Sunday, 26 August 2018

quantum mechanics - Quantization of energy in semi-infinite well


Consider an electron with total energy E>V2 in a potential with V(x)={x<0V10<x<LV2x>L

where V2>V1>0.


We can determine that ϕE(x)={0x<0Asin(kx)+Bcos(kx)0<x<LCeqx+Deqxx>L

where k2=2meE and q=kV1E.


We can also apply the boundary condition at x=0 to determine that ϕe(x)=Asin(kx)

for x[0,L].


We can also apply boundary conditions at x=L to find that Asin(kL)=DeqL

Akcos(kL)=DqeqL
(since C=0 due to the corresponding positive exponent), and kcot(kL)=q


I'm stuck with the question: are the energy states with E>V2 quantized?


I can see that, because the boundary condition at x=L is not homogeneous, we cannot determine the eigenvalues in discrete form. Does this mean that the energy states are not quantized in this case?


Would appreciate some help.




Answer



Asin(kiL)=DeqL

Akicos(kiL)=DqeqL
kicot(kiL)=q
Insert the values for ki and q: [2m(V1Ei)/2]1/2cot[2m(V1Ei)/]1/2L=[2mEi/2]1/2
The allowed energy levels ($E_i

Rectangular potential well.


But particles with energy E>V2 can not be bound (contained in the well). Such a particle, coming in from the right e.g., would simply bounce off the infinite potential wall.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...