Eigenfunction & Potential Barrier

[ZedCar]

Homework Statement

A particle of total energy E is incident on a potential barrier V0 (E<V0) between x=0 and x=a. Write down the allowed eigenfunctions in the regions x<0, 0<x<a and x>a in terms of five unknown constants A, B, C, D and F where A and F are the amplitudes of the incident and transmitted eigenfunctions.

Homework Equations

The Attempt at a Solution

The solution as provided with this question is as below, though you'll notice it uses A, B, C, D, F & G. So either the question mistakenly excluded 'G' or there is some error in the provided solution.

ψI = Aexp(ikx) + Bexp(-ikx)

ψII = Cexp(αx) + Dexp(-αx)

ψIII = Fexp(ikx) + Gexp(-ikx)

If the solution provided above is correct, is 'G' therefore the amplitude of the transmitted eigenfunction, not 'F' as stated in the question?

 

[TSny]

The solution gives the general form of the solution of the Schrödinger equation for each region. If you then invoke the additional information that the particle is incident on the barrier traveling from left to right, you should be able to argue that G must be zero.

 

[ZedCar]

The solution gives the general form of the solution of the Schrödinger equation for each region. If you then invoke the additional information that the particle is incident on the barrier traveling from left to right, you should be able to argue that G must be zero.

Thanks TSny.

From further advice I have been able to obtain I believe you are correct in stating that G must be zero.

May I ask how can it be deduced that G must be zero?

What exactly is it that the G represents that is zero?

 

[TSny]

Does the term Gexp(-ikx) represent a particle traveling to the right or to the left?

If the particle makes it to region III, which direction would it have to be traveling?

 

[ZedCar]

Does the term Gexp(-ikx) represent a particle traveling to the right or to the left?

From the way your question is worded I'm guessing the term Gexp(-ikx) represents a particle traveling to the left. Hence the reason it equals zero.

If this is correct, does this mean that the first term in each expression, ie A, C, F represents the particle moving to the right, and the second term in each expression ie B, D, G represents the particle moving to the left?

If the particle makes it to region III, which direction would it have to be traveling?

To the right.

 

[TSny]

From the way your question is worded I'm guessing the term Gexp(-ikx) represents a particle traveling to the left. Hence the reason it equals zero.

Yes.

If this is correct, does this mean that the first term in each expression, ie A, C, F represents the particle moving to the right, and the second term in each expression ie B, D, G represents the particle moving to the left?

That's true for A, B, F, and G. However, the C and D terms are exponentially growing or damping terms that do not correspond to traveling waves.

 

[ZedCar]

That's true for A, B, F, and G. However, the C and D terms are exponentially growing or damping terms that do not correspond to traveling waves.

And for C and D, do you know these are exponential growing/damping terms, and not a traveling wave, because they are inside the barrier?

 

[TSny]

And for C and D, do you know these are exponential growing/damping terms, and not a traveling wave, because they are inside the barrier?

They are exponentially growing/damping because the arguments of the exponentials do not have a factor of $i$. And this is due to the fact that inside the barrier, the barrier height is greater than the total energy of the particle.

 

[ZedCar]

They are exponentially growing/damping because the arguments of the exponentials do not have a factor of $i$. And this is due to the fact that inside the barrier, the barrier height is greater than the total energy of the particle.

Thanks very much for that TSny.