There's no paradox in the statement that the eigenstate of least energy of a non-relativistic quantum particle in a box doesn't have zero energy. The value of the energy of any eigenstate is the result of a calculation, that's perfectly consistent.

A particle in a box doesn't have potential energy, since the potential in the box is equal to zero; it only has kinetic energy. So the total energy of a particle in a box is kinetic-the Hamiltonian is given by the square of the momentum operator (up to constants).

A particle in a harmonic potential, also, has an eigenstate of least energy, whose eigenvalue is equal to hbar omega/2, where the potential energy is m^2omega^2 x^2/2. This isn't a paradox either and, in this case it's not possible to assign this energy exclusively to the kinetic or the potential part.

The same holds however for the value of the energy eigenvalue of any eigenstate of the Hamiltonian, not, just the ground state. So, for a quantum particle, in a potential, it's not possible to say what part of the expectation value of its energy is due to the kinetic and which to the potential term. And the reason for that is that the two terms don't commute: So eigenstates of the Hamiltonian aren't, in general common eigenstates of the kinetic and of the potential terms-they can't be, since no such common eigenstates exist. The states of definite momentum-i.e. of kinetic energy-are superpositions of eigenstates of definite position-i.e. of potential energy; and both are, different superpositions of states of definite total energy. It's possible to use either of the three bases to describe the system, because they are bases in function space; but It's the Hamiltonian that defines time evolution.

The ``proportions'' are defined by the square of the modulus of the corresponding coefficients in the expansion of the wavefunction in the different bases.

This isn't, incidentally, anything new.

Here is an introductory response first to thank our friend from France for his illustrative response and then to shed light on the question and its sought-after answer.

The response was excellent but incomplete.

The reason is that it relies entirely on the Schrödinger SE equation which is a subset of physics but not physics as a whole.

A. Einstein once said that the Schrödinger equation with the Bohr/Copenhagen interpretation was incomplete.

We assume that Einstein's reasoning is that SE does not describe time precisely:

It views time as a non-woven separable controller in 4D x-t space.

In other words, the time step or jump dt is chosen arbitrarily, independently of the spatial increment dx.

It is true that the macroscopic potential inside a closed box is zero, which is predicted by the B matrix chain statistics of the Cairo techniques in the article:

[Is it time to reformulate the Poisson and Laplace partial differential equations? I, Abbas, Researchgate and IJISRT Journal, June 2023]

However, stating that the two terms "KE and PE" do not commute is a matter of debate since both can be measured at the same time.

Furthermore, we see proof that the energy of a quantum particle in a potential box cannot be zero, but we do not see mathematical or physical proof that its minimum energy is hf/2, which is at heart of the current question.

?

To be continued.

Either one is working within quantum mechanics-which means using one of many equivalent formalisms, among which is solving the Schrödinger equation-or one isn’t.

That the kinetic energy operator doesn’t commute with the potential energy operator is a trivial consequence. It doesn’t have anything to do’with the interpretation of quantum mechanics.

If one isn’t working within quantum mechanics, the discussion of quantum particles doesn’t make sense.

Ismail Abbas,

The potential energy of a quantum particle is a perfect zero inside the box not because of Schrödinger equation or Dirac equation or the like but because this is how the model of a particle in a box is constructed.Thus,your question about whether the minimum energy is kinetic or potential or both reflects your ignorance about what the particle-in-box model is.

Answer II, continued.

This is a brief response to shed light on the issue and to thank our French colleague for his illustrative response.

We speak two completely different languages:

I-QM and the Shrödinger equation operate in Hamiltonian space where KE+PE is a constant of motion, this constant is obviously equal to the total energy of the system.

II-The space of the statistical transition matrix B which is the statistical equivalence of the Schrödinger equation and its derivatives.

The space of matrix B is defined by the formula,

PE/E is a constant of motion.

We assume that the solution of matrix chains B is more informative than SE itself.

In other words, the SE solution is a subset of the chain-to-matrix solution B.

For example, the matrix form of the time-independent solution of the quantum particle in an infinite potential well is given by:

Ψ=[M1 + C1 V(x,t)] Ψ

(M1 is the transition matrix resulting from the Nabla^2 operator.)

would be reformulated as follows,

Ψ=[M1 + C1 V(x,t)+C2 b] Ψ

Where b is the vector of boundary conditions.

It is worth mentioning that B-matrix string theory is not entirely new and has been working effectively since 2020.

To be continued.

Please go to,

A statistical numerical solution for the time-independent Schrödinger equation -II

It is more complete and more understandable.

Dear Dr. Ismail Abbas There is no single one dimension and static equation that has been written in past can describe three dimension of nature that it is changing constantly through temperature, and pressure.

Thus none of our icons in past that assumed their equation is presenting nature is incorrect.

This fact supported by science.

we would recommend the research article on Researchgate:

A statistical numerical solution for the time-independent Schrödinger equation,-II, November 2023.

It is Dark Energy

Ismail Abbas Alaya Kouki Javad Fardaei Issam Mohanna

The "zero-point energy" or "vacuum energy" paradox, relates to the minimum energy of a quantum particle confined to a box. In this case, the energy referred to as Emin = hf/2 represents the lowest possible energy state of the particle.

It's important to note that this minimum energy is not solely kinetic energy or potential energy but rather a combination of both. In quantum mechanics, the total energy of a particle is described by its Hamiltonian operator, which includes both kinetic and potential energy terms.

In the case of a particle in a box, the Hamiltonian operator includes a kinetic energy term related to the particle's momentum and a potential energy term related to the confinement within the box. The minimum energy state, Emin, arises due to the wave nature of the particle, where its wavefunction cannot be completely localized to a single point within the box. As a result, even in the absence of external forces, the particle exhibits quantum fluctuations that contribute to its minimum energy.

Therefore, the minimum energy of a quantum particle in a box, Emin = hf/2, encompasses both kinetic and potential energy contributions, reflecting the inherent quantum nature of the system.

Hope it helps: partial credit AI