Energy, Heat, and Work

Energy at a time is represented by a Hamiltonian. Heat and work are different: they describe how energy is transferred during a physical process. In small quantum systems this distinction is not a semantic nicety. Measurements can disturb the state, reservoirs can be structured, interaction energy can be non-negligible, and a “work distribution” usually refers to a protocol rather than to a single Hermitian operator.

This page fixes the standard weak-coupling bookkeeping convention used across this chapter. It is the local canonical home for the definitions of internal energy, heat, work, heat currents, and power in reduced open-system models. Driven Many-Body Systems applies that ledger to extensive absorption, symmetry-constrained heating, and periodic many-body regimes. Fluctuation relations and entropy-production inequalities have their own canonical pages.

Basic Convention

For a system described by a density operator $\rho(t)$ and a chosen system Hamiltonian $H(t)$, the internal energy is

$$E(t) = \operatorname{Tr}\!\left[ \rho(t)H(t) \right].$$

If $\rho(t)$ and $H(t)$ are differentiable, then

$$\frac{dE}{dt} = \operatorname{Tr}\!\left[ \dot\rho(t)H(t) \right] + \operatorname{Tr}\!\left[ \rho(t)\dot H(t) \right].$$

The standard weak-coupling sign convention in this volume is:

$$\dot Q = \operatorname{Tr}\!\left[ \dot\rho(t)H(t) \right], \qquad \dot W = \operatorname{Tr}\!\left[ \rho(t)\dot H(t) \right].$$

Thus

$$\frac{dE}{dt} = \dot Q+\dot W.$$

Here $\dot Q$ is heat current into the system and $\dot W$ is power supplied to the system by externally changing the Hamiltonian. With the opposite sign convention for work, some thermodynamics books write power delivered by the system as $-\dot W$.

Quantity	Meaning in This Convention	Typical Source
$E=\operatorname{Tr}(\rho H)$	internal energy assigned to the chosen reduced system	state and system Hamiltonian
$\dot Q=\operatorname{Tr}(\dot\rho H)$	energy change from state evolution at fixed Hamiltonian	reservoirs, measurements, coarse-grained dissipation
$\dot W=\operatorname{Tr}(\rho\dot H)$	energy change from explicit Hamiltonian control	external drive, changing fields, moving boundaries

The split is a convention with physical assumptions. It is most transparent when the Hamiltonian change is controlled by a macroscopic drive and the state change at fixed Hamiltonian is caused by a reservoir.

What Is Being Assumed

The formulas above are exact as an identity for the selected pair $(\rho,H)$, but their thermodynamic interpretation assumes more:

• The system Hamiltonian is the energy observable whose expectation is being tracked.
• The external drive appears as explicit time dependence in $H(t)$.
• Dissipative state changes are generated by modeled reservoirs or measurement apparatuses.
• Interaction energy with untracked degrees of freedom is either negligible, included consistently in $H(t)$, or accounted for separately.
• The zero of energy and the frame used for $H(t)$ have been fixed.

These assumptions are usually appropriate for weak system–bath coupling and a controlled Hamiltonian drive. They can fail for strong coupling, finite reservoirs, non-Markovian dynamics, time-dependent rotating frames, or ambiguous system–environment partitions. For the microscopic starting point, see System–Bath Hamiltonians.

Energy Is an Observable, Work Is a Process Quantity

At a fixed time, energy measurements are described by the spectral projectors of $H(t)$. If

$$H(t) = \sum_n E_n(t)\Pi_n(t),$$

then an ideal energy measurement has outcomes $E_n(t)$ with probabilities $\operatorname{Tr}[\Pi_n(t)\rho(t)]$.

Work is not usually represented by a single observable measured at one time. It is energy transferred by a process. For a closed driven protocol from $0$ to $\tau$, one often compares an initial energy measurement of $H(0)$ with a final energy measurement of $H(\tau)$. That two-point measurement scheme defines random values

$$W_{m n} = E_m^\tau-E_n^0,$$

but it also defines a measurement protocol with backaction. If the initial state has coherence between energy eigenspaces, the initial energy measurement changes the state used in the distribution. For this reason the slogan “work is not an observable” means:

• there is no universal Hermitian operator $W$ whose spectral measure gives work for arbitrary driven quantum processes;
• work distributions depend on the operational protocol used to define them;
• the average energy change can be meaningful even when no unique work distribution has been specified.

