quantum tunnelling and "borrowing" energy?

[gib65]

Is quantum tunnelling the phenomenon I've heard of that requires particles to "borrow" energy from the universe? I mean, if a particle is to penetrate a barrier, one would think it needs a large amount of energy to do so. So is this where the concept of "borrowing" energy comes from?

[fredrik]

In the quantum description the classical conservations laws applies to expectations/mean values. Which means energy is not really conserved, but it's expectation value is, which in turns translates into that fluctuations in the "classical conservation" is allowed. This is to say that energy is on average conserved, but there may be fluctuations. The larger away from the mean you get, the more unlikely is such a fluctuation to be observed.

 

So when you compare classical energy with quantum energy, the more accurate comparasion would be to compare the classical energy to the quantum expectation value.

 

[math]E_{classical} = <Energy.QM> = \sum_{i} p(i)E_{i}[/math]

[math]\sum_{i} p(i) = 1[/math]

Where the sum is over the possibile energy states.

 

This is not unlike thermodynamics where the temperature T kind of measures the mean energy per particle. But the actually energy of any particular molecule will vary according to a distribution. Chemical reactions proceed even though the mean energy (temperature) is really below the activation energy.

 

In this sense, QM is really more "humble" by realising that we think we "know" are nothing but our expectations. In the classical domain, the expectations are stable enough to qualify for effective facts.

 

/Fredrik

[swansont]

As fredrik implies, it's the concept of needing to conserve energy because you want the particle to over the barrier, that drives that description. Energy conservation over short time spans only needs to satisfy [math]\Delta E\Delta t >\hbar/2[/math] so you can violate conservation of energy as long as it happens over a short enough time span, which is related to how high and wide the energy barrier is.

 

Of course, in a more detailed treatment by QM it's just a matter of the wave function not going to zero at an interface where the potential changes, while classically it would.

[Severian]

I think (in my opinion) this is easier to understand in terms of virtual particles.

 

'Real' particles obey Einstein's famous formula:

 

[math]E^2=p^2c^2+m^2c^4[/math]

 

where E is energy, p is momentum and m is mass. All well and good. When a particle obeys this equation we say that it is 'real' or 'on mass-shell' (the latter terminology can be understood by realising that m is an invariant length in 4-momentum space, so the equation defines a spherical surface in 4-d momentum space, or a 'shell').

 

But particles can be 'off-shell' too. Then the above equation is violated - the energy is not equal to the summed squares of the momentum and mass. This is analagous to the HUP, but here we are redefining what we mean by energy such that it is still conserved (so the E here isn't the E in the HUP, the [math]\sqrt{p^2c^2 +m^2c^4}[/math] is). We instead violate the mass-shell relation and that allows us to tunnel through the barriers.

 

However, how far off-shell (i.e. how much the relation is violated) is inversely related to the lifetime of the particle. To be exact, the particle's lifetime is proportional to

 

[math] \left| \frac{1}{E^2-p^2c^2-m^2c^4} \right|^2[/math]

 

(neglecting the natural decay width for the moment), so a virtual particle cannot live very long. This is the equivalence of the HUP only allowing the borrowing of energy for a limited time, because the particle must decay very quickly if it is far off shell.