Quantum Mechanics Particle moving in 1-D potential

[20930997]

1. Homework Statement
I've been attempting this one for a while now and so far it has been to no avail. The question itself is in the attached document.

I am given an expression that defines the wavefunction moving in 1-D over an interval and the time component is fixed with t=0.

2. Homework Equations

〖|ψ(x,0)|〗^2 = prob dist, in which likelihood of finding a particle at that point can be inferred.


3. The Attempt at a Solution

first of all I assume that (A) is a complex constant and as a result i multiply the wavefunction by its conjugate to get the following,

〖|ψ(x,0)|〗^2=|A|^2 (x^4-2x^3 a+x^2 a^2) which I am not sure is correct and furthermore have no idea how to sketch it.

any guidance would be appreciated!

 

[vela]

first of all I assume that (A) is a complex constant and as a result i multiply the wavefunction by its conjugate to get the following,

〖|ψ(x,0)|〗^2=|A|^2 (x^4-2x^3 a+x^2 a^2) which I am not sure is correct and furthermore have no idea how to sketch it.

That's correct, but it would probably be better not to multiply it out initially. Just leave it as
$$|\Psi(x,0)|^2 =
\begin{cases}
|A|^2 (x-a)^2 x^2 & 0 \le x \le a \\
0 & \text{otherwise}
\end{cases}$$ Surely, you must have done curve sketching in algebra and calculus. Where does the function vanish? Where is it positive? Where is it negative? Where does it attain a maximum or a minimum?

 

[20930997]

ahh yes that makes it a lot clearer, thanks for the help!

 

[20930997]

I'm also having difficulty with part b of this question.

I tried using a symmetry approach to try and deduce the value of |A| but in my working the |A|^2 term ends up being canceled which is of no use to me.

again any pointers would be much appreciated.

 

[vela]

You need to use the normalization condition to determine |A|.