For fluctuation relations based on two energy measurements, see Fluctuation Theorems.

Closed Driven Systems

For a closed system with unitary dynamics,

$$\dot\rho = - \frac{i}{\hbar}[H(t),\rho],$$

the Hamiltonian part does not change the expectation value of $H(t)$ at fixed time:

$$\operatorname{Tr}\!\left[ H(t)\dot\rho(t) \right] = - \frac{i}{\hbar} \operatorname{Tr}\!\left[ H(t)[H(t),\rho(t)] \right] = 0.$$

Thus, in the weak-coupling convention,

$$\dot Q=0, \qquad \frac{dE}{dt}=\dot W.$$

A closed system can still absorb or release energy when $H(t)$ is explicitly changed. The energy transfer is work because it is caused by a controlled parameter, not by exchange with an untracked thermal reservoir.

If the Hamiltonian is time-independent, closed unitary dynamics conserves $E$. If $H(t)$ changes cyclically so that $H(\tau)=H(0)$, the net work need not vanish: the drive may leave the state in a different energy distribution.

Open Systems and Heat Currents

For a Markovian master equation

$$\dot\rho = - \frac{i}{\hbar}[H(t),\rho] + \sum_\alpha \mathcal L_\alpha(\rho),$$

the heat current into the system from channel or reservoir $\alpha$ is commonly defined as

$$\dot Q_\alpha = \operatorname{Tr}\!\left[ H(t)\mathcal L_\alpha(\rho) \right],$$

so that

$$\frac{dE}{dt} = \operatorname{Tr}\!\left[ \rho\dot H \right] + \sum_\alpha \operatorname{Tr}\!\left[ H\mathcal L_\alpha(\rho) \right].$$

This expression is thermodynamic only if $\mathcal L_\alpha$ really represents a reservoir or monitored channel whose energy exchange is being modeled. A Lindblad equation is mathematically trace preserving and completely positive, but it is not automatically a thermal master equation.

For a thermal reservoir at inverse temperature $\beta_\alpha$, transition rates should obey detailed-balance constraints with respect to the Hamiltonian whose energy is being counted. See Thermal Master Equations and Detailed Balance.

Qubit Example

Consider a two-level system with ground energy $0$ and excited energy $\hbar\omega(t)$:

$$H(t) = \hbar\omega(t) |e\rangle\langle e|.$$

Let

$$p_e(t) = \langle e|\rho(t)|e\rangle.$$

Then the internal energy is

$$E(t) = \hbar\omega(t)p_e(t).$$

Differentiating gives

$$\frac{dE}{dt} = \hbar\omega(t)\dot p_e(t) + \hbar\dot\omega(t)p_e(t).$$

In the standard convention,

$$\dot Q = \hbar\omega(t)\dot p_e(t), \qquad \dot W = \hbar\dot\omega(t)p_e(t).$$

The interpretation is direct:

• changing the excited-state population at fixed gap exchanges heat with whatever caused the transition;
• changing the level spacing at fixed population performs work through the external control of $\omega(t)$.

For a thermal amplitude-damping model,

$$\dot p_e = \gamma_\uparrow p_g - \gamma_\downarrow p_e, \qquad p_g=1-p_e,$$

the heat current is

$$\dot Q = \hbar\omega \left( \gamma_\uparrow p_g - \gamma_\downarrow p_e \right).$$

Absorption from the bath gives positive heat into the system. Spontaneous or stimulated emission into the bath gives negative heat into the system.

Strong Coupling and Interaction Energy

For a closed system plus bath,

$$H_{\mathrm{tot}} = H_S+H_B+H_I,$$

the total energy is unambiguous once $H_{\mathrm{tot}}$ is specified. The reduced system energy $E_S=\operatorname{Tr}_S(\rho_S H_S)$ is less automatic because part of the energy can sit in the interaction term $H_I$.

