# Probability of finding particle in 1D finite potential well

1. Jun 10, 2012

### Ryomega

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. The attempt at a solution

This is to check I'm not going insane:

P = ∫ lψxl2 dx = 1

2. Jun 10, 2012

### vela

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

3. Jun 10, 2012

### 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.

4. Jun 10, 2012

### vela

Staff Emeritus
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?

5. Jun 10, 2012

### Ryomega

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.

6. Jun 10, 2012

### vela

Staff Emeritus
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?

7. Jun 10, 2012

### Ryomega

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

8. Jun 10, 2012

### Norfonz

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

ave[f(x)] = ⌠f(x)dx / ⌠dx

Food for thought.

9. Jun 10, 2012

### HallsofIvy

Staff Emeritus
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"!

Last edited: Jun 11, 2012
10. Jun 11, 2012

### 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?

11. Jun 11, 2012

### vela

Staff Emeritus
How do you normalize a wave function?

12. Jun 11, 2012

### Ryomega

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

13. Jun 11, 2012

### Ryomega

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

14. Jun 11, 2012

### vela

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

15. Jun 11, 2012

### Ryomega

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

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

16. Jun 11, 2012

### vela

Staff Emeritus
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?

17. Jun 12, 2012

### Ryomega

-∞ to ∞ ∫ lψ(x)l2 dx = ϕ(x)

18. Jun 12, 2012

### vela

Staff Emeritus
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.

19. Jun 12, 2012

### 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.

20. Jun 12, 2012

### vela

Staff Emeritus