Spectral Decomposition of the Propagator

For a time-independent Hamiltonian, the propagator is determined by the spectrum of $H$. In the energy basis,

$$U(T)=e^{-iHT/\hbar}$$

is diagonal: each energy component acquires the phase $e^{-iET/\hbar}$. In coordinate space this becomes the spectral decomposition of the propagator kernel.

For a nondegenerate discrete spectrum,

$$K(x_b,T;x_a,0) = \sum_n \psi_n(x_b)\psi_n^*(x_a) e^{-iE_nT/\hbar}.$$

Continuous spectra replace sums by integrals, and mixed spectra contain both. This page is the canonical home for that organization of the propagator.

Spectral Theorem Viewpoint

If $H$ is self-adjoint, the spectral theorem defines functions of $H$. In finite dimensions or a purely discrete setting,

$$H=\sum_n E_nP_n,$$

where $P_n$ projects onto the eigenspace with energy $E_n$. Then

$$U(T)=e^{-iHT/\hbar} = \sum_n e^{-iE_nT/\hbar}P_n.$$

This is the operator-level statement behind the coordinate-space spectral sum. The propagator kernel is just the position-basis matrix element:

$$K(x_b,T;x_a,0) = \langle x_b\rvert U(T)\lvert x_a\rangle.$$

The spectral decomposition answers a different question from the short-time path-integral derivation. The path-integral route builds propagation from many small time steps. The spectral route diagonalizes the Hamiltonian and applies the phase $e^{-iET/\hbar}$ to each energy component.

Discrete Spectra

Suppose the Hamiltonian has a complete orthonormal set of discrete eigenstates:

$$H\lvert n\rangle=E_n\lvert n\rangle, \qquad \langle m\vert n\rangle=\delta_{mn}, \qquad \sum_n\lvert n\rangle\langle n\rvert=I.$$

Then

$$U(T) = \sum_n e^{-iE_nT/\hbar} \lvert n\rangle\langle n\rvert.$$

Taking coordinate matrix elements gives

$$\begin{aligned} K(x_b,T;x_a,0) &= \sum_n e^{-iE_nT/\hbar} \langle x_b\vert n\rangle \langle n\vert x_a\rangle\\ &= \sum_n \psi_n(x_b)\psi_n^*(x_a) e^{-iE_nT/\hbar}. \end{aligned}$$

This expression is a coherent sum of amplitudes. The terms are not probabilities for different energies unless an energy measurement has actually been performed.

Degeneracy

If an energy $E_n$ is degenerate, use a basis $\{\lvert n,\alpha\rangle\}$ for that eigenspace:

$$P_n = \sum_{\alpha} \lvert n,\alpha\rangle\langle n,\alpha\rvert.$$

The propagator contribution of that eigenspace is

$$e^{-iE_nT/\hbar} \langle x_b\rvert P_n\lvert x_a\rangle = e^{-iE_nT/\hbar} \sum_{\alpha} \psi_{n,\alpha}(x_b)\psi_{n,\alpha}^*(x_a).$$

Thus the discrete spectral kernel is more invariantly written as

$$K(x_b,T;x_a,0) = \sum_n e^{-iE_nT/\hbar} P_n(x_b,x_a),$$

where

$$P_n(x_b,x_a) = \langle x_b\rvert P_n\lvert x_a\rangle.$$

This avoids pretending that a particular basis inside a degenerate subspace is physically preferred.

Continuous Spectra

For a continuum, generalized eigenstates are usually delta-normalized. A simple notation is

$$H\lvert E,\alpha\rangle = E\lvert E,\alpha\rangle,$$

with identity resolution

$$I = \int dE \sum_\alpha \lvert E,\alpha\rangle \langle E,\alpha\rvert.$$

Then

$$U(T) = \int dE \sum_\alpha e^{-iET/\hbar} \lvert E,\alpha\rangle \langle E,\alpha\rvert,$$

and

$$K(x_b,T;x_a,0) = \int dE \sum_\alpha \psi_{E,\alpha}(x_b)\psi_{E,\alpha}^*(x_a) e^{-iET/\hbar}.$$

This notation hides important normalization choices. In some problems the natural label is momentum $p$, wave number $k$, angular momentum, channel index, or a mix of discrete and continuous labels. The measure must match the chosen normalization.

