Probability for finding particle outside potential well

[Eric_meyers]

Homework Statement

A particle in the harmonic oscillator potential is in the first excited state. What is the probability of finding this particle in the classically forbidden region?

Homework Equations

probability of finding particle = integral of abs[psi squared] [a,b]

The Attempt at a Solution

So, I'm using mathematica to find the intersection point of my x probability distribution with the oscillator potential function.

Solve[ ((m*w)/(\[Pi]*h))^(1/4)*(2*m*w/h)^(1/2)*
Exp[-((m*w)/(2*h))*x^2] - (1/2) (w^2)*m*x == 0, x]

but I get the answer
{{x -> -(Sqrt[h] Sqrt[
ProductLog[(8 Sqrt[(m w)/h])/(h^2 Sqrt[\[Pi]] w^2)]])/(
Sqrt[m] Sqrt[w])}, {x -> (
Sqrt[h] Sqrt[
ProductLog[(8 Sqrt[(m w)/h])/(h^2 Sqrt[\[Pi]] w^2)]])/(
Sqrt[m] Sqrt[w])}}

and then when I use this in my integral I get the answer
1/Sqrt[\[Pi]]2 ((m w)/h)^(3/2) If[Re[1/h] Re[m w] > 0

which obviously isn't a number as expected and seems to contain imaginary numbers. I don't understand what I'm doing wrong but this problem doesn't seem like it should be so complicated.

 

[Dick]

It doesn't make any sense at all to intersect the probability distribution with the potential. They don't even have the same units. The classically allowed region of x is the region where E>V(x), where E is the energy of the first excited state.

 

[Eric_meyers]

Right, I've corrected that part using the energy of the first excited state and finding those x intersection points. However the integral I get still involves an error function, and I'm wanting to get a numerical approximation.

 

[Dick]

Right, I've corrected that part using the energy of the first excited state and finding those x intersection points. However the integral I get still involves an error function, and I'm wanting to get a numerical approximation.

Try doing a change of variables so the exponential part becomes exp(-u^2). What's u in terms of x? If you do that the other constants in the integral should drop out too.

 

[paranormal]

I would like to point out that the probability of finding a particle outside of a potential well is not well-defined. This is because the concept of a potential well implies that the particle is confined within a certain region, and therefore the probability of finding it outside of that region is essentially zero.

In this specific scenario, the particle is in the first excited state of a harmonic oscillator potential, which means that it has a non-zero probability of being found in the classically forbidden region. However, this probability is not a fixed value and will depend on the specific parameters of the system.

The attempt at a solution using Mathematica is incorrect because it is trying to find the exact location of the particle outside of the potential well, which is not a well-defined concept. Instead, one could use statistical methods to estimate the probability of finding the particle in the classically forbidden region. This could involve calculating the average position of the particle over a large number of measurements, or using numerical simulations to approximate the probability distribution of the particle's position.

In conclusion, the probability of finding a particle outside of a potential well is not a straightforward calculation and requires careful consideration of the specific system and experimental procedures.