Thermodynamics of small systems is of contemporary interest, ranging from (bio)molecular and nano-scales to even scales of only a few particles. From the viewpoint of energetics, one of primary issues is to understand how the works are extracted from such systems. Just a little more than a decade ago, a pure-state quantum-mechanical analog of the Carnot engine reversibly operating at vanishing temperatures has been proposed in [

1]. It is the smallest engine consisting only of a single quantum particle confined in the one-dimensional infinite square-well potential. It has been shown that the work can be extracted through the controls of the quantum states of the particle and the width of the well in a specific manner. The “efficiency” of the engine has been calculated to be:

where

${E}_{\text{\hspace{0.05em}}H}$ (

${E}_{\text{\hspace{0.05em}}L}$) is the value of the system energy fixed along the analog of the isothermal process at high (low) “temperature” of the cycle. (Since this engine operates without finite-temperature heat baths, it should not be confused with quantum heat engines discussed [

2,

3,

4,

5,

6,

7]). It is of physical interest to examine how universal the efficiency of this form is.

To understand the physics behind the mechanism of such a Carnot-like engine, it is useful to note a structural similarity between quantum mechanics and thermodynamics. Let

H and

$|\psi \rangle $ be the Hamiltonian and quantum state of the system under consideration, respectively. An analog of the internal energy in thermodynamics is the expectation value of the Hamiltonian:

This is in fact the ordinary thermodynamic internal energy in the vanishing-temperature limit, if

$|\psi \rangle $ is chosen to be the ground state. Under changes of both the Hamiltonian and the state along a certain “process”, it varies as

$\delta E=\left(\delta \langle \psi |\right)H|\psi \rangle +\langle \psi |\delta H|\psi \rangle \langle \psi |H\left(\delta |\psi \rangle \right)$. This has a formal analogy with the first law of thermodynamics [

8]:

where

$\left(\delta \langle \psi |\right)H|\psi \rangle +\langle \psi |H\left(\delta |\psi \rangle \right)$ and

$\langle \psi |\delta H|\psi \rangle $ are identified with the analogs of the changes of the quantity of heat,

$\delta \prime Q$, and the work,

$-\delta \prime W$, respectively.

H depends on the system volume,

V, which changes in time very slowly in an equilibrium-thermodynamics-like situation. More precisely, the time scale of the change of

V is much larger than that of the dynamical one,

$~\hslash /E$. Then, in the adiabatic approximation [

9], the instantaneous Schrödinger equation,

$H(V)|{u}_{\text{\hspace{0.05em}}n}(V)\rangle ={E}_{\text{\hspace{0.05em}}n}(V)|{u}_{\text{\hspace{0.05em}}n}(V)\rangle $, holds, provided that the energy eigenvalues naturally satisfy the inequality

${E}_{\text{\hspace{0.05em}}n}({V}_{\text{\hspace{0.05em}}1})>{E}_{\text{\hspace{0.05em}}n}({V}_{\text{\hspace{0.05em}}2})$ for

${V}_{\text{\hspace{0.05em}}1}<{V}_{\text{\hspace{0.05em}}2}$. Assuming that

${\left\{|{u}_{\text{\hspace{0.05em}}n}(V)\rangle \right\}}_{\text{\hspace{0.05em}}n}$ forms a complete orthonormal system, an arbitrary state

$|\psi \rangle $ is expanded as

$|\psi \rangle ={\displaystyle {\sum}_{\text{\hspace{0.05em}}n}{c}_{\text{\hspace{0.05em}}n}(V)|{u}_{\text{\hspace{0.05em}}n}(V)\rangle}$, where the expansion coefficients satisfy the normalization condition,

${{\displaystyle {\sum}_{\text{\hspace{0.05em}}n}\left|{c}_{\text{\hspace{0.05em}}n}(V)\right|}}^{\text{\hspace{0.05em}}2}=1$. Accordingly, the adiabatic scheme allows us to write:

It has been shown in [

8] that imposition of the Clausius equality on the Shannon entropy (not the von Neumann entropy) and the quantity of heat makes pure-state quantum mechanics transmute into equilibrium thermodynamics at finite temperature.

Here, the following question is posed. An analog of the working material in thermodynamics is the shape of the potential that confines a particle. Then, is the efficiency in Equation (1) universal independently of the potential?

In this paper, we answer this question by generalizing the quantum-mechanical Carnot-like engine with the one-dimensional infinite square-well potential to the case of an arbitrary confining potential. We present the most general formula for the efficiency. In marked contrast to the genuine thermodynamic Carnot engine, the efficiency of the quantum-mechanical Carnot engine depends generically on the shape of a potential as the analog of the working material, implying that the engine is not universal. This non-universality is due to the absence of an analog of the second law of thermodynamics in pure-state quantum mechanics, where the von Neumann entropy identically vanishes. We also identify a class of spectra that yields the efficiency of the form in Equation (1).