When $H_I$ is weak and the bath is large, it is often accurate to neglect interaction energy in the reduced energy balance. When $H_I$ is not negligible, several inequivalent conventions appear:

• assign $H_I$ to neither side and track it separately;
• include a fraction of $H_I$ in an effective system Hamiltonian;
• use a Hamiltonian of mean force for equilibrium thermodynamics;
• enlarge the “system” to include strongly coupled environmental modes.

These conventions can give different heat and work assignments while agreeing on the total energy budget. A mature calculation states which convention is used and why it is appropriate for the model.

Time-Dependent Energy Offsets and Frames

Adding a scalar offset changes the bookkeeping:

$$H'(t) = H(t)+c(t)I.$$

Then

$$E'(t) = E(t)+c(t), \qquad \dot W' = \dot W+\dot c(t),$$

while transition frequencies and closed-system state evolution are unchanged up to a global phase. This shows why the zero of energy must be fixed when quoting work. A time-dependent scalar offset can look like power in the bookkeeping even when it has no observable effect on the isolated system.

Changing rotating frames can introduce similar hazards. The Hamiltonian in a rotating frame may be the generator of transformed dynamics, not the laboratory energy observable. Thermodynamic heat and work should be assigned using the energy operator appropriate to the physical energy budget, not merely whatever operator appears in the most convenient equation of motion.

Measurements, Information, and Energy

Measurements can change a system’s energy. Whether that energy change is called heat, work, or measurement backaction depends on the physical implementation.

Examples:

• A projective energy measurement of a nondegenerate Hamiltonian changes no energy statistics if the state is already diagonal in the energy basis.
• A projective measurement in a non-energy basis can inject or remove energy because it changes the state at fixed Hamiltonian.
• Continuous monitoring can add diffusion or jumps to the conditional state; the detector and any feedback controller must be included in the energy budget.
• Feedback can extract work only when the measurement record, memory, controller, and erasure costs are treated consistently.

The safest language is operational: specify the apparatus, the record, and the external controller. Then assign energy flows according to that enlarged model. For monitored dynamics, see Quantum-Jump Trajectories and Stochastic Master Equations.

Heat Is Not Entropy

Heat and entropy are related but not identical. Heat is energy transfer. Entropy is a state function, such as the von Neumann entropy

$$S(\rho) = -k_B\operatorname{Tr}(\rho\ln\rho).$$

For an ideal thermal reservoir at temperature $T_\alpha$, heat current into the system contributes reservoir entropy change

$$\frac{dS_{\alpha}}{dt} = - \frac{\dot Q_\alpha}{T_\alpha}.$$

The system entropy production rate is then typically organized as

$$\dot\Sigma = \frac{dS}{dt} - \sum_\alpha \frac{\dot Q_\alpha}{T_\alpha},$$

with $\dot Q_\alpha$ defined as heat into the system. The inequality $\dot\Sigma\ge0$ requires a thermodynamically consistent model, not merely any equation that decreases coherence. See Entropy Production.

Common Mistakes

1. Treating work as a universal Hermitian operator. Energy is an observable at a time; work is assigned to a process.

2. Calling every change in $\rho$ heat. A measurement, a noisy classical drive, or coarse graining can change $\rho$ without being a thermal reservoir.

3. Using a local dissipator as a heat bath without checking detailed balance for the interacting Hamiltonian.

4. Ignoring interaction energy at strong coupling.

5. Forgetting sign conventions. In this volume, positive $\dot Q$ and positive $\dot W$ both increase the system energy.

6. Quoting a work distribution without specifying the measurement protocol.

7. Confusing decoherence with thermalization. Dephasing can suppress off-diagonal elements without exchanging energy.

Minimal Checklist

Before using heat or work in a calculation, record:

• the system Hamiltonian used in $E=\operatorname{Tr}(\rho H)$;
• the sign convention for heat and work;
• which parameters in $H(t)$ are externally controlled;
• which terms in $\dot\rho$ are assigned to which reservoirs or detectors;
• whether coupling energy is negligible, tracked, or absorbed into an effective Hamiltonian;
• whether the master equation has a thermal stationary state when it is supposed to describe a thermal bath;
• whether a claimed work distribution comes from two projective measurements, full counting statistics, weak measurements, trajectories, or another protocol.