For a one-dimensional free particle,

$$H=\frac{p^2}{2m},$$

and the momentum identity is

$$I=\int_{-\infty}^{\infty} \frac{dp}{2\pi\hbar} \lvert p\rangle\langle p\rvert$$

when

$$\langle x\vert p\rangle=e^{ipx/\hbar}.$$

The spectral integral is

$$K_0(x_b,T;x_a,0) = \int_{-\infty}^{\infty} \frac{dp}{2\pi\hbar} \exp\left[ \frac{ip(x_b-x_a)}{\hbar} - \frac{ip^2T}{2m\hbar} \right],$$

which evaluates to the free-particle propagator after the real-time convergence prescription is specified.

Mixed Spectra

Many Hamiltonians have both bound states and continuum states. A typical identity resolution has the schematic form

$$I = \sum_{n\in\mathrm{bound}} \lvert n\rangle\langle n\rvert \;+ \int dE \sum_\alpha \lvert E,\alpha\rangle\langle E,\alpha\rvert.$$

The propagator then splits into

$$\begin{aligned} K(x_b,T;x_a,0) &= \sum_{n\in\mathrm{bound}} \psi_n(x_b)\psi_n^*(x_a) e^{-iE_nT/\hbar}\\ &\quad+ \int dE \sum_\alpha \psi_{E,\alpha}(x_b)\psi_{E,\alpha}^*(x_a) e^{-iET/\hbar}. \end{aligned}$$

The bound-state terms are square-normalizable contributions. The continuum integral describes scattering and dispersive components. The details of the continuum eigenfunctions encode boundary conditions, phase shifts, channels, and normalization conventions.

Bound-State Contributions

Bound states contribute isolated phases. If the initial wavefunction has overlap

$$c_n=\langle n\vert\psi(0)\rangle,$$

then the bound-state part of the evolved wavefunction is

$$\psi_{\mathrm{bound}}(x,T) = \sum_{n\in\mathrm{bound}} c_n\psi_n(x)e^{-iE_nT/\hbar}.$$

For a purely bound discrete spectrum, this is the whole evolution. For a mixed spectrum, it is only the part that remains in the bound subspace.

In real time, these terms do not decay merely because time passes. Their phases rotate. Decay or relaxation requires additional physics, such as coupling to other degrees of freedom, resonances, coarse graining, or an open-system description.

Scattering-State Contributions

Scattering states are usually not square-normalizable. They are delta-normalized and must be integrated against wave-packet amplitudes. If continuum states are labeled by channel $\alpha$ and energy $E$, then a continuum wave packet has amplitudes

$$c_\alpha(E) = \langle E,\alpha\vert\psi(0)\rangle,$$

and evolves as

$$\psi_{\mathrm{cont}}(x,T) = \int dE \sum_\alpha c_\alpha(E) \psi_{E,\alpha}(x) e^{-iET/\hbar}.$$

The scattering eigenfunctions carry physical information in their asymptotic form. Phase shifts, reflection and transmission amplitudes, and channel mixing appear in $\psi_{E,\alpha}(x)$, not in the universal phase factor alone.

This is why continuum normalization must be stated before interpreting coefficients. A delta-normalized amplitude, a box-normalized amplitude, and a flux-normalized amplitude answer different practical questions.

