Probability of finding particle

[Aziza]

This is example from my book:

For some particle, let ψ(x,0) = \frac{1}{\sqrt{a}}exp^(-|x|/a).

Finding the probability that the particle is found between -x₀ and x₀ yields a probability of 86.5%, independent of x₀! But how can this be, since as x₀ tends to infinity, the probability of finding the particle between negative infinity and infinity must be 1...so the probability suddenly jumps from 86% to 100%?

I am thinking that maybe this is not a valid wavefunction since it has a sharp point at x=0 and my professor said that the wavefunction cannot have any sharp bends...?

 

[Vanadium 50]

It's not independent of x0. What is the role of a here?

 

[Aziza]

It's not independent of x0. What is the role of a here?

It is just a constant, sorry I forgot to mention that.

 

[Aziza]

It's not independent of x0. What is the role of a here?

But yes it is independent: here is the exact problem:

http://books.google.com/books?id=Yf...ymmetric, decaying exponentially from&f=false

(example 6.2 which uses the wavefunctiongiven in example 6.1)

 

[Vanadium 50]

It's not just a constant. It plays a very important role. What is it?

Hint: what are its dimensions?

Note also that your original problem doesn't specify that x0 is a special value of x.

 

[Aziza]

It's not just a constant. It plays a very important role. What is it?

Hint: what are its dimensions?

Note also that your original problem doesn't specify that x0 is a special value of x.

ohhh i think i see...a is related to the max height of the curve which is itself related to x₀ so as x₀ increases the max height decreases, thus keeping the probability constant in that interval...right?

 

[Bill_K]

What you're calling 'a' is called x₀ in the book. 'a' and x₀ are exactly the same thing.