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Doesn't quantum tunneling contradict the 1st Law of thermodynamics? As far as I remember my school physics teacher told us that quantum tunneling is the reason why we can observe the alpha decay, as the energy gained from the mass deficit is not enough to overcome the potential barrier of the nucleus. How can one explain this paradox without giving up the 1st Law of thermodynamics?

Qmechanic
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MrYouMath
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Doesn't quantum tunneling contradict the 1st Law of thermodynamics?

Thermodynamics is an emergent theory, based on underlying statistical mechanics. Conservation of energy is a general law in classical mechanics and is carried over into the formulation of quantum mechanics in a consistent manner.

As far as I remember my school physics teacher told us that quantum tunneling is the reason why we can observe the alpha decay, as the energy gained from the mass deficit is not enough to overcome the potential barrier of the nucleus. How can one explain this paradox without giving up the 1st Law of thermodynamics?

So it is not the first law of thermodynamics, because tunneling is a quantum mechanical phenomenon on individual quantum mechanical particles/entities and not a collective effect. The question correctly stated is saying: is the law of conservation of energy violated by tunneling?

Here is tunneling for particles described by a quantum mechanical wavefunction in a potential well.

tunneling

This is the quantum mechanical solution of "particle in a potential well". If the potential well does not have infinite walls, the energy state solutions carry through and out of the walls in quantum mechanics. Energy conservation is not violated because the particle occupies the same energy level, inside and out side. For the probability of it to exist at any place, the wavefunction must be non-zero and open- that is what "probability" means: Out of N such particles in the same potential well ( N alpha+ rest of nucleus) n alpha may be found outside the potential well is what the solution tells us.

That is why quantum mechanics was developed, to explain phenomena that cannot be reconciled with classical mechanics.

(Please note that if this were a classical barrier with the same parameters, lets say a car climbing a hill and coming down to the exact same level, the energy spent to go to the top would be regained when reaching the same level on the other side of the hill, so again no energy would be spent in total. It is not energy conservation that classically is paradoxical, but the probabilistic disappearance of a barrier.)

anna v
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  • Ah ... is energy in Quantum mechanics also a wave functiona? Hence the 1st Law is not violated and still the particle can escape the potential wall. – MrYouMath Mar 21 '16 at 08:34
  • energy is found by applying the energy operator on the wavefunction. The reason it is not violated is because the wavefunction has the inside the wall and outside levels equal to the level in the potential well. Please read carefully the partenthesis in my answer. Classically also energy conservation is not violated. What is violated is the classical intuition that to cross a barrier you have to spend extra energy , as in digging a tunnel. – anna v Mar 21 '16 at 11:33
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There is no violation of conservation of energy.

Quantum mechanics simply allows for a non-zero probability of a particle with energy $E$ to be detected in or behind a potential wall with potential $V>E$. The particle doesn't magically gain "enough energy" to go through the barrier, it's just that our classical idea of the region being completely forbidden for particles with smaller energy is false in quantum mechanics - quantumly, the wavefunction inside such a classically forbidden region decays exponentially, but does not immediately vanish and stops decaying after entering a classically allowed region again, so you get a non-zero probability to detect the particle outside the potential well or even inside the barrier.

ACuriousMind
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  • I know the probability concept, but still it makes no sense. How can a particle overcome that potential wall without additional energy? Stating that the probability is there, does not explain the fact that it does not have the requiered energy to get to that position. That is where i see a contradiction to the 1st law. – MrYouMath Mar 17 '16 at 11:55
  • @MrYouMath: There is no energy "required" in quantum mechanics. Particles can be detected in regions with higher potential energy than their own energy. That just comes out if you solve the Schrödinger equation for such a setup. You're trying to insist that the world should work as you classically expect it. It doesn't. – ACuriousMind Mar 17 '16 at 11:58
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    Hmm ... i am not insisting on something i just want to understand it :D. QM is also true for large objects (but with much smaller probabilities). Why is it that the same question in classical mechanics requiers some change in energy. If i don't need the energy concept for the movement of small particles, why should i need it for large ones? – MrYouMath Mar 17 '16 at 12:11
  • @MrYouMath: Large objects are mostly forced into localized positions by the constant interaction with their environment (air molecules, light, you name it) - their wavefunctions don't have enough undisturbed time to "spread" through barriers, while the "standard" state of the nucleons in an atom that can $\alpha$-decay is already pretty spread out (compared to the length scale of the barrier/nucleus). – ACuriousMind Mar 17 '16 at 12:14
  • @ACuriousMind Is the use of $\Delta E \Delta t \ge \frac{\hbar}{2} $ frowned on in this instance? – Farcher Mar 18 '16 at 06:13
  • @Farcher: What do you want $\Delta t$ to be in this instance? – ACuriousMind Mar 18 '16 at 10:44
  • @ACuriousMind Loan period. – Farcher Mar 18 '16 at 15:29
  • @Farcher: Sorry, what? Perhaps have a look at this answer as to what $\Delta t$ represents, I don't know what you want to do here. – ACuriousMind Mar 18 '16 at 15:44
  • @ACuriousMind Thank you for the informative link. In one of the answers there is the following statement "Energy exchange (ΔE) and time-frame (Δt) during which this can happen.". Is that a reasonable interpretation of what Δt is? – Farcher Mar 18 '16 at 16:07
  • @Farcher: I don't think so, as that would mean that the more energy you exchange, the faster that exchange can happen, which seems patently absurd to me. – ACuriousMind Mar 18 '16 at 16:12
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    @Farcher Your problem is that you are tying to think what happens with energy during the tunneling. This can be done because energy do not commute with the position of particle. If you are measuring position you can't measure energy and actually this is the reason why tunneling is possible. If you detect a particle tunneling a barrier you can say nothing about the energy. In other words, the inital eigenstate of energy is a suporposition of eigenstates of position, a wave function. The barrier need to be very high to prevent tunneling. – Nogueira Mar 19 '16 at 13:49