On the energy density in quantum mechanics

There are several definitions of energy density in quantum mechanics. These yield expressions that differ locally, but all satisfy a continuity equation and integrate to the value of the expected energy of the system under consideration. Thus, the question of whether there are physical grounds to choose one definition over another arises naturally. In this work, we propose a way to probe a system by varying the size of a well containing a quantum particle. We show that the mean work done by moving the wall is closely related to one of the definitions for energy density. Specifically, the appropriate energy density, evaluated at the wall corresponds to the force exerted by the particle locally, against which the work is done. We show that this identification extends to two and three dimensional systems.


I. INTRODUCTION
In many textbooks on quantum mechanics [1][2][3] we are taught that an important consequence that emerges from the fact that the wave function that describes a quantum particle satisfies the Schrodinger equation, is that the probability density for finding the particle at a given position satisfies a continuity equation.This reflects the fact that probability is locally conserved [1][2][3].However, and rather surprisingly, there is very little discussion in the literature regarding the concept of energy density in quantum mechanics, how to define it, how it evolves, whether it is conserved, and what is its physical significance [4][5][6][7][8][9][10][11][12].In these few works, different expressions for the energy density have been proposed.These densities differ locally from one another, though they all integrate to the same value of the expected energy.It has been argued that there is no "physical" way to choose which of these different densities is physically meaningful, and that it essentially comes down to a matter of preference [7,8].In this paper we present what we believe are the minimum conditions that the energy density must satisfy and we discuss two different possible definitions that satisfy the conditions.We illustrate a way to probe the systems locally in order to identify which of the energy densities has a physical manifestation.To do this, we consider particles trapped in simple infinite well systems and probe the systems by changing the size of the well.We show that the expected work done to change the size of the well is directly related to the local value of one of the energy densities at the boundary.This density can be interpreted as the effective force exerted by the particle in the well.The other density is irrelevant in this respect as it vanishes identically at the walls.

II. THE PROBABILITY DENSITY AND THE ENERGY DENSITY
If we consider a quantum particle in a region I, in the position representation it is described by a wave function Ψ(x, t) that satisfies the Schrodinger equation where V (x) is a potential and the wave function is normalized: The integrand |Ψ(x, t)| 2 ≡ ρ D (x, t) is interpreted as the probability density of finding the particle at position x at time t.Then, as is well-known, the probability density fulfills a continuity equation where the flux J D is given by This implies that the probability density is locally conserved [1,3].By the same token, the expected energy of a quantum particle can be expressed as This expression suggests that the integrand might be interpreted as an energy density, akin to the probability density discussed above.If this was the case, the energy density in the system would be given by Unfortunately, the choice ρ E 1 above is not satisfactory as there is no guarantee that this density is real at every point where it is defined, though it is easy to see that the imaginary part always integrates to zero [10].
It would be reasonable to expect that a physically acceptable definition of quantum energy density should, at least, meet the following conditions: 1. it should be a local, real-valued function in the position representation; 2. its integral over the system must be equal to the expected energy E .
However, an infinite number of different expressions for the energy density satisfying these conditions can still be defined.
One possible definition of energy density, that does satisfy the above conditions, is which is obviously real, and integrates to E .We cannot guarantee that ρ E 2 is non-negative, even when the potential is positive for all x.Nevertheless, it has been argued that the negative kinetic energies that this entails are not necessarily unphysical [13], and may even be measured [14].Another possible choice for the energy density is which is real and also integrates to E .In contrast to ρ E 2 , this choice is non negative if the potential V (x) is non negative itself.
The energy densities defined above are locally conserved in the sense that they satisfy a continuity equation for appropriate energy fluxes: Clearly, however, both expressions for the energy density differ locally, so the question of whether there are physical grounds to choose one over the other, or neither, arises naturally.Further, the physically meaningful density will be associated to an energy flux, which is central to the description of energy transport in quantum systems In what follows, we propose a simple scheme to probe locally a particle in a box.Specifically, we calculate the work done by moving one of the box's walls a distance dℓ.The idea is that if this movement is done quickly, then the work will be due to interaction of the wall with the system at its immediate vicinity.As it turns out, the work done is independent of the speed at which the wall moves.
A. One dimensional system Following one of the procedures described in [15,16], we begin by considering a particle of mass µ in a one dimensional infinite square well of size L(t) (the extension to higher dimension "cubic" wells is straight forward).We can write the wave function of the particle as where φ n (x, t) are the instantaneous eigenfunctions of the Hamiltonian, corresponding to eigenvalues (instantaneous energies) and the coefficients b n (t) are chosen so that Ψ(x, t) satisfies the time dependent Schrodinger equation.Substituting Ψ(x, t) into eq.( 1) and using the ortonormality of the eigenfuctions φ n (x, t), we find where The initial values of the b n (t) are fixed when we prepare the initial state: The expected energy of the particle is given by We are interested in the change in energy of the particle due to the motion of the wall after a time δt, i.e.E(δt)−E(0) ≈ Ė(0)δt.From eq.(II A) we can calculate where Ėn (0) is L(0) is the initial length of the well, and v ≡ L(0) is the initial speed at which the wall moves.The ḃn (0) can be calculated using eq.( 13), where the time derivatives of the instantaneous eigenfunctions can be obtained from equation (11) and the resulting integrals appearing in eq. ( 13) can be evaluated directly Then, after a little algebra, Ė(0) can be expressed in terms of the initial conditions of the system as Now we compare this expression with the value of the value of energy density ρ E 2 and ρ E 3 evaluated at the wall.For this system we have Evaluating at the wall, x = L(t), we see that ρ E 2 (L(t), t) = 0 trivially for all time.On the other hand, for ρ 3 (x, t) we have Evaluating this expression at the wall, x = L(t), at t = 0 we get (21) Thus, the average amount of work δW done by the the motion of the wall during a time interval δt is where δℓ = vδt is the distance the wall moves.Both the particle density ρ D and ρ E 2 vanish at the wall, thus, this result suggests that the wall transfers energy to, or receives energy from the particle as if the wall interacted with the system through the energy density ρ E 3 .In particular, in this one dimensional example, ρ E 3 (L(0), 0), being an energy per unit length, can be thought of as the mean force exerted locally and instantaneously by the particle on the wall.

