## 2 Modeling
### New Model Based on Quantum Mechanical Formulation  
This section proposes a new numerical model of the IR distribution based on the Schrodinger equation. An IR lamp was combined with an elliptical reflector, as shown in Figure 2; \(f,l\), and \(d\) denote the focal length of the ellipse, entrance length of the reflector, and the height between the target material and top of the ellipse, respectively. In this case, the IR rays are distributed over the entrance length \(l\). This work replaces this condition with the one-dimensional infinite potential well problem, where an imaginary energy particle is located probabilistically [33]. As the heat flux from the lamp does not depend on time, the time-independent Schrodinger equation can be used:  
\[\frac{d^{2}\psi}{dx^{2}}+\frac{2 mE}{\hbar^{2}}\psi(x)=0. \tag{1}\]  
\(m\) is the mass of the particle, \(E\) is the energy, \(\hbar\) represents the reduced Planck's constant, and \(\psi\) denotes the wave function. In this analogy, the energy particles are imaginary. Figure 1: Focused IR heating method: (**a**) Illustration of the focused IR heating; (**b**) Example of a specimen heated by the IR heating [22]. Figure 2: Geometrical parameter of the elliptical reflector. Figure 3 presents the concept drawing of the energy particle in the infinite potential well. The particle cannot exist outside the well; therefore, the boundary condition is given by:  
\[\psi(x)=0\ (x\leq 0\ or\ x\geq a). \tag{2}\]  
where \(a\) is the length of the infinite potential well. The solution of \(\psi(x)\) is well-known [33], given by:  
\[\psi(x)=A\cos kx+B\sin kx,\ \text{where}\ k^{2}=\frac{2 mE}{\hbar^{2}}. \tag{3}\]  
By applying the boundary condition of Equation (2) to Equation (3), the value of \(A\) becomes 0 for \(x=0\), while \(B\sin ka=0\) for \(x=a\). This leads to \(k=\frac{n\pi}{a}\). By applying the parameter \(k\) to Equation (3), the wave function at the \(n\)th eigenstate \(\psi_{n}(x)\) is determined by:  
\[\psi_{n}(x)=B\sin\frac{n\pi x}{a}\ \ (n=1,\ 2,\ 3,\ \cdots). \tag{4}\]  
\(\psi_{n}(x)\) should satisfy the below probability density condition and orthogonal conditions, respectively. \[\left|\psi_{n}\right|^{2}=\int_{-\infty}^{\infty}\psi_{n}^{*}(x)\psi_{n}(x)dx =1\ \ \text{ and }\int_{-\infty}^{\infty}\psi_{n}^{*}(x)\psi_{m}(x)dx=\delta_{nm}. \tag{5}\]  
As \(\psi_{n}(x)\) is a time-independent function in the IR heating condition, \(\psi_{n}^{*}(x)=\psi_{n}(x)\). Combining Equations (4) and (5) determines the value of \(\left(B=\sqrt{\frac{2}{l}}\right)\), then the wave function of the \(n\)th eigenmode is represented by:  
\[\psi_{n}(x)=\sqrt{\frac{2}{l}}\sin\frac{n\pi x}{l}\ \ \ (n=1,\ 2,\ 3,\ \cdots). \tag{6}\]  
where \(a=l\) because the length of the reflector and the length of the potential well are the same in this study. Then the energy of the \(n\)th eigenmode is determined simply by:  
\[E_{n}=\frac{n^{2}\pi^{2}\hbar^{2}}{2\eta l^{2}}\ \ \ (n=1,\ 2,\ 3,\cdots). \tag{7}\]  
Figure 3: Infinite potential well. Next, an arbitrary state of the wave function can be given by the superposition of several eigenmode functions, as follows:  
Figure 1:0  
The coefficients, \(c_{n}\), should satisfy the below condition:  
Figure 1:1  
The \(\left|c_{n}\right|^{2}\) value of each eigenmode in Equation (9) represents the coefficient of each \(\left|\psi_{n}\right|^{2}\). Then, the energy distribution can be given by:  
Figure 1:2  
By using Equation (10), the IR distribution, \(q_{\text{total}}^{{}^{\prime\prime}}(x)\), of Figure 1a can be simply modeled as below:  
Figure 1:3  
Note that \(q_{\text{total}}^{{}^{\prime\prime}}(x)\) is the magnitude of the heat flux incident in the opposite direction of the outward unit normal vector of the heated surface. To use the model in engineering applications, \(E_{n}\) and \(c_{n}\) should be specified using a model calibration process. For the calibration, the integration of the \(q_{\text{total}}^{{}^{\prime\prime}}(x)\) should be the same as the power of the lamp \((I_{0})\).