Consider a box with width L centered at x = 0, so that it extends from x = -L/2

Question

Consider a box with width L centered at x = 0, so that it extends from x = -L/2 to x = +L/2. Note that this box is symmetrical about x = 0.
(a) Consider possible wave functions of the form ?(x) = A sin kx. Apply the boundary conditions at the wall to obtain the allowed energy levels.
(b) Another set of possible wave functions are functions of the form ? (x) = A cos kx. Apply the boundary conditions at the wall to obtain the allowed energy levels.
(c) Propose a combination of all the energies in parts (a) and (b).

Explanation / Answer

for
a) boundary conditions are at y=Asinkx is

0 at x=-l/2

and

0 at x=l/2

so

putting these,

we get

0=Asink*l/2

k*l/2 is a multiple of

hence k=2n/l

k=2/l for lowest energy

to find A,

we need to normalize it

-l/2l/2A2sin2kx=1

so we get A

b)

with similar boundaries

Acos(k*l/2)=0

k=/l

normalizing,

we get A

:)

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