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I know that in quantum mechanics there is no "time operator", so such a question is ill-posed. Anyway if the tunneling is instantaneous, this would imply an information transmission faster than $c$. On the other hand, how could someone define such a "time"?

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Possibly duplicate of: physics.stackexchange.com/q/38237 –  Ruslan Dec 15 '13 at 20:07

2 Answers 2

Assume a particle moves in a potential, in one dimension, of the form $V(x) = \infty$ if $x^2 > a^2$ else $V(x) = \gamma \delta(x)$,where $\gamma >0$. Let $E_0$ be the energy of the particle. Then $\langle v \rangle = \sqrt{\frac{2E_0}{m}}$

Frequency of collisions = $\frac{\langle v \rangle}{a}$

Frequency of tunelling = $\frac{\langle v \rangle}{a} * T$

$T$ is the transmission probability which can be calculated by solving the Schrodinger equation.

Time required for a particle to tunnel = $\frac{a}{\langle v \rangle * T} $

Edit: Corrected Typo. $V(x)=\infty$

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I'm more interested in a finite size potential, say with FWHM width $a$. Then I don't like your dimensionality argument because in the potential barrier the velocity is imaginary, thus $t = \langle v \rangle / a$ is an imaginary time –  Mattia Jan 24 at 13:53

I think what you're asking here is:

"If a particle is about to tunnel through a barrier, how long will it take to get from one side to the other?"

If so then you should consider how the particle is tunnelling through the barrier. It tunnels through the barrier because it's wavefunction "leaks" through the barrier - which means that it has a non-zero probability to be located outside of the barrier (i.e. it's already outside the barrier "when it starts to tunnel through it"). Because of this, the question unfortunately doesn't have a well defined answer.

If you're asking:

"How long will it take until a particle is transmitted through a barrier in a 1-dimensional system with two confining potentials"

See ramanujan-dirac's answer.

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I like when you say that the problem doesn't have a well defined answer. Indeed in QM, while position, momentum, energy etc. can be calculated taking the expectation value of the corresponding operator, you can't define the tunneling time as an expectation value of some operator, because there's no time operator in QM! –  Mattia Jan 24 at 13:50
    
Alas the nature of QM is that some questions have fuzzy answers :) –  jk88 Jan 25 at 18:49

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