Before giving a brief discussion of the solution for the space-fractional Schrödinger equation, we would like to remind that the space or time-fractional processes can be caused by spatial deformation, entropic restrictions, or deformed potentials. It is well known that the differential equations which correspond to these processes can be represented with a non-integer derivative value25,26,27,28,29,30,31,32.
It is known that a discrete stochastic dynamics is categorized by the finite characteristic waiting time (T=int _{0}^{infty } t psi (t) dt) and the finite jump length variance (Sigma ^{2} =int _{-infty }^{infty } x^{2} lambda (x) dx) where (psi (t)) and (lambda (x)) respectively denote the waiting time and jump length probability distributions. For instance, for a Markovian process, (psi (t)) is of Poisson form and (lambda (x)) has Gaussian form. However, for the non-Markovian process, the waiting time T diverges, and the jump length variance (Sigma ^{2}) is still kept finite. In a such process, the long-tailed waiting time probability distribution takes an asymptotic form and which leads to the time-fractional equation in the continuum limit30. Conversely, in the other case, it leads to the space-fractional equation in the continuum limit. As a result, space and time exponents are represented by (alpha) and (beta), respectively. The space-fractional exponent is between (0<alpha le 2), however, the time-fractional exponent is between (0<beta le 1).
In a physical process, the value of these exponents which affect the solution of the differential equation can be controlled by changing the effects of the spatial deformation, entropic restrictions, or deformed potentials.
Now, to proceed discussion, we consider a one-dimensional space-fractional Schrödinger equation for a single particle in a box of length 2a under the infinite well potential. This problem was discussed by Laskin in several papers32,33,34. Here we briefly review the fractional Schrödinger equation and its results for infinite well potential. In the Refs. 32,33,34,35, the Hamiltonian of the system is given by
$$begin{aligned} H_{alpha } psi _{n}(x) = E_{alpha n} psi _{n},(x) quad n = 0,1,2,… end{aligned}$$
(1)
where (alpha) denotes the fractional index (0 < alpha le 2). For the infinite square well with a length of 2a, the space-fractional Schrödinger equation is given by
$$begin{aligned} D_{alpha } left( – hbar ^{2} frac{d^{2}}{dx^{2}} right) ^{alpha /2} psi _{n}(x) + V psi _{n}(x) = E_{alpha } psi _{n}(x) end{aligned}$$
(2)
where the coefficient (D_alpha = chi m c^{2} / (mc)^{alpha }) with (chi) a positive real number, and c is the speed of the light. When (alpha =2), taking (chi =1/2) and hence (D=1/(2m))35. The first term in Eq. (2) corresponds to the fractional kinetic energy
$$begin{aligned} T_{alpha } = D_{alpha } |p|^{alpha } =frac{1}{2} m c^{2} left( frac{ |p| }{m c} right) ^{alpha } = D_alpha left( – hbar ^{2} frac{d^{2}}{dx^{2}} right) ^{alpha /2} . end{aligned}$$
(3)
On the other hand, the potential for infinite well is defined as
$$begin{aligned} V = left{ begin{array}{ll} 0 &{} hbox { if} |x| < a \ +infty &{} hbox { if} |x| > a end{array} right. end{aligned}$$
(4)
the potential takes infinite value at the (x=pm a). Finally, the solutions of the fractional Schrödinger equation are given by
$$begin{aligned} psi _{n} (x) = left{ begin{array}{ll} frac{1}{sqrt{2}}sin frac{npi }{2a}x + a&{} hbox { if} |x| < a \ 0 &{} hbox { if} |x| > a end{array} right. end{aligned}$$
(5)
and
$$begin{aligned} E_{alpha n} = D_{alpha } left( frac{n pi hbar }{2 a} right) ^{alpha } end{aligned}$$
(6)
where (psi _{n} (x)) denotes the wave function solution and (E_{alpha n}) is the energy eigenvalues of Eq. (2). Here we note that some authors claim that the solution does not satisfy the ground state solution of the space-fractional Schrödinger equation (2). However, it is shown that this problem can be solved by assuming that the solution is the limit of the finite square well problem35. Thus, the solution
$$begin{aligned} lim _{V_{0}rightarrow infty } psi _{n}^{finite} (x) = psi _{n} (x) qquad lim _{V_{0}rightarrow infty } E_{alpha n}^{finite} = E_{alpha n} end{aligned}$$
(7)
satisfies the ground state requirement of the infinite square well35.
To carry out the work and efficiency of the fractional quantum Szilard like-engine we need the canonical partition function and heat exchanges for all stages. Therefore, in the next, we discuss these quantities for the infinite well potential given in Eq. (4) following the method given in Ref. 23.
