130_notes.dvi

or
d^2 u(x)
dx^2

+

2 m

̄h^2

(E−V)u(x) = 0

and thegeneral solution, forE > V, can be written as either

u(x) =Aeikx+Be−ikx

or
u(x) =Asin(kx) +Bcos(kx)

, withk=

√

2 m(E−V)
̄h^2. We will also need solutions for the classically forbidden regions wherethe
total energy is less than the potential energy,E < V.

u(x) =Aeκx+Be−κx

withκ=

√

2 m(V−E)
̄h^2. (Bothkandκare positive real numbers.) The 1D scattering problems are
often analogous to problems where light is reflected or transmittedwhen it at the surface of glass.

First, we calculate the probability the a particle of energyEis reflected by a potential step (See

section 9.1.2) of heightV 0 : PR=

(√

E−

√

√ E−V^0

E+

√

E−V 0

) 2

. We also use this example to understand the

probability currentj= 2 ̄him[u∗dudx−du

∗

dxu].

Second we investigate the square potential well (See section 9.1.3)square potential well (V(x) =−V 0
for−a < x < aandV(x) = 0 elsewhere), for the case where the particle is not boundE >0.
Assuming a beam of particles incident from the left, we need to matchsolutions in the three regions
at the boundaries atx=±a. After some difficult arithmetic, the probabilities to be transmitted
or reflected are computed. It is found that the probability to be transmitted goes to 1 for some
particular energies.

E=−V 0 +

n^2 π^2 ̄h^2

8 ma^2

This type of behavior is exhibited by electrons scattering from atoms. At some energies the scattering
probability goes to zero.

Third we study the square potential barrier (See section 9.1.5) (V(x) = +V 0 for−a < x < aand
V(x) = 0 elsewhere), for the case in whichE < V 0. Classically the probability to be transmitted
would be zero since the particle is energetically excluded from being inside the barrier. The Quantum
calculation gives the probability to be transmitted through the barrier to be

|T|^2 =

(2kκ)^2

(k^2 +κ^2 )^2 sinh^2 (2κa) + (2kκ)^2

→(

4 kκ

k^2 +κ^2

)^2 e−^4 κa

wherek=

√

2 mE

̄h^2 andκ=

√

2 m(V 0 −E)
̄h^2. Study of this expression shows that the probability to
be transmitted decreases as the barrier get higher or wider. Nevertheless, barrier penetration is an
important quantum phenomenon.

We also study the square well for the bound state (See section 9.1.4) case in whichE <0. Here
we need to solve a transcendental equation to determine the bound state energies. The number of
bound states increases with the depth and the width of the well butthere is always at least one
bound state.