Unitarity and Conservation of Probability

Closed-system quantum time evolution is unitary. This means that the map from one time to another preserves inner products, norms, and total probability. In the dynamics volume, unitarity is the structural reason a state can change in time without probability being created or destroyed.

The statement is global and representation-independent. In wave mechanics it appears locally as a continuity equation for probability density and current. In density-operator language it appears as trace preservation under $\rho\mapsto U\rho U^\dagger$. In open-system language, unitarity generally holds only for the larger system plus environment, not for the reduced subsystem alone.

Unitary Operators

An operator $U$ is unitary when

$$U^\dagger U = UU^\dagger = I.$$

It preserves inner products:

$$\langle U\phi\rvert U\psi\rangle = \langle\phi\rvert U^\dagger U\lvert\psi\rangle = \langle\phi\rvert\psi\rangle.$$

Taking $\phi=\psi$ gives norm preservation:

$$\lVert U\psi\rVert^2 = \lVert\psi\rVert^2.$$

Since probabilities are computed from squared norms and Born-rule inner products, unitary evolution preserves the total probability carried by a normalized closed-system state.

From Self-Adjoint Hamiltonians to Unitary Evolution

Let two state vectors obey the same Schrödinger equation,

$$i\hbar\frac{d}{dt}\lvert\psi(t)\rangle = H(t)\lvert\psi(t)\rangle, \qquad i\hbar\frac{d}{dt}\lvert\phi(t)\rangle = H(t)\lvert\phi(t)\rangle.$$

If $H(t)$ is self-adjoint, then

$$\frac{d}{dt} \langle\phi(t)\rvert\psi(t)\rangle = \frac{i}{\hbar} \langle\phi(t)\rvert H(t)\lvert\psi(t)\rangle - \frac{i}{\hbar} \langle\phi(t)\rvert H(t)\lvert\psi(t)\rangle =0.$$

Thus inner products are constant in time. Norm preservation is the special case $\phi=\psi$.

The same result can be written directly for the time-evolution operator. If

$$i\hbar\frac{\partial}{\partial t}U(t,t_0) = H(t)U(t,t_0), \qquad U(t_0,t_0)=I,$$

then self-adjointness of $H(t)$ implies

$$\frac{d}{dt} \left[ U^\dagger(t,t_0)U(t,t_0) \right] =0.$$

The initial condition gives $U^\dagger(t_0,t_0)U(t_0,t_0)=I$, so

$$U^\dagger(t,t_0)U(t,t_0)=I$$

throughout the evolution.

What Probability Conservation Means

For a normalized state and a projective measurement with projectors $\{P_a\}$,

$$p(a;t) = \langle\psi(t)\rvert P_a\lvert\psi(t)\rangle, \qquad \sum_a P_a=I.$$

The probabilities can change with time if the state rotates relative to the measurement projectors. But their sum remains

$$\sum_a p(a;t) = \langle\psi(t)\rvert\psi(t)\rangle =1.$$

So unitarity does not say that every measurement probability is constant. It says that the state remains normalized and that the transformation between times is reversible within the closed-system Hilbert space.

If two states are evolved by the same unitary operator, their transition probability is preserved:

$$\left\lvert \langle\phi(t)\rvert\psi(t)\rangle \right\rvert^2 = \left\lvert \langle\phi(t_0)\rvert\psi(t_0)\rangle \right\rvert^2.$$

This is why unitary evolution preserves distinguishability relations between closed-system pure states.

Wave-Mechanics Form: Continuity Equation

For a single nonrelativistic particle with wavefunction $\psi(\mathbf r,t)$, define

$$\rho(\mathbf r,t) = \lvert\psi(\mathbf r,t)\rvert^2.$$

For the usual Hamiltonian with a real scalar potential, the probability current is

$$\mathbf j = \frac{\hbar}{2mi} \left( \psi^*\nabla\psi - \psi\nabla\psi^* \right),$$

and the continuity equation is

$$\frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf j =0.$$

This is the local version of probability conservation: probability in a region can change only by flowing through its boundary. Integrating over a region $\Omega$ gives

$$\frac{d}{dt} \int_\Omega \rho\,d^3r = - \int_{\partial\Omega} \mathbf j\cdot d\mathbf S.$$

If the region is all space and the boundary flux vanishes, then the total norm is conserved:

$$\frac{d}{dt} \int_{\mathbb R^3} \lvert\psi(\mathbf r,t)\rvert^2\,d^3r =0.$$

The detailed current formulas, boundary conditions, and scattering applications belong to Probability Current and Continuity Equation.

Boundary Conditions and Domains

In finite-dimensional examples, “Hermitian Hamiltonian” and “self-adjoint Hamiltonian” are often used interchangeably. In infinite-dimensional wave mechanics, the domain matters. A differential expression such as

$$- \frac{\hbar^2}{2m} \frac{d^2}{dx^2}$$

does not define a complete quantum Hamiltonian until its domain and boundary conditions are specified.

The probability-conservation test is practical: acceptable closed-system boundary conditions make the Hamiltonian self-adjoint and make the net boundary flux consistent with unitary evolution. Infinite-wall, periodic, and suitable self-adjoint boundary conditions achieve this in different ways.

