QM - one-dimensional barrier - a simple step

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In summary: This allows for the possibility of tunnels--a phenomenon where particles are reflected more than once.
  • #1
maria clara
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analyzing the simple-step scattering problem for E<V, we find that the solution to the schroedinger equation is:

PHI(left) = Aexp(ikx)+Bexp(-ikx)
PHI(right) = Cexp(-qx)

Continuity of the function and it's derivative at x=0 gives the relations between the parameters A, B and C.

Solving the appropriate equations we obtain

R = |B\A|^2 = 1
meaning that there is total reflection; hence the transmission must be zero.

But if we calculate the probability of finding a particle, say in the interval a<x<2a, we get a non-zero probability (because PHI(right) does not venish).
How is it possible, if every particle must be reflected?
 
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  • #2
maria clara said:
Solving the appropriate equations we obtain

R = |B\A|^2 = 1
meaning that there is total reflection; hence the transmission must be zero.

But if we calculate the probability of finding a particle, say in the interval a<x<2a, we get a non-zero probability (because PHI(right) does not venish).
How is it possible, if every particle must be reflected?
If you obtain a reflection value of one then you have incorrectly determined the coefficients and/or the R value.
 
  • #3
The calculation of R is taken from a book, there shouldn't be a mistake there.

And this is also a quotation from the book:
"In classical physics region 2 is a "forbidden" domain. In quantum mechanics, however, it is possible for particles to penetrate the barrier."

Region 2 is the region of the step (where V=V0).
 
  • #4
maria clara said:
The calculation of R is taken from a book, there shouldn't be a mistake there.
Then you must have incorrectly determined the relationships between the coefficients and/or wave numbers. What is the formula you have for the R value?
maria clara said:
And this is also a quotation from the book:
"In classical physics region 2 is a "forbidden" domain. In quantum mechanics, however, it is possible for particles to penetrate the barrier."

Region 2 is the region of the step (where V=V0).
That is correct...
 
  • #5
R = |B\A|^2 = |(1-iq/k)/(1+iq/k)| = 1

Anyway, if the particles can penetrate the barrier, it's logical that the probability to find a particle in a certain interval on the "other side" is non-zero;
But still R=1. Does it mean that the particles are actually reflected from the inside of the barrier?
 
  • #6
maria clara said:
R = |B\A|^2 = |(1-iq/k)/(1+iq/k)| = 1
Sorry my bad, I thought you were talking about the case where E>V. In the case where E<V, then yes, the R value is unity.
maria clara said:
Anyway, if the particles can penetrate the barrier, it's logical that the probability to find a particle in a certain interval on the "other side" is non-zero;
But still R=1. Does it mean that the particles are actually reflected from the inside of the barrier?
So yes, although the wavefunction is 'totally reflected' at the boundary, if we calculate the probability density inside the potential step, we find it to be non-zero. Therefore, even though the entire wavefunction is reflected, there is still some non-zero probability to find the particles inside the potential. This is what is referred to as tunnelling.
 
  • #7
thanks:smile:
 
  • #8
What Hootenanny said in different words:

Very roughly, unlike classical reflection, the position at which quantum reflection occurs is uncertain; the particle can sometimes penetrate into the forbidden region before being reflected.
 

FAQ: QM - one-dimensional barrier - a simple step

What is the concept of a one-dimensional barrier in quantum mechanics?

In quantum mechanics, a one-dimensional barrier refers to a potential energy barrier that is present in a one-dimensional space, such as a single line or a one-dimensional graph. This barrier can be modeled as a step function, where the potential energy is constant on either side of the barrier and changes abruptly at the barrier.

How does a particle behave when encountering a one-dimensional barrier in quantum mechanics?

When a particle encounters a one-dimensional barrier, it can either be transmitted through the barrier or reflected back. This behavior is described by the Schrödinger equation and depends on the energy of the particle and the height and width of the barrier.

What is the significance of the transmission and reflection coefficients in one-dimensional barrier problems?

The transmission and reflection coefficients represent the probabilities of a particle being transmitted or reflected when encountering a one-dimensional barrier. They are important in understanding the behavior of particles in quantum mechanics and can be calculated using the Schrödinger equation and the properties of the barrier.

How does the height and width of the barrier affect the behavior of particles in one-dimensional barrier problems?

The height and width of the barrier determine the energy levels at which particles can be transmitted or reflected. A higher barrier will result in a lower transmission coefficient and a higher reflection coefficient, while a wider barrier will result in a lower reflection coefficient and a higher transmission coefficient.

What are some real-life examples of one-dimensional barriers in quantum mechanics?

One-dimensional barriers are commonly used in the study of semiconductors, where particles encounter potential barriers in the form of energy barriers between different materials. They are also found in quantum tunneling, where particles can tunnel through one-dimensional barriers even if they do not have enough energy to overcome the barrier.

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