Question

Consider a finite square well, with V = 1.2 eV outside. It holds several energy states, but we are only interested in two:

• E₁ = 1.15 eV
• E₂ = 1.1 eV

1) For E = 1.15 eV, what is the decay constant, κ, outside the well in nm^-1(i.e., where V = 1.2 eV)?

κ =

2) For E = 1.1 eV, what is the decay constant, κ, outside the well in nm^-1 (i.e.,) where V = 1.2 eV)?

κ =

3) For E = 1.15 eV, suppose the probability density at some position,x, outside the well is P(x) and the probability density 1 nm farther from the well is P(x+1 nm). What is the ratio, P(x+1 nm)/P(x), of these two probailities?

Ratio =

4) For E = 1.1 eV, suppose the probability density at some position,x, outside the well is P(x) and the probability density 1 nm farther from the well is P(x+1 nm). What is the ratio, P(x+1 nm)/P(x), of these two probailities?

Ratio =

5) If we squeeze the well (decrease L), the energies of the states will increase. What is the limiting value κ_limit of an energy state's κ as its energy approaches the top of the well in nm^-1(i.e., as E → 1.2 eV).

κ_limit =

Answer 1

In the region outside the box, the decay constant is given by:

Here

1) for E=1.15eV

Note: though this may seem to be a very large value, indicating that wavefunction must decay almost instantaneously, one must remember that the masses of particles exhibiting quantum behavior in the well is ~10^-20 to 10^-30 kg. thus k' will be small for such particles, thereby allowing tunneling.

2) for E=1.1eV

Outside the well, the wavefunction is given by

Thus for x>L/2,

for x<-L/2 (Assuming x+1nm is also outside the well)

3) for E=1.15eV

if x>L/2,

if x<-L/2

4) for E=1.1eV

if x>L/2,

if x<-L/2