Probability of finding particle in 1D finite potential well

[Ryomega]

Homework Statement

ψ_x is the function of postion for a particle inside a 1D finite square well. Write down the expression for finding the particle a≤x≤b. Do not assume that ψ_x is normalised.

Homework Equations

The Attempt at a Solution

This is to check I'm not going insane:

P = ∫ lψ_xl² dx = 1

Is that about right?

 

[vela]

So you're saying you will always find the particle between x=a and x=b? That's what a probability of 1 means.

 

[Ryomega]

Err... not sure why I typed in the "= 1" bit since I don't know what the probability is.
But you understood that the probability of finding the particle between a and b is expressed through my equation. So I rest my case.

 

[vela]

Well, I wouldn't say I understood that's what you meant. I guessed that is what you probably meant. For all I know, you could have been writing down the normalization condition. What you wrote is too vague to be meaningful, and it isn't correct. Can you flesh it out a bit?

 

[Ryomega]

P = ∫ lψ_xl² dx (integral from a to b)

This is the formal expression for probability of finding a particle within a boundary.

For normalisation, we say that the particle must be somewhere since we take the integral from -∞ to ∞.

1 = ∫ lψ_xl² dx (integral -∞ to ∞)

Two totally separate equations.

 

[vela]

Both of those equations hold only if the wave function is normalized. If the wave function isn't normalized, what would be the expression for the probability?

 

[Ryomega]

...the awkward moment when I don't know the answer has arrived.
Please illiterate.

 

[Norfonz]

Consider finding the average value of a function over its domain.

ave[f(x)] = ⌠f(x)dx / ⌠dx

Food for thought.

 

[HallsofIvy]

If you are given that the object is in an infinite square well, then of course there is a probability of 1 that it is in the square well. I would have expected the problem to ask for the probability distribution of its position in that well. You can get that by solving Shrodinger's equation.

Oh, dear! I just reread the title and realized it says finite well, not infinite"!

 

[Ryomega]

So I'm getting... do a double integral or solve the schödinger equation or do a barrel roll.
Care to go a tiny bit further for me please?

 

[vela]

How do you normalize a wave function?

 

[Ryomega]

square and integrate it. So... double integration and power of 4?

 

[Ryomega]

That's given that the particle is between a and b and that isn't specified

 

[vela]

square and integrate it. So... double integration and power of 4?

I don't know what you're trying to say. Show your work so that your meaning is clear.

 

[Ryomega]

∫∫ llψ_xl²l² dx dx (inner from a to b, outer from -∞ to ∞)

seems strange...but that's what I'm imagining at the minute

 

[vela]

Let's say you have the unnormalized wave function $\psi(x)$, and let $\phi(x)$ be its normalized counterpart. How are $\psi$ and $\phi$ related?

 

[Ryomega]

-∞ to ∞ ∫ lψ(x)l² dx = ϕ(x)

 

[vela]

The lefthand side of your equation is a definite integral, so the result is a number, not a function of x. Try again. Surely your textbook has an example.

 

[Ryomega]

This is why I'm here, I've looked through 3 books and numerous sites and I can't seem to find this particular example where the wave function is NOT normalised.

 

[vela]

Take a look at this:

http://panda.unm.edu/Courses/Fields/Phys491/Notes/TISEInfiniteSquare.pdf

It covers solving the Schrödinger equation for the infinite square well, including normalizing the wave function.

 

[Ryomega]

I've been reading this a while now and I finally understand what you were trying to teach me. Page two of the document shows that the normalisation process, of the wave equation. I see now, where I've been going wrong.

Thank you so much!