Failure Modes and Effective Descriptions

If the Hamiltonian used for the explicit degrees of freedom is not self-adjoint, norm need not be conserved. A common effective form is

$$H_{\mathrm{eff}} = H-\frac{i}{2}\Gamma, \qquad H=H^\dagger, \qquad \Gamma=\Gamma^\dagger.$$

Then

$$\frac{d}{dt} \lVert\psi(t)\rVert^2 = - \frac{1}{\hbar} \langle\psi(t)\rvert\Gamma\lvert\psi(t)\rangle.$$

If $\Gamma$ is positive semidefinite, the norm decreases. This can be useful for absorption, loss, no-jump conditional evolution, or outgoing boundary conditions, but it is not closed-system unitary evolution on the retained Hilbert space.

Other nonunitary descriptions include selective measurement updates, reduced dynamics after tracing out an environment, Lindblad master equations, and numerical time-steppers that do not preserve unitarity. Some of these are physically correct in their own domains; the point is that their probability bookkeeping is different. For reduced dynamics, the density-operator trace may be preserved even though the subsystem map is not unitary.

Example: Spin Precession

For

$$H = \frac{\hbar\omega}{2}\sigma_z,$$

the time-evolution operator is

$$U(t) = \begin{pmatrix} e^{-i\omega t/2} & 0\\ 0 & e^{i\omega t/2} \end{pmatrix}.$$

For an initial state

$$\lvert\psi(0)\rangle = \alpha\lvert+\rangle + \beta\lvert-\rangle, \qquad \lvert\alpha\rvert^2+\lvert\beta\rvert^2=1,$$

the evolved state is

$$\lvert\psi(t)\rangle = \alpha e^{-i\omega t/2}\lvert+\rangle + \beta e^{i\omega t/2}\lvert-\rangle.$$

The norm is still

$$\lvert\alpha\rvert^2+\lvert\beta\rvert^2=1.$$

However, measurement probabilities in a different basis, such as the $\sigma_x$ basis, can oscillate because the relative phase changes. This is the typical pattern: total probability is conserved while particular measurement probabilities can evolve.

Common Mistakes

• Thinking unitarity means every observable or measurement probability is conserved.
• Proving norm preservation but forgetting that inner products are also preserved.
• Treating a merely symmetric differential expression as a self-adjoint Hamiltonian without checking boundary conditions.
• Calling nonunitary effective evolution “fundamental loss of probability” when untracked degrees of freedom or conditioning are present.
• Using a numerically convenient time stepper and mistaking norm drift for physical dynamics.
• Assuming open-system trace preservation is the same thing as unitary evolution on the subsystem.

Cross-Links

• Unitary Time Evolution
• Time-Evolution Operator
• Hamiltonians as Generators
• Liouville–von Neumann Equation
• Continuity Equation
• Probability Current
• Reduced Dynamics
• Lindblad–GKSL Equation

References

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.

Exercises

Inner products

Show directly from the Schrödinger equation that two states evolved by the same self-adjoint Hamiltonian keep a constant inner product.

Solution

Use

$$\frac{d}{dt}\lvert\psi(t)\rangle = - \frac{i}{\hbar}H(t)\lvert\psi(t)\rangle$$

and

$$\frac{d}{dt}\langle\phi(t)\rvert = \frac{i}{\hbar}\langle\phi(t)\rvert H(t).$$

Then

$$\frac{d}{dt} \langle\phi(t)\rvert\psi(t)\rangle = \frac{i}{\hbar} \langle\phi(t)\rvert H(t)\lvert\psi(t)\rangle - \frac{i}{\hbar} \langle\phi(t)\rvert H(t)\lvert\psi(t)\rangle =0.$$

From local to global conservation

Assume $\partial_t\rho+\nabla\cdot\mathbf j=0$ and that the surface flux at infinity vanishes. Show that total probability is conserved.

Solution

Integrate over all space:

$$\frac{d}{dt} \int_{\mathbb R^3}\rho\,d^3r = - \int_{\mathbb R^3} \nabla\cdot\mathbf j\,d^3r.$$

Using the divergence theorem, the right side becomes a surface integral at infinity:

$$- \int_{\partial\mathbb R^3} \mathbf j\cdot d\mathbf S.$$

If that flux vanishes, then

$$\frac{d}{dt} \int_{\mathbb R^3}\rho\,d^3r =0.$$

Effective loss

Let $H_{\mathrm{eff}}=H-i\Gamma/2$ with $H=H^\dagger$ and $\Gamma=\Gamma^\dagger$. Derive the norm-change formula.

Solution

The evolution equation gives

$$\frac{d}{dt}\lvert\psi\rangle = - \frac{i}{\hbar}H_{\mathrm{eff}}\lvert\psi\rangle.$$

Its adjoint gives

$$\frac{d}{dt}\langle\psi\rvert = \frac{i}{\hbar}\langle\psi\rvert H_{\mathrm{eff}}^\dagger.$$

Therefore

$$\frac{d}{dt} \langle\psi\rvert\psi\rangle = \frac{i}{\hbar} \langle H_{\mathrm{eff}}^\dagger-H_{\mathrm{eff}}\rangle.$$

Since

$$H_{\mathrm{eff}}^\dagger-H_{\mathrm{eff}} = i\Gamma,$$

one obtains

$$\frac{d}{dt} \lVert\psi\rVert^2 = - \frac{1}{\hbar} \langle\psi\rvert\Gamma\lvert\psi\rangle.$$