Thread: Particle in a box (wavefunction problem)

There is a particle in a box from -a to a (i.e. length of box = 2a) with the following wavefunction:

ψ(x,0) = C(a²−x²) when |x| < a
ψ(x,0) = 0 when |x| > a

where C is a constant.

Find a value of C such that ψ(x,0) is normalised and hence write down the corresponding probability density.

To do this I tried ∫|ψ(x,0)|² = 1 from -a to a

= ∫ (ψ*(x,0) times ψ(x,0)) = C²(a²−x²)(a²−x²) and followed on from here.
Does this look right?

Originally Posted by Unoticed

There is a particle in a box from -a to a (i.e. length of box = 2a) with the following wavefunction:

ψ(x,0) = C(a²−x²) when |x| < a
ψ(x,0) = 0 when |x| > a

where C is a constant.

Find a value of C such that ψ(x,0) is normalised and hence write down the corresponding probability density.

To do this I tried ∫|ψ(x,0)|² = 1 from -a to a

= ∫ (ψ*(x,0) times ψ(x,0)) = C²(a²−x²)(a²−x²) and followed on from here.
Does this look right?

Yes.

Thanks, just to make sure; is the probability density in this case just:
C²(a²−x²)(a²−x²) ?

Originally Posted by Unoticed

Thanks, just to make sure; is the probability density in this case just:
C²(a²−x²)(a²−x²) ?

Yes. Note that in this case it is a "linear probability density" not a "volume probability density," though you probably already knew that.

-Dan