130_notes.dvi

k′=

√

2 m(E−V 0 )

̄h^2

E=−V 0 +

n^2 π^2 ̄h^2

8 ma^2

There are analogs of this in 3D. The scattering cross section oftengoes to zero for certain particular
energies. For example, electrons scattering off atoms may have nearly zero cross section at some
particular energy. Again this is a wave property.

9.1.4 Bound States in a Potential Well*.

We will work with the same potential well as in the previous section butassume that−V 0 < E <0,
making this abound state problem. Note that this potential has a Parity symmetry. In the left
and right regions the general solution is

u(x) =Aeκx+Be−κx

with

κ=

√

− 2 mE

̄h^2

.

Thee−κxterm will not be acceptable at−∞and theeκxterm will not be acceptable at +∞since
they diverge and we could never normalize to one bound particle.

u 1 (x) = C 1 eκx

u 3 (x) = C 3 e−κx

In the center we’ll use the sine and cosine solutions anticipating parity eigenstates.

u 2 (x) =Acos(kx) +Bsin(kx)

k=

√

2 m(E+V 0 )

̄h^2

Again we will have 4 equations in 4 unknown coefficients.