How Does Expanding a Quantum Potential Well Affect Particle Probability?

[evildarklord1985]

Quantum mechanics- help!?

An innitely deep one-dimensional potential well runs from x = 0 to x = a. Let a particle be placed in the
ground state corresponding to this system. Then, within innitly short time, expand the potential well so that it now runs from x = 0 to x = 2a. If the energy of this particle is now measured, what is the probability of nding it in the ground state corresponding to this new system.? What is the probability of nding it in the rst excited state state?

 

[malawi_glenn]

As I wrote in the second question of yours, show work done and forumlas/relations that you know, then someone will try to help you.

 

[evildarklord1985]

sorry...i thought of using the same method we normally do for particle-in-the-box problem, where psi(x) = A sin(kx) + B cos(kx) . Then, using boundary condition of psi(x)=0 at x=0 , and x=2a...

From, psi(0) = 0, B =0

From , psi(2a) =0 ; I got : A sin(2ka) = 0
So, 2ka = n * pi
or, k = n*pi/2a

but can k be a non-interger? I'm confused... is this right way to approach the solution for the above problem?

 

[malawi_glenn]

Why CANT k be a non-integer? give me a reason..

2ka = n*pi, where n=1,2,3,... does k have to be an integer?

 

[evildarklord1985]

oh ok...so with this expression of k , I can write the energy level equation as :
E= n^2 * h^2 / 16m a^2...but the question asks about finding the probability of finding the particle. Isn't the total probability just 1 in the whole space?

 

[malawi_glenn]

No, that was not the question...

 

[evildarklord1985]

so, how do i go from what i had to get the solution? I'm confused...how do you actually calculate the probability of finding a particle in its certain state of energy level? All I know is probability of finding the particle in a region of the box...

 

[malawi_glenn]

You have that the particle is in the ground state of an infinite well with length a. The suddenly the well gets length 2a. What is the probablity that the particle is in the ground state of that well?

now you have two wave functions, and you shall find out how much they "overlap"

You must have covered this in your course..

 

[evildarklord1985]

sorry i don't think i encountered any kinds of problem like this in the course...this is a bonus homework of my professor to toture us for Spring break...anyway, what kind of calculation can you do to calculate the overlapping probability you mentioned?

 

[malawi_glenn]

You don't have a textbook either?