130_notes.dvi

9.9 Sample Test Problems

1. A beam of 100 eV (kinetic energy) electrons is incident upon apotential stepof height
   V 0 = 10 eV. Calculate the probability to be transmitted. Get a numericalanswer.

2.*Find the energy eigenstates (and energy eigenvalues) of a particleof massmbound in the

1D potentialV(x) =−V 0 δ(x). AssumeV 0 is a positive real number. (Don’t assume thatV 0

has the units of energy.) You need not normalize the state.

Answer

κ=

√

− 2 mE

̄h^2

du

dx

∣

∣

∣

∣

+

−

du

dx

∣

∣

∣

∣

−

=

− 2 mV 0

̄h^2

eκ^0

−κ−(+κ) =

− 2 mV 0

̄h^2

κ=

mV 0

̄h^2

E=−

̄h^2 κ^2

2 m

=−

̄h^2

2 m

m^2 V 02

̄h^4

=

mV 02

2 ̄h^2

3.*A beam of particles of wave-numberk(this meanseikx) is incident upon a one dimensional

potentialV(x) =aδ(x). Calculate the probability to be transmitted. Graph it as a function

ofk.

Answer

To the left of the origin the solution iseikx+Re−ikx. To the right of the origin the solution is

Teikx. Continuity ofψat the origin implies 1+R=T. The discontinuity in the first derivative

is

∆

dψ

dx

=

2 ma

̄h^2

ψ(0).

ikT−(ik−ikR) =

2 ma

̄h^2

T

2 ik(T−1) =

2 ma

̄h^2

T

(

2 ik−

2 ma

̄h^2

)

T= 2ik

T=

2 ik

2 ik+^2 ̄hma 2

PT=|T|^2 =

4 k^2

4 k^2 +^4 m

(^2) a 2
̄h^4
Transmission probability starts at zero fork= 0 then approachesP= 1 asymptotically for
k >ma ̄h 2.
4.*A beam of particles of energyE >0 coming from−∞is incident upon a delta function
potential in one dimension. That isV(x) =λδ(x).
a) Find the solution to the Schr ̈odinger equation for this problem.