130_notes.dvi

9 One Dimensional Potentials

9.1 Piecewise Constant Potentials in 1D

Several standard problems can be understood conceptually usingtwo or three regions with constant
potentials. We will find solutions in each region of the potential. Thesepotentials havesimple
solutions to the Schr ̈odinger equation. We must then match the solutions at the boundaries
between the regions. Because of the multiple regions, these problems will requiremore work with
boundary conditionsthan is usual.

9.1.1 The General Solution for a Constant Potential

We have found the general solution of the Schr ̈odinger Equation ina region in which the potential
is constant (See section 7.6.1). Assume the potential is equal toV 0 and the total energy is equal to
E. Assume further that we are solving the time independent equation.

− ̄h^2

2 m

d^2 u(x)

dx^2

+V 0 u(x) =Eu(x)

d^2 u(x)

dx^2

=−

2 m(E−V 0 )

̄h^2

u(x)

ForE > V 0 , the general solution is

u(x) =Ae+ikx+Be−ikx

withk=

√

2 m(E−V 0 )
̄h^2 positive and real. We could also use the linear combination of the abovetwo
solutions.
u(x) =Asin(kx) +Bcos(kx)

We should use one set of solutions or the other in a region, not both. There are only two linearly
independent solutions.

The solutions are also technically correct forE < V 0 butkbecomes imaginary. For simplicity,

lets write the solutions in terms ofκ=

√

2 m(V 0 −E)
̄h^2 , which again is real and positive. The general
solution is
u(x) =Ae+κx+Be−κx.

These are not waves at all, but real exponentials. Note that theseare solutions for regions where
the particle is not allowed classically, due to energy conservation; the total energy is less than the
potential energy. Nevertheless, we will need these solutions in Quantum Mechanics.

9.1.2 The Potential Step

We wish to study the physics of a potential step for the caseE > V 0.

V(x) =

{

0 x < 0

+V 0 x > 0