Transmission coefficient limit

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SUMMARY

The discussion centers on the behavior of the transmission coefficient (T) for a particle encountering a potential barrier as the potential (V) approaches the energy (E) of the particle. It is established that as V approaches E, the term V-E approaches zero, leading to the denominator of the transmission coefficient equation tending towards infinity, which results in T approaching zero. The mathematical limit indicates that the denominator behaves as 1 + (mEa²)/(2ħ²), highlighting the significance of the sinh²(k₁a) term as k₁ approaches zero when V₀ approaches E.

PREREQUISITES
  • Understanding of quantum mechanics, specifically potential barriers.
  • Familiarity with the concept of transmission coefficients in quantum tunneling.
  • Knowledge of mathematical limits and their implications in physics.
  • Basic grasp of hyperbolic functions, particularly sinh.
NEXT STEPS
  • Study the derivation of the transmission coefficient for quantum tunneling.
  • Explore the implications of potential barriers in quantum mechanics.
  • Learn about the role of hyperbolic functions in quantum physics.
  • Investigate the physical interpretations of limits in quantum mechanics.
USEFUL FOR

Students and professionals in physics, particularly those focusing on quantum mechanics and particle physics, will benefit from this discussion. It is also relevant for researchers exploring quantum tunneling phenomena.

TheCanadian
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I've attached the equation for the transmission coefficient of a particle going through a potential barrier and E < V. I was simply wondering in the limit V --> E, why does T --> 0 (i.e. the V-E term --> 0 and thus the denominator would approach infinity, making T --> 0)? Shouldn't it be approaching 1?
 

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TheCanadian said:
I've attached the equation for the transmission coefficient of a particle going through a potential barrier and E < V. I was simply wondering in the limit V --> E, why does T --> 0 (i.e. the V-E term --> 0 and thus the denominator would approach infinity, making T --> 0)? Shouldn't it be approaching 1?
There is also the ##sinh²(k_1 a)## to take into account, since ##V_0 \rightarrow E## implies ##k_1 \rightarrow 0##.

Purely mathematically, you will get as limit ##1+\frac{mEa²}{2 {\hbar}^2}## in the denominator.
(Physically maybe ##E \rightarrow V_0## makes more sense, so in the limit, replace ##E## by ##V_0##.)
 
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