Undergrad Question about the solution of TISE (finite potential barrier)

[Clara Chung]

Let's say the solution on the left hand side is a sinkx + b coskx = 0
We get one solution for each E if we omitted a and another solution if we omitted b. However, how to conclude that they are all the solutions? Will there be any solution of the 5 coefficients such that a and b are both not zero?
Thank you

 

[drvrm]

Let's say the solution on the left hand side is a sinkx + b coskx = 0

If you are dealing with a physical situation, the five coefficients are necessary- however, the continuity condition gives you four relations;
Therefore the ratio of coefficients with incident amplitude carries physical interpretations-like reflection, transmission.etc.
making a and b both zero does not carry any meaning.
moreover, the total wave function can be zero on the left side if there is infinite wall

 

[Clara Chung]

If you are dealing with a physical situation, the five coefficients are necessary- however, the continuity condition gives you four relations;
Therefore the ratio of coefficients with incident amplitude carries physical interpretations-like reflection, transmission.etc.
making a and b both zero does not carry any meaning.
moreover, the total wave function can be zero on the left side if there is infinite wall

Sorry, but I mean will there be solution where a and b are both NOT zero?

 

[drvrm]

Sorry, but I mean will there be a solution where a and b are both NOT zero?

yes, such solutions exist on the left of the potential barrier- gen solution is
Psi(x) =a exp( i.k.x) +b exp(- i.k.x)
where
k = Sqrt( 2mE)/ h_bar

 

[Clara Chung]

yes, such solutions exist on the left of the potential barrier- gen solution is
Psi(x) =a exp( i.k.x) +b exp(- i.k.x)
where
k = Sqrt( 2mE)/ h_bar

yes, but I mean is it possible for all 6 coefficients be non zero?

 

[drvrm]

yes, but I mean is it possible for all 6 coefficients be non zero?

When one is treating the problem from the view of a particle incident from left with E, < V(x) you can have only five constants

say {A.B} {C.D } and E in the three regions as there will be transmitted amplitude in the third region.- the most interesting is (E/A)
physically as it represents tunneling. As no wave is coming from right the sixth amplitude will be zero.

depending on the energy and corresponding width of the barrier certain approximations can be made.
pl. see the detailed treatment in the reference below-that may help you.

Ref.-http://www.cse.salford.ac.uk/physics/gsmcdonald/pp/PPLATOResources/h-flap/p11_1t.pdf