in units so that it comes to 1. (in some other probs. where real lengths are given you may not have this freedom.)

Then draw a picture of what that function looks like. You can probably then guess the answer.

However the question asks you about probability and you really must know what the relation of your psi is to the probability of finding a particle in a certain space - in this case in a certain length. So do another picture of what that 'probability density' looks like in your problem. Then basically the regularity or symmetry of your pic will give you the answer.

You are going to have to think smart like this for other problems (calculus does some of the thinking for you).

 

[bamboy192]

thanks for the reply, can you show me using calculus because even though the class doesn't require it, I've already taken it so maybe that would be the best approach.

 

[epenguin]

thanks for the reply, can you show me using calculus because even though the class doesn't require it, I've already taken it so maybe that would be the best approach.

It would be bad if my general comments influenced you to not just solve that problem in the way I indicated, making you think you need to wait till you have studied something else. Just draw those two figs and I hope it will be obvious.

If you have done a bit then you will know the bit of math I suggested you use. Which is almost not calculus though it gets a lot treated in calculus courses and physical applications.

The calculus comes in in getting that equation that the Prof gave you. It is obtained from the 1-dimensional quantum wave equation aka the Schrödinger equation. If you know that equation a bit of calculus you can then verify that it is a solution of the Schrödinger equation in the particular case of Particle in a Box. It is an exercise in differentiation.