For weak-coupling assumptions, use the Approximation Checklist.

Exercises

Driven Level Spacing

A two-level system has $H(t)=\hbar\omega(t)|e\rangle\langle e|$ and excited-state population $p_e(t)$. Show that changing $\omega$ at fixed $p_e$ is work in the convention of this page, while changing $p_e$ at fixed $\omega$ is heat.

Solution

The internal energy is

$$E(t)=\hbar\omega(t)p_e(t).$$

Differentiation gives

$$dE = \hbar p_e\,d\omega + \hbar\omega\,dp_e.$$

At fixed population, $dp_e=0$, so

$$dE=\hbar p_e\,d\omega.$$

This energy change comes from moving the level spacing through an external control parameter and is assigned to work. At fixed level spacing, $d\omega=0$, so

$$dE=\hbar\omega\,dp_e.$$

This energy change comes from changing the state populations at fixed Hamiltonian and is assigned to heat if the population change is caused by a reservoir.

Thermal Qubit Current

For the qubit master equation

$$\dot p_e = \gamma_\uparrow p_g-\gamma_\downarrow p_e, \qquad p_g=1-p_e,$$

with $H=\hbar\omega|e\rangle\langle e|$, compute the heat current into the system. What condition makes the stationary heat current vanish?

Solution

The heat current into the system is

$$\dot Q = \operatorname{Tr}(H\dot\rho) = \hbar\omega\dot p_e.$$

Using the rate equation,

$$\dot Q = \hbar\omega \left( \gamma_\uparrow p_g - \gamma_\downarrow p_e \right).$$

At stationarity, $\dot p_e=0$, so

$$\gamma_\uparrow p_g^{\mathrm{ss}} = \gamma_\downarrow p_e^{\mathrm{ss}}.$$

The stationary heat current then vanishes. For a thermal bath, detailed balance gives

$$\frac{\gamma_\uparrow}{\gamma_\downarrow} = e^{-\beta\hbar\omega},$$

so the stationary population ratio is Gibbsian:

$$\frac{p_e^{\mathrm{ss}}}{p_g^{\mathrm{ss}}} = e^{-\beta\hbar\omega}.$$

Energy Offset

Let $H'(t)=H(t)+c(t)I$. Show how $E$ and $\dot W$ change under this replacement. Why is this a warning about work bookkeeping?

Solution

Since $\operatorname{Tr}\rho=1$,

$$E' = \operatorname{Tr}[\rho H'] = \operatorname{Tr}[\rho H]+c(t) = E+c(t).$$

The work rate changes as

$$\dot W' = \operatorname{Tr}[\rho\dot H'] = \operatorname{Tr}[\rho\dot H] + \dot c(t)\operatorname{Tr}\rho = \dot W+\dot c(t).$$

If $c(t)I$ has no observable effect except a global phase in an isolated system, then counting $\dot c(t)$ as physical work would be a convention-dependent artifact. A thermodynamic calculation must fix the energy zero and state which Hamiltonian represents the physical energy budget.

References

• R. Alicki, “The quantum open system as a model of the heat engine,” Journal of Physics A 12, L103, 1979.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• M. Esposito, U. Harbola, and S. Mukamel, “Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,” Reviews of Modern Physics 81, 1665, 2009.
• M. Campisi, P. Hänggi, and P. Talkner, “Colloquium: Quantum fluctuation relations: Foundations and applications,” Reviews of Modern Physics 83, 771, 2011.
• P. Talkner, E. Lutz, and P. Hänggi, “Fluctuation theorems: Work is not an observable,” Physical Review E 75, 050102, 2007.
• J. Gemmer, M. Michel, and G. Mahler, Quantum Thermodynamics, Springer, 2nd ed., 2009.
• S. Vinjanampathy and J. Anders, “Quantum thermodynamics,” Contemporary Physics 57, 545, 2016.
• S. Deffner and S. Campbell, Quantum Thermodynamics: An Introduction to the Thermodynamics of Quantum Information, Morgan & Claypool, 2019.