## 1. Introduction

## 2. Derivation of the General Formula for the Efficiency

**Figure 1.**The quantum-mechanical analog of the Carnot cycle depicted in the plane of the volume V and pressure P. During $A\to B$ and $C\to D$, the expectation values of the Hamiltonian are fixed, whereas during $B\to C$ and $D\to A$, the quantum states are kept unchanged.

## 3. Potential Dependence of the Efficiency: Examples

## 4. Conclusion

## Acknowledgment

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