What is the Most Likely Position of a Particle in 1-Dimensional Wavefunction?

[nadeemo]

Homework Statement

Find the most likely position of the particle.

Homework Equations

$$\Psi$$ = A[(x+1)$$^{2}$$ - 1)]
between x = 0 and x = 1
$$\Psi$$ = 0 anywhere else

The Attempt at a Solution

I found A to equal $$\sqrt{15 / 38}$$... but I am not sure how to do the rest of it

 

[Dick]

You didn't even have to do that if you just want the most likely position. That's at the maximum of psi*conjugate(psi), the probability density.

 

[nadeemo]

that gives you expected value of the particles position <x>, which is the next question,
i need to find the most likely position of the particle...
is there another way to do this?

 

[Dick]

that gives you expected value of the particles position <x>, which is the next question,
i need to find the most likely position of the particle...
is there another way to do this?

No, <x> is the integral of x*psi*conjugate(psi) over the integral of psi*conjugate(psi). The 'most likely position' is the maximum of psi*conjugate(psi).

 

[nadeemo]

so take the derivative of |psi*cojugate(psi)| and set it to 0?

the conjugate would be A[x+1)^2 +1] ?

 

[Dick]

so take the derivative of |psi*cojugate(psi)| and set it to 0?

the conjugate would be A[x+1)^2 +1] ?

Yep. And since psi is real, conjugate(psi)=psi. You'll probably notice that the maximum of psi*psi is the same as the maximum of |psi|.

 

[nadeemo]

thanks a bunch =)