The glucose solution was prepared using a Sigma Aldrich reagent with a purity of + 99% (molecular weight of 180.16 g/mol) dissolved in phosphate buffer at pH 7.4. Glucose determination was obtained by UV-visible spectrophotometric techniques using a Perkin Elmer Lambda XLS + model spectrophotometer, which works in a spectral scanning range from 200 to 950 nm. Measurements were made at 5, 10, 15, 20, 25, and 30 mM glucose, and a blank was used as a reference (Buffer without glucose). A quartz cuvette was used for measurements of all concentrations of interest. Subsequently, a calculation was made using Mirror Boundary Condition MBC in solution of the Schrὃdinger equation to relate the characteristic glucose peaks in the UV region with the fundamental analysis developed in this work.

### Theoretical description of the electronic transitions in a glucose molecule

The glucose molecule C6H12O6 can be considered as a quantum dot (QD) having various geometric shapes (**see** Fig. 2), one of which is nearly linear, others nearly cylindrical (α and β shapes). Glucose is a molecule characterized by the absence of color, so the main electronic energy transitions that define its optical properties most likely correspond to the UV spectral region (195–400 nm).

For the calculation, we use the approximation developed previously in our group of the Mirror Boundary Condition (MBC) [19, 20], where we equate the function Ψ of the electron near the boundary with that of its reflection in the boundary that acts as a mirror. Since the physical meaning is related to the square of the function module, the MBC has two versions depending on the type of boundary: the non-penetrable boundary-mirror (odd MBC or OMBC) when we equate the real function of the electron and that of its reflection with opposite signs (obviously, it gives function value zero at the boundary) and the boundary with possibility of tunneling when the equalization occurs with the same sign (even MBC or EMBC). In the case of glucose in an aqueous medium, we cannot be sure of the impossibility of electrons crossing the border, so both versions are considered.

One of the approximations of a glucose molecule is a cylinder with a diameter *a* = 5.2 Å and height *c* = 7.3 Å. We consider a particle (electron) inside a cylindrical quantum well for the approximation. According to common procedure, we will assume that the longitudinal and transversal motion of the confined particle are independent of each other, allowing us to separate the wave function into:

\(\psi \left(x,y,z\right)=\psi \left(x,y\right)\psi \left(z\right)=\psi (r,\varphi )\psi \left(z\right)\) (1

Under these conditions, the variables can be separated in the time-independent Schrodinger equation. The motion along the z-axis results in the one-dimensional particle in the box problem:

\(\frac{-{\hslash }^{2}}{2m}\frac{{d}^{2}\psi \left(z\right)}{d{z}^{2}}=E\psi \left(z\right)\) (2

The solution for the particle in the box is well-known, resulting in the wave function:

\(\psi \left(z\right)=\sqrt{\frac{2}{c}}\text{sin}\left(\frac{{n}^{2}{\hslash }^{2}}{8m{c}^{2}}\right)\) (3

The quantized energy levels for the z-axis are given by:

\({E}_{n}=\frac{{n}^{2}{\hslash }^{2}}{8m{c}^{2}}\) (4

For the circular cross-section with radius a, the polar wavefunction can be separated into:

\(\psi \left(r,\varphi \right)=AF\left(r\right){e}^{ik\varphi }\) (5

Where \(k\) is the quantized wavenumber associated with the angular momentum \(L=\hslash k\). For the radial component, the time-independent Schrodinger equation satisfies Bessel’s equation:

\({r}^{2}\frac{{d}^{2}F\left(r\right)}{{dr}^{2}}+r\frac{dF\left(r\right)}{dr}+\left({n}^{2}{r}^{2}-{k}^{2}\right)F\left(r\right)=0\) (6

The expression for energy levels in MBC approximation is [19, 20]:

\({E}_{n}=\frac{2{h}^{2}}{{ma}^{2}} {S}_{\lfloor P\rfloor i}^{2}= \frac{{h}^{2}}{{2\pi }^{2}{ma}^{2}} {S}_{\lfloor P\rfloor i}^{2}\) (7

Where \({E}_{n}\) is the quantized energy level, h is Planck’s constant, m is the molecule’s mass, a is the molecular diameter, and c is its height. The parameter **s**|*p*|i takes the q|p|i values for OMBC and *t*|*p*|i for EMBC, which are the nodes and extremes of the Bessel function (See Table 1). Considering quantization along *the z-axis*, one will obtain the total energy.

\(E=\frac{{h}^{2}}{2m} \left(\frac{{S}_{\lfloor P\rfloor i}^{2}}{{\pi }^{2}{a}^{2}}+\frac{{n}_{z}^{2}}{{4c}^{2}}\right)\) (8

Table 1

Argument values at nodes and extremes of cylindrical Bessel function The Even Mirror Boundary Conditions or EMBC establish that a particle may penetrate the barrier, returning to the confined volume. Meanwhile, the Odd Mirror Boundary Condition establishes strong confinement. The calculated ground state and first state are shown in Table 2. For the OMBC, the ground state is established for q|p|1 = 2.4, and the first excited state is for **q****|p|2**=3.7. The energy levels for each state are 3.95 and 8.42 eV, respectively. The first transition for the OMBC between the base and first states is 4.47 eV or 277 nm. Meanwhile, the experimental results show an absorption peak at 263 nm, only a 14 nm deviation from the OMBC model. The next excited state corresponds to the value q|p|2 = 5.1, gives E2 = 15.37 eV, and ΔE = 11.41 eV, wavelength 109 nm. The higher excited states will give transitions in the deep ultraviolet that are out of the limits of our measurements, so we will not consider them. An intertransition between the first and second stages has a ΔE = 6.94 eV, wavelength of 179 nm. For the EMBC, the first transition from the base state to its first level is 3.25 eV or 382 nm. The difference between experimental and theoretical in this model is 119 nm, wider than the OMBC model.

q|p|1 | t|p|1 | q|p|2 | t|p|2 | q|p|3 | t|p|3 | q|p|4 | t|p|4 |

2.4 | 0 | 5.5 | 3.7 | 8.5 | 7.1 | 11.6 | 10.2 |

0 | 1.63 | 3.7 | 5.4 | 6.9 | 8.6 | 10.3 | 11.6 |

0 | 2.9 | 5.1 | 6.8 | 8.4 | 10.0 | 11.7 | 13.2 |

0 | 4.3 | 6.4 | 8.1 | 9.9 | 11.4 | 13.2 | 14.2 |

Table 2

Calculated energy levels and transition for the OMBC and EMBC model

OMBC Model | EMBC Model |

Ground state q|p|1=2.4 | First state q|p|2=3.7 | Ground state t|p|1=1.63 | First state t|p|1=2.9 |

E (eV) | 3.95 | E (eV) | 8.42 | E (eV) | 2.20 | E (eV) | 5.45 |

**ΔE (eV)** | 4.47 | **ΔE (eV)** | 3.25 |

**λ(nm)** | 277 | **λ(nm)** | 382 |

**Experimental (nm)** | 263 | **Experimental (nm)** | 263 |