B. Two and three dimensional systems
For completeness, we show that these results extend to a quantum particle in a circular box in two dimensions and in a spherical box in three dimensions.
In the two dimensional case we consider a particle with mass µ confined in an infinite circular potential where the radius varies in time R(t) = R 0 + vt.The wave function takes the form where z m,n is the nth the zero of Following the same procedure as above (see details in Appendix A), we find that the average work done to move the wall a distance δℓ = vδt is given by Now we check that this work is done against the mean total force exerted by the particle on the wall through the energy density ρ E 3 : where Evaluating at t = 0 and performing the integral in eq. ( 27), we find so indeed we have δW = −Fδℓ Finally, for the three dimensional case we consider a particle with mass µ in an infinite spherical moving well potential of radius R(t) = R 0 + vt.Now the wave function takes the form where z l,n is the nth zero of j l (x), the spherical Bessel function of order l, Y l,m (θ, φ) are the spherical harmonics, and θ l,n (t) = 1 t 0 E l,n (t ′ )dt ′ , as before.For this case, the change in the total energy of the system caused by moving the boundary at a speed v for a time δt is (See Appendix B): As in the previous examples, this change of energy is due to work done against the total force exerted by the particle on the boundary.In this case this is given by where Here Y l,m = Y l,m (θ, ψ)r and Ψ l,m = r∇Y l,m (θ, ψ) are the vector spherical harmonics [17].These satisfy the orthogonality relation Ψ l,m • Y l,m =0, and the integral relations Thus, evaluating ρ 3 ( r, t) at t = 0 and integrating, we have Finally, using the properties of the Bessel functions, we get Thus, we again have dW = −Fdℓ, or equivalently, dW = −PdV , where the pressure is P = F/A, the differential of volume is dV = Adℓ, and A = 4πR 2 0 is the area of the surface.

III. DISCUSSION
If we have a quantum particle in a infinite well, changing the size of the well changes the energy of the system.Conservation of energy implies that this change of energy must be due to work done when moving the walls against the force exerted by particle.We have shown that the mean amount of work done when moving the wall is directly related to the energy density ρ E 3 at the wall.This energy density also satisfies the physical requirements of being local, and, of course, integrates to the expected value of energy.Further, this expression yields a positive definite value everywhere if the potential is also positive, and it satisfies a continuity equation, implying that it is locally conserved.In contrast, due to the boundary condition ρ E 2 vanishes identically at the wall (as does the probability density ρ D ), so its value cannot be probed by moving the wall, and though it is also conserved, it is not necessarily positive everywhere it is defined, even when the potential is nowhere negative, which implies that the particle may have negative kinetic energy in some regions of the system.We contend that these results imply that ρ E 3 can be considered as the actual physical energy density of the system, in the sense that it is measurable as the mean instantaneous local force (pressure) exerted by the particle on the wall.This would also imply that the energy flux associated to ρ E 3 is the relevant flux to describe energy transport in quantum systems.From a different perspective, ρ E 3 is analogous to the electromagnetic radiation pressure on a perfect absorber, as in that case too P rad = u, where u is the energy density of the wave.Of course, if all that is wanted is the integral of the density to obtain the expected energy, then any one of the expressions discussed in this work can be used, even ρ E 1 , which may have complex values locally.
and its derivative at t = 0 where ḃ(t) and Ėm,n (t) are respectively.The integral in the last expression when which implies that these terms in Ė(0) vanish, moreover when n = n ′ we have using [19] and the Leibnitz rule, we obtain By the recurrence relation analogously we have On the other hand, the energy density is given by in this case we take Analogously, we need to know ρ(R 0 , 0) = ρ 3 ( r, t)δ(r − R 0 )rdrdφ.We know that 2π 0 e (m−m ′ )iφ dφ = 2πδ m,m ′ , thus we have Applying the same recurrence relation as above we get, where R(t) = R 0 + vt.The Hamiltonian operator is where L is the orbital angular momentum operator, which is given by in this case, the instant eigenfunctions are where z l,n is the nth zero of spherical Bessel function of order l (j l (x)) and Y l,m (θ, φ) are the spherical harmonics.The wave function takes the form of where θ l,n (t) = 1 t 0 E l,n (t ′ )dt ′ , with Then, the expected energy is given by where ḃ(t) and Ėl,n (t) are respectively where dΩ is the solid angle differential.Analogously to the 2-dimensional case, we only work when n ′ = n because when n = n ′ the above integral vanishes.Now, dr (B13) using the same relations as above for this type of integrals we have, By the recurrence relations The energy density in this case is 2b n,l,m (t)b * n ′ ,l ′ ,m ′ (t) R(t) 3 j l+1 (z l ′ ,n ′ ) j l ′ +1 (z l ′ ,n ′ ) e i(θ l,n ′ (t)−θ l,n (t))