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Dimensionless equations in non-relativistic quantum mechanics

We discuss the numerous advantages of using dimensionless equations in non-relativistic quantum mechanics. Dimensionless equations are considerably simpler and reveal the number of relevant parameters in the models. They are less prone to round-off errors when applying numerical methods because all the quantities are of the other of unity. A dimensionless equation facilitates the application of perturbation theory and provides a glimpse of the sort of solution we are going to obtain beforehand.


Introduction
Solving the Schrödinger equation in non-relativistic quantum mechanics is greatly facilitated if we first convert that equation into a dimensionless form. The reason is that fundamental constants likeh (Plank constant divided 2π) the electronic mass m e and charge e, as well as other model parameters, are removed leaving a much simpler equation [1]. The algebraic manipulation of the dimensionless equation is considerably less laborious and its numerical treatment, if it is not exactly solvable, exhibits less round-off errors after getting rid of such small numbers.
It is a common practice, though in our opinion quite undesirable, to state that "we choose units so thath = m = e = c = 1" or similar expressions [2]. This is specially so in the case of pedagogical papers where one should teach the students to carry out the procedure of making dimensionless equations in detail [3]. For this reason, in this paper we show how to derive dimensionless equations and discuss the advantages of the approach as well as valuable information about the physical result we are looking for.
In section 2 we discuss several one-dimensional examples, in section 3 we focus on atoms and molecules, in section 4 we outline the utility of dimensionless equations in the application of perturbation theory and in section 5 we summarize the main results and draw conclusions.

One-dimensional models
In order to illustrate how to convert quantum-mechanical equations into dimensionless differential equations we begin with some simple one-dimensional models in non-relativistic quantum mechanics. For simplicity, we first focus on time-independent problems. The Hamiltonian operator for a particle of mass m in a potential V (x) in the coordinate representation is given by where −∞ < x < ∞. The strategy is simple: we first define a dimensionless coordinatex = x/L, where L is a unit of length that we choose conveniently for each problem. If we take into account that d/dx = (dx/dx) d/dx = L −1 d/dx we conclude that d 2 /dx 2 = L −2 d/dx 2 . Second, we define the dimensionless Hamiltonian operatorH as Therefore, if ψ is an eigenfunction of H with eigenvalue E (Hψ = Eψ) then the corresponding eigenvalueẼ ofH is the dimensionless energy and both eigenval- whereh 2 / mL 2 is the unit of energy.
As a first example we choose a particle of mass m in an impenetrable box In this case the unit of length is given by the box length and we defineψ (x) = ψ (Lx) (the normalization factor is irrelevant for present discussion) that is a solution to and satisfies the boundary conditionsψ (0) =ψ (1) = 0. The eigenvalues of this equation areẼ n = n 2 π 2 2 , n = 1, 2, . . . , so that equations (6) and (3) yield the well known energies of the particle in a box [4]. It is clear that when we make the quantum-mechanical eigenvalue equation dimensionless we can predict the dependence of the energies on the model parameters and physical constants (h, m and L in the present case) without solving the equation. Besides, equation (5) makes the statement "we choose units so thath = m = L = 1" self-evident (specially for pedagogical purposes). In addition to it, we also predict that the eigenfunctions for the particle in a box will depend on the variablex = x/L and this is actually so as shown by [4] ψ n = 2 L sin nπx L .
It is always convenient to tell the students that the arguments of functions like sin(z), ln(z), exp(z), etc should be dimensionless and that a result that does not follow this rule is wrong. In our opinion, the dimensionless energyẼ is more convenient than ǫ = 2mE/h 2 [4] that exhibits units of length −2 .
The example above may look unimpressive and we will discuss some others below. Another simple one is the harmonic oscillator with potential where k > 0 is the force constant. In this case we have Therefore, if we choose then the dimensionless Hamiltonian operator will bẽ and the unit of energy results to bē We realize that the energies of the harmonic oscillator are of the form E n = hωẼ n , whereẼ n is dimensionless. If we solve the eigenvalue equation we obtain the allowed energies E n =hω (n + 1/2), n = 0, 1, . . . that agree with the previous equation becauseẼ n = (n + 1/2) are the eigenvalues of the dimensionless Hamiltonian operator (11) [4]. Once again, we realize the form of the energies in terms of the model parameters and physical constantsh, m and k without solving the eigenvalue equation. Besides, we also know that the states of the harmonic oscillator will be functions ofx. It is well known that in this case is an Hermite polynomial and N n a normalization factor [4].
In some cases the potential-energy function is defined in terms of a length parameter, for example: and we have two possibilities. If we choose L = a we havẽ If, on the other hand, we choose Although the dimensionless Hamiltonians (15) and (17)  Up to now we have been tacitly assuming that our interest was the calculation of the bound states supported by the potential. Suppose that we are interested in the calculation of the scattering states for the potential (13). In this case the present analysis tells us that the transmission T and reflection R coefficients can be expressed in terms of only two quantities: λ andẼ. For instance, consider the textbook example given by the tunnel effect through the rectangular potential barrier This problem can be solved exactly and the result[M76] rewritten as that confirms our prediction that T depends on only two dimensionless parameters when other strategies produce results in terms of more dimensional quantities [4].
Another interesting and well known example is the Morse oscillator with potential where D e , a > 0. In this case there are also two obvious possibilities and we choose L = 1/a so that The bound-state energies of the Morse oscillator are known to be [5] E n = hc ω e (n + 1/2) − χ e ω e (n + 1/2) 2 , that can be easily rewritten as in agreement with the prediction of present approach.
In closing this section we briefly focus on the time-dependent Schrödinger and proceed as before with respect to the Hamiltonian operator. In addition to it we define the dimensionless timet = ωt, where ω is an arbitrary frequency.
the Schrödinger equation becomes In the case of the harmonic oscillator, for example, it follows from equation (25) that ω = k/m is the oscillator frequency.
The Hamiltonian operator for a system of K particles of masses m i , charges q i at the positions r i , i = 1, 2, . . . , K is given by [5,6] where ǫ 0 is the vacuum permittivity and r ij = |r i − r j | is obviously the distance between particles i and j. In order to obtain a dimensionless Schrödinger equation we proceed as before and introduce a length unit L and the dimen- If m e and −e denote the electronic mass and charge, respectively, then we define the dimensionless quantities Therefore, if we choose the dimensionless Hamiltonian becomes It is worth noticing that L = a 0 is the well known atomic unit of length and is the atomic unit of energy. As argued above, one of the most noticeable advantages of this procedure is that we get rid of small numbers likeh, e, m e , ǫ 0 , etc. It is equivalent to setting these quantities equal to unity. Such small numbers may increase the round-off errors in a numerical calculation of atomic and molecular properties and for this reason atomic units are used throughout [5,6].
The potential-energy function of this system of particles is invariant under space translations and, consequently, we should remove the free motion of the center of mass before applying any approximate method to the Schrödinger equation [7,8] (and references therein). However, this issue is not relevant to present discussion because we do not solve any equation here.
As an example, consider the Hamiltonian operator for the hydrogen atom [5,6] where m = m e m n / (m e + m n ) is the reduced mass of the system, m n the nuclear mass and r the distance between the nucleus and the electron. The reduced mass appears when we remove the motion of the center of mass as mentioned above.
In atomic units the Hamiltonian (32) becomes If instead of the unit of length (29) we choose the Hamiltonian operator for hydrogen takes an even simpler form In the case of atoms it is common usage to resort to the so called clampednucleus approximation, which for hydrogen can be expressed as follows: Within this approximation it is not necessary to remove the motion of the center of mass because it is located at the nucleus that remains fixed at origin [5,6].
However, in the case of highly accurate calculations (which may also include relativistic effects) the mass-polarization terms due to the nuclear motion should be taken into consideration.

