Is Negative Kinetic Energy Real in Quantum Systems?

[Adrian Baker]

Negative Kinetic Energy?

I have heard that in processes like quantum tunneling and other quantum models, the KE can be considered to be negative when compared to a classical system. (A very odd concept!)

I've tried searching the web, but only found one or two very technical sites that go way over my head..

Can anyone enlighten me on this please, or perhaps point me in the right direction?

 

[jcsd]

If it were possible to measure the energy of a particle inside of a potential barrier (ewhich had tunnelled there) it would indedd be negative.

 

[Adrian Baker]

errr... thanks for the comments guys, but we are still a bit over my head!

Any chance of some more detail. [?]

Thanks

 

[xeguy]

Originally posted by Adrian Baker
I have heard that in processes like quantum tunneling and other quantum models, the KE can be considered to be negative when compared to a classical system. (A very odd concept!)

I've tried searching the web, but only found one or two very technical sites that go way over my head..

Can anyone enlighten me on this please, or perhaps point me in the right direction?

Quantum mechanics can be pretty counterintuitive sometimes! Classically, if you have a potential barrier of height V and a particle incident on that barrier with E < V, the particle would reflect off the barrier completely.

The same system in quantum mechanics gives a non-zero probability that the particle will be transmitted through the barrier. This is a wave phenomenon, but in quantum mechanics particles exhibit wave-like properties.

So while the particle is tunneling, does it have negative KE? This would be in violation of classical physics, so there has to be a way to rectify this. What it comes down to is the condition required to observe the negative kinetic energy leads to an exceptionally small tunneling probability. The argument follows from the uncertainty principle. Even if you set up your system to maximize your chance to measure the negative kinetic energy, you would minimize the probability of the particle tunneling through your barrier!

The wavefunction of the tunneling particle decreases exponentially in the barrier. The tunneling probability is strongly dependent on the width of the barrier, the mass of the particle, and the quantity V-E. For instance, the ratio of tunneling probability for protons to electrons is around a factor of 10^-91. So electrons are much more likely(!) than protons at the same energy to tunnel through the same barrier. This also precludes you tunneling through your front door. :-)

Experimentally, tunneling does occur. Examples include nuclear alpha decay and Josephson junctions.

 

[jby]

So while the particle is tunneling, does it have negative KE? This would be in violation of classical physics, so there has to be a way to rectify this. What it comes down to is the condition required to observe the negative kinetic energy leads to an exceptionally small tunneling probability. The argument follows from the uncertainty principle. Even if you set up your system to maximize your chance to measure the negative kinetic energy, you would minimize the probability of the particle tunneling through your barrier!

Is there an uncertainty principle that says this? Attempt to observe/measure negative KE => small tunneling probability? Elaboration please.

 

[Adrian Baker]

Thanks for that xeguy - it is appreciated.

I wouldn't say I fully understand it now , but I certainly now understand what it is that I don't know!

 

[xeguy]

You're welcome, Adrian. If you take a QM course you'll have a better understanding of the concept.

 

[xeguy]

Originally posted by jby
Is there an uncertainty principle that says this? Attempt to observe/measure negative KE => small tunneling probability? Elaboration please.

You can do a calculation from the Heisenberg uncertainty relations. If tunneling occurs, you have to localize the particle in a range [del]x << (barrier width). From this you get a condition on [del]p, and hence [del]E. The uncertainty must be much less than |E-V|, where V is the height of your potential barrier, if you want to observe negative KE. The final relation you get is a condition on the exponent of the decaying exponential. The exponent must be >> 1 in order to observe this negative KE. Since the transmission probability depends directly on this term, it approaches zero.

 

[zare]

it is possible to have negative energy. in some spatial anomalies like star going supernova and stuff, some particles were expelled at above c speed. then the tachyon theory came up, and (by that theory) if it has negative mass if must have negative energy (e=mc2, and mass isn't vector but scalar.)

 

[ranyart]

Originally posted by zare
it is possible to have negative energy. in some spatial anomalies like star going supernova and stuff, some particles were expelled at above c speed. then the tachyon theory came up, and (by that theory) if it has negative mass if must have negative energy (e=mc2, and mass isn't vector but scalar.)

In a quaint way what you don't see is what you get!

In e=mc2, space/ENERGY transforms into mass/ENERGY without the c2. The normal E=MC2 relates to 3-dimensional transformations, a deeper inquiry and one finds that Space(1-d negative matter) can transform into Matter(2-d negative space) or the Vacuum Force.

 

[zare]

i see, thanks for replying. but i didnt invent tachyon theory, and one of reports from mathematical construction model of tachyon cleary stated that e=mc2 statement. and if something has a mass and velocity it has kinetic energy. in non-relativistic energy ranges we could apply e=(mv2)/2. but I am sure that more quantum laws interfere in that example than simple explanation between mass and energy.

if tachyons were real, it would be pretty handy to have'em around. lift your car into space, sit in it, and if it weights 1.5 tons, load 1.6 tons of tachyons into trunk. your mass is -100 kg, and off you go ;) (warp 9)

 

[ranyart]

Originally posted by zare
i see, thanks for replying. but i didnt invent tachyon theory, and one of reports from mathematical construction model of tachyon cleary stated that e=mc2 statement. and if something has a mass and velocity it has kinetic energy. in non-relativistic energy ranges we could apply e=(mv2)/2. but I am sure that more quantum laws interfere in that example than simple explanation between mass and energy.

if tachyons were real, it would be pretty handy to have'em around. lift your car into space, sit in it, and if it weights 1.5 tons, load 1.6 tons of tachyons into trunk. your mass is -100 kg, and off you go ;) (warp 9)

Humm..not so I am afraid! The technical difficulties would far outway the practical fuel consumption savings!

The main thing would be how do you "head-them-off-at-the-pass", as you would have to intercept Tachyons, you cannot 'load them prior'.

You would have to know well in advance 'where and when' they be?

 

[zare]

heisenberg's postulate dictates that on that level of quantum engineering you can't know both where and when. and of course, i was joking about the faster-than-light car...

as for "engage" negative mass and "disengage" it wouldn't be so hard maybe. i read that bond between two tachyons is so strong that twists (?) laws of physics and mt=mt1+mt2=2mt1 doesn't apply. what applies is some strange stuff like mt=(mt1)^2. so they have positive mass because negative mass is squared. so for negative mass you would need to decouple them.

about tachyons they arent considered to be real. it's more like mathematical model of particle that has negative mass because it exists only in faster-than-light environment. even at fermilab they are ruled out.