Please help Problems on particle-in-a-box models

[wee00x]

The normalized wave function for a particle in a 1D box in which the potential energy is zero
between x= 0 and x= L and infinite anywhere else is

normalized wave function = sqrt(2/L)*sin(npix/L)

What is the probability that the particle will be found between x= L/4 and x= L/2 if the particle is in the state characterized by the quantum number n= 1?




Hint given:

indefinite integral of sin^2(ax)dx = .5x - .25asin(2ax)

 

[susskind_leon]

What you probably don't know (but should know) is that
P(particle between a and b)=∫_a^b |ψ(x)|^2 dx
where ψ is the wavefunction of the particle.

 

[wee00x]

I am confused as to what I use the second formula for... How do I find the probability of the particle being located in the region bounded by x=l/4 and x=l/2? How do i get an actual value for this since L is being used, and not a real number?

 

[susskind_leon]

How about you do the calculation and see what you end up with? There is a possibility that L cancels out...
Btw.: Which one is the second formula for you?

 

[wee00x]

indefinite integral of sin^2(ax)dx = .5x - .25asin(2ax)

that is the second formula. and I tried working it out but I couldn't get L to cancel out.. i just got an ugly answer...

 

[susskind_leon]

Please write down your entire calculation then I can help you with it.