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I just have a hypothetical question:

If we have two beams of electrons which have the same energy, they approach some potential barrier of same fixed height. One of the barriers have a width of $d$ and the other has a width of $2d.$

My conclusion is this:

If both electrons do emerge from each of the two barriers, so electron $1$ passes through barrier $1$, and electron $2$ passes through barrier $2$, then both electrons will have different probabilities of them passing the respective barrier by the formula: $$p=ke^{-d\sqrt{m\Delta E}}$$

Since the width of the barriers are different, both must have different probabilities of successfully tunneling the barrier. However, the energy of the electrons from each barrier before tunneling is the same as after it has tunneled, hence the De-Broglie's wavelength of the electrons is the same after the tunneling has occurred from each of the two barriers.

Is my conclusion correct?

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Yes, all of that is correct.

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    $\begingroup$ Yes but the OP wonders why the transmission probability is different. $\endgroup$ – Gert Apr 12 at 14:58
  • $\begingroup$ @Gert The question, to my reading, contains none of that. $\endgroup$ – Emilio Pisanty Apr 12 at 15:07

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