What Is the Probability of Finding a Particle in the Interval [0,d]?

[Nugso]

Homework Statement

Suppose we have a particle in 1-dimension, with wavefunction $$Ae^{-\frac{|x|}{2d}}$$ . What is the probability to find the particle in the interval [0,d]?
Please provide your answer in terms of A, d, mathematical constants such as π (entered as pi) or e (entered as e). (Assume that A is real)

Homework Equations

$$∫ψ²dx = 1$$

The Attempt at a Solution

I think I need to find A by normalizing it. $$∫ψ²dx = 1$$

By integrating it, I get $$A= 1/\sqrt{2d}$$

Now, I have to integrate it again, but this time with the interval of [0,d]

$$∫1/sqrt(2d)*e^{-\frac{|x|}{2d}}*1/sqrt(2d)*e^{-\frac{|x|}{2d}}dx$$

and the answer I'm finding is, $$1/2*(1-e^{-1/d})*d$$

But somehow the answer is wrong. How do I correct it?

 

[BOYLANATOR]

I would check your solution for A again.

 

[Dick]

Homework Statement

Suppose we have a particle in 1-dimension, with wavefunction $$Ae^{-\frac{|x|}{2d}}$$ . What is the probability to find the particle in the interval [0,d]?
Please provide your answer in terms of A, d, mathematical constants such as π (entered as pi) or e (entered as e). (Assume that A is real)

Homework Equations

$$∫ψ²dx = 1$$

The Attempt at a Solution

I think I need to find A by normalizing it. $$∫ψ²dx = 1$$

By integrating it, I get $$A= 1/\sqrt{2d}$$

Now, I have to integrate it again, but this time with the interval of [0,d]

$$∫1/sqrt(2d)*e^{-\frac{|x|}{2d}}*1/sqrt(2d)*e^{-\frac{|x|}{2d}}dx$$

and the answer I'm finding is, $$1/2*(1-e^{-1/d})*d$$

But somehow the answer is wrong. How do I correct it?

Check it again. How did you wind up with a -1/d in the exponent?

 

[Nugso]

Sorry for the late reply. I checked it and corrected the mistake.

 

[paranormal]

I would like to clarify that the probability to find a particle in a specific interval is given by the integral of the square of the wavefunction over that interval. In this case, the wavefunction is given by Ae^{-\frac{|x|}{2d}} and the interval is [0,d]. Therefore, the probability to find the particle in this interval is given by:

P = ∫0d (Ae^{-\frac{|x|}{2d}})^2 dx

= A^2 ∫0d e^{-\frac{|x|}{d}} dx

= A^2 (-d(e^{-1}-1))

= A^2 (1-e^{-1})

Therefore, the probability to find the particle in the interval [0,d] is given by A^2 (1-e^{-1}). This can also be expressed in terms of d and mathematical constants as A^2 (1-e^{-1}) = 1/2πd (1-e^{-1}).