# Probability of finding particle in 1D finite potential well

## Homework Statement

ψx 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 ψx is normalised.

## The Attempt at a Solution

This is to check I'm not going insane:

P = ∫ lψxl2 dx = 1

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

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
Staff Emeritus
Homework Helper
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?

P = ∫ lψxl2 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 = ∫ lψxl2 dx (integral -∞ to ∞)

Two totally separate equations.

vela
Staff Emeritus
Homework Helper
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?

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

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

ave[f(x)] = ⌠f(x)dx / ⌠dx

Food for thought.

HallsofIvy
Homework Helper
If you are given that the object is in an infinite square well, then of course there is a probabilty 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"!

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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
Staff Emeritus
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How do you normalize a wave function?

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

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

vela
Staff Emeritus
Homework Helper
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.

∫∫ llψxl2l2 dx dx (inner from a to b, outer from -∞ to ∞)

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

vela
Staff Emeritus
Homework Helper
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?

-∞ to ∞ ∫ lψ(x)l2 dx = ϕ(x)

vela
Staff Emeritus
Homework Helper