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. A state can therefore change in time without total 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,

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

Assume the Schrödinger initial-value problem has solutions for the interval in both directions, and that the following derivatives are valid on their common domain. For a finite-dimensional piecewise-continuous Hermitian matrix these assumptions hold. 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. This establishes an isometry. To conclude unitarity in an infinite-dimensional space one also needs surjectivity; the backward propagator supplies the inverse and gives $UU^\dagger=I$. Norm preservation alone does not exclude a nonsurjective isometry.

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 are developed in 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. These descriptions can preserve total probability while redistributing it among measurement outcomes or unobserved degrees of freedom. 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.

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• 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.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.

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 $\Gamma=\Gamma^\dagger$. Here $\Gamma$ has units of energy; a decay-rate operator is $\Gamma/\hbar$. 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

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