Perturbation theory
In this section we show that suitable dimensionless Schrödinger equations may facilitate the application of perturbation theory [1]. Since, as already pointed out above, we do not solve the Schrödinger equation in this paper we will not be concerned with the convergence properties of the perturbation series.
The first example is the widely discussed quartic anharmonic oscillator that we write in the following way On applying the strategy outlined in section 2 we have In this case we can try two choices of the length unit L, the first one is the harmonic oscillator length L = h 2 /(mk 2 ) 1/4 that leads tõ If we apply perturbation theory we obtain the λ-power series that is suitable for sufficiently small values of λ. There are several efficient methods for the calculation of the coefficientsẼ (j) in exact analytical form to any desired order j[F01].
A second choice is that leads toH This equation suggests that we can expand the energies as In this case we cannot obtain the expansion coefficientsẽ (j) exactly but the mere knowledge of the existence of this series is useful in the application of some resummation methods [1]. It is worth pointing out that a calculation for a single value of λ is equivalent to an infinite number of calculations based on variations of m, k 2 and k 4 such that k 4 / mk 3 2 1/2 is constant. This fact is an obviously useful advantage of resorting to a dimentionless equation.
Let us now consider a one-dimensional Hamiltonian operator with the potential (13), where V 0 > 0 and f (q) exhibits a minimum at q = 0 such that We assume that f (q) can be expanded in a Taylor series The dimensionless Hamiltonian reads If we choose then the dimensionless Hamiltonian becomes which shows that the energies can be expanded as There are efficient approaches for the exact analytical calculation of the coeffi-cientsẼ (j) [1].
The last example is the Hamiltonian operator for an atom with N electrons and nuclear charge Ze in the clamped-nucleus approximation where r i is the distance between the electron i and the nucleus and r ij the distance between a pair of electrons.
In this case we choose and obtainH It is clear that the result of the application of perturbation theory as indicated above will be a series of the form Since the perturbation coefficientsẼ (j) only depend on N , then if the number of electrons remains constant we conclude that the rate of convergence of this perturbation series will improve with Z.
This paper shows the advantages of using dimensionless equations in non-relativistic quantum mechanics. The dimensionless Schrödinger equation is simpler than the original one which facilitates the process of obtaining the desired solutions.
If one has to resort to a numerical method the dimensionless equation (with all its quantities of the order of unity) is considerably less prone to round-off errors. When we derive a dimensionless equation we realize which are the relevant parameters that should appear in the solution beforehand. In the case of perturbation theory we can predict the general form of the solution and obtain a suitable perturbation parameter. We believe that it is worthwhile to teach this approach in undergraduate as well as graduate courses on quantum mechanics.
It is worth adding that dimensionless equations are also useful in other areas of physics, such as, for example, classical physics [9,10].