(a) Solve the Schrödinger equation for the particle-in-a-box problem (0
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Need help on part d only. All parts included for context
Transcribed Image Text:(a) Solve the Schrödinger equation for the particle-in-a-box problem (0 <1 < l),
that is, obtain
n°h?
En -
sin
8ml?'
where m is the mass of the particle.
(b) For the above eigenfunctions V,„(r) of the particle-in-a-box, show that they are
orthonormal, that is
(V„L.) = [ V(1)¥,(x)dr = dm,n
where dmn is the Kronecker delta function defined as
Sm,n =1 when m = n;
dm,n = 0 when m+n
(c) (For the sake of variable substitution, replace variable z by z in the above equa-
tion. You are going to need it for this problem.) Now consider a particle in a
new box defined by -<1< What would be the new eigen-energy En and
eigen-finction V,(x)?
Hint: substitute z = r+ and compare the equation and the boundary condition
to the problem above where you made the z-substitution. Of course, this time
you obtain the solution in variable z, but you need to make a back transformation
I = z - to express the solution in r.
(d) What if the boundary condition becomes a <r< b? What would be the new
eigen-energy E, and eigen-function ,(1)?
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