# Does the wavelength of an individual quantum particle affect its chances of quantum tunneling through a barrier?

Physics Asked on January 2, 2022

I’ve read elsewhere about the energy of a wave or wave packet, and the amplitude of its probability amplitude, affecting its odds of quantum tunneling…

But what about the wavelength or frequency of a particle(s)?

I've read elsewhere about the energy of a wave or wave packet, and the amplitude of its probability amplitude, affecting its odds of quantum tunneling...

Lets be clear if we are talking of quantum mechanical or classical physics wave packets. As you are talking of tunneling, we are in the realm of quantum mechanics.

Here is a detailed quantum mechanical solution which shows how the probability of tunneling depends on the energy, the barrier and the length of the barrier.

Then lets clear whether you are talking of elementary particles described by a wavepacket, or of complicated quantum mechanical entities, like phonons, that obey quantum mechanical equations.

For elementary particles the wavefunction is a probability wavefunction, not an energy in space one. For the elementary particles of the standard model the wave packet form is necessary to describe in quantum field theory a free particle. Otherwise everything is interactions described by Feynman diagrams.

Tunneling is described in potential well solutions:

Note that $$Ψ$$ is a probabiity wave, and there is a smaller probability to find the wave after the barrier, and the energy is unchanged.

But what about the wavelength or frequency of a particle(s)?

In general particle in quantum mechanics calculations cannot be described with a wavelength or frequency, only its probability to manifest has a wave dependence.

Maybe this analysis will help , but remember the wave behavior described is the probability amplitude behavior.

Answered by anna v on January 2, 2022

In most situations wave length is related to the momentum (and therefore to the energy) by the De Broglie relation: $$p = hbar k = frac{h}{lambda}.$$

It is important to note however that here we speak about the wave length of a particle incident on the barrier (or outgoing after the tunneling). In the barrier itself the kinetic energy of the particle is negative, so that the wave length is replaced by the decay length, roughly estimated as $$lambda = frac{hbar}{sqrt{2m(V_0 - E)}},$$ where $$V_0$$ is the height of the barrier, and $$E$$ is the energy of the particle.

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