# Negative kinetic energy in tunneling

• gulsen
In summary: This is not a physical state, as it cannot be measured. And it goes against the fundamental principles of quantum mechanics. In summary, the conversation discusses the concept of negative kinetic energy in quantum mechanics, particularly in relation to potential barriers and the conservation of energy. It also touches on the importance of physical reasoning and the implications of imaginary eigenvalues.
gulsen
Say I have a particle with a kinetic energy of 5 (in some units). And I have a potential barrier of 10 between 0<x<1 (again, in ome unit system), and 0 elsewhere. According to quantum theory, the partcile may be found between 0 and 1. And in this region, if the energy is conserved (5 = T + 10), shouldn't the kinetic energy be -5?!? So that $$\hat T \psi_2 = -5 \psi_2$$.

What's going on here??

gulsen said:
Say I have a particle with a kinetic energy of 5 (in some units). And I have a potential barrier of 10 between 0<x<1 (again, in ome unit system), and 0 elsewhere. According to quantum theory, the partcile may be found between 0 and 1. And in this region, if the energy is conserved (5 = T + 10), shouldn't the kinetic energy be -5?!? So that $$\hat T \psi_2 = -5 \psi_2$$.

What's going on here??

How do you know that your $\psi$ inside the barrier is an eigenfunction of your T operator in the first place? Go ahead and solve for the wavefunction inside the barrier, and see if it is an eigenfuntion of your operator.

Furthermore, why should KE be conserved? Shouldn't you be more concerned with H?

Zz.

I didn't say kinetic energy should be conserved. Note that I'm conserving energy and not kinetic energy, by saying: 5 = T + 10 (initial energy = energy inside bump)
$$\psi_2$$ should be eigenfunction of T because $$H = T + V_0$$ in this case. Also, the solution for 2nd region is
$$(T + V_0)\psi_2 = E\psi_2$$
$$T\psi_2 = -5\psi_2$$

There's indeed such a problem. In regions where energy of particle is smaller than the minumum of particle, the solutions are exponential rather than oscialating, and kinetic energy operator becomes negative. For instance, (in nuclear physics) in a spherical potential well, the solution in the sphere (r<R) is $$F \sinh(qr)$$ where $$q^2 > 0$$.

$$<T> = <\psi | T | \psi> = <F \sinh(qr) | -\frac{\hbar^2}{2m} \frac{\partial ^2}{\partial r^2} | F \sinh(qr)>$$

$$<T> = -\frac{\hbar^2}{2m} q^2 <F \sinh(qr) | F \sinh(qr)>$$
Which is a negative number since $$<F \sinh(qr) | F \sinh(qr)>$$ is the probability of finding the particle within the sphere. I don't know whether such a problem arises in the field theory, so I expect some educated ones shed light on the topic.

Last edited:
This all goes together:

- evanescent wave
- imaginary wavevector
- negative "kinetic energy"

why would that be a problem ?

The important thing is not the "kinetic energy" would be positive.
The important thing is the Noether theorem: a symmetry implies a conserved quantity, time invariance implies energy conservation.

Dealing with (quantum) waves brings the possibility of evanescent wave and tunelling and this translates in negative kinetic energy "during tunneling".

lalbatros said:
why would that be a problem ?

Because eigenvalues of an observable, namely momentum, becomes imaginary then.

## What is negative kinetic energy in tunneling?

Negative kinetic energy in tunneling refers to the phenomenon where a particle has a negative kinetic energy when passing through a potential barrier in quantum tunneling. This is in contrast to classical mechanics where kinetic energy is always positive.

## How does negative kinetic energy affect tunneling?

Negative kinetic energy can have different effects on tunneling depending on the specific conditions. In some cases, it can increase the probability of tunneling by decreasing the energy barrier. In other cases, it can cause the particle to be reflected back instead of passing through the barrier.

## What is the significance of negative kinetic energy in quantum mechanics?

Negative kinetic energy is a consequence of the wave-like nature of particles in quantum mechanics. It allows particles to pass through energy barriers that would be impossible to overcome in classical mechanics. This phenomenon is crucial in understanding the behavior of particles on a microscopic scale.

## Can negative kinetic energy be observed in everyday life?

No, negative kinetic energy is a purely quantum mechanical phenomenon and cannot be observed in everyday life. It is only observable at the subatomic level and is not relevant to macroscopic objects.

## What are some real-world applications of negative kinetic energy in tunneling?

Negative kinetic energy in tunneling has many applications in modern technology, such as in quantum computing and tunneling microscopy. It is also important in understanding and manipulating the behavior of electrons in semiconductors, which is crucial for the development of electronic devices.

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