Examples

For an infinite square well on $0\lt x\lt L$,

$$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\frac{n\pi x}{L}, \qquad E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}.$$

The kernel is

$$K_{\mathrm{box}}(x_b,T;x_a,0) = \sum_{n=1}^{\infty} \frac{2}{L} \sin\frac{n\pi x_b}{L} \sin\frac{n\pi x_a}{L} e^{-iE_nT/\hbar}.$$

For the harmonic oscillator,

$$E_n=\hbar\omega\left(n+\frac12\right),$$

so

$$K_{\mathrm{ho}}(x_b,T;x_a,0) = \sum_{n=0}^{\infty} \psi_n(x_b)\psi_n^*(x_a) e^{-i\omega(n+1/2)T}.$$

Evaluating this sum with Mehler’s formula gives the closed oscillator kernel. That derivation belongs to Harmonic-Oscillator Propagator.

Relation to Green Functions

The time-domain propagator and the energy-domain resolvent are built from the same spectral data but apply different scalar functions to energy.

For the propagator,

$$E\longmapsto e^{-iET/\hbar}.$$

For the resolvent,

$$E\longmapsto \frac{1}{z-E}.$$

Thus a discrete Green function has the form

$$G(x,x';z) = \sum_n \frac{\psi_n(x)\psi_n^*(x')}{z-E_n},$$

while the propagator has phases $e^{-iE_nT/\hbar}$ instead. The spectral poles and branch cuts of Green functions encode the same bound and continuum structure that appears in the propagator’s sums and integrals.

A common bridge is the retarded time-domain Green function

$$G_R(t) = -\frac{i}{\hbar}\Theta(t)e^{-iHt/\hbar}.$$

The step function and prefactor encode a response convention; they are not part of the basic unitary kernel by themselves. The resolvent and response-function conventions are developed in Spectral Representation of Green Functions.

Common Mistakes

• Writing a discrete sum when the Hamiltonian has a continuum or mixed spectrum.
• Forgetting degeneracy labels or replacing projectors by a basis-dependent expression without checking completeness.
• Treating continuum eigenstates as normalizable states.
• Using the wrong measure for the chosen normalization convention.
• Reusing full-line free-particle eigenfunctions when boundary conditions demand a different domain.
• Confusing the time-domain propagator with the energy-domain Green function.
• Interpreting spectral terms as probabilities after no energy measurement has been performed.

Cross-Links

• Propagator Kernel
• Composition Law
• Free-Particle Propagator
• Harmonic-Oscillator Propagator
• Propagator Table
• Energy Eigenstates
• Spectral Decomposition
• Discrete and Continuous Spectra
• Spectral Theorem, Practical Version
• Spectral Representation of Green Functions

References

• 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.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
• G. F. Roach, Green’s Functions, 2nd ed., Cambridge University Press, 1982.

Exercises

1. Starting from $H=\sum_nE_nP_n$, derive $U(T)=\sum_ne^{-iE_nT/\hbar}P_n$.

Solution

The exponential is a function of $H$. By functional calculus,

$$f(H)=\sum_n f(E_n)P_n.$$

Taking

$$f(E)=e^{-iET/\hbar}$$

gives

$$U(T)=e^{-iHT/\hbar} = \sum_n e^{-iE_nT/\hbar}P_n.$$

2. Show how degeneracy modifies the coordinate-space spectral kernel.

Solution

If the eigenspace of $E_n$ has basis $\{\lvert n,\alpha\rangle\}$, then

$$P_n = \sum_\alpha \lvert n,\alpha\rangle\langle n,\alpha\rvert.$$

The kernel contribution is

$$\begin{aligned} \langle x_b\rvert e^{-iE_nT/\hbar}P_n \lvert x_a\rangle &= e^{-iE_nT/\hbar} \sum_\alpha \langle x_b\vert n,\alpha\rangle \langle n,\alpha\vert x_a\rangle\\ &= e^{-iE_nT/\hbar} \sum_\alpha \psi_{n,\alpha}(x_b) \psi_{n,\alpha}^*(x_a). \end{aligned}$$

3. Use momentum states to write the free-particle spectral integral.

Solution

For $H=p^2/(2m)$ and

$$\langle x\vert p\rangle=e^{ipx/\hbar}, \qquad I=\int\frac{dp}{2\pi\hbar}\lvert p\rangle\langle p\rvert,$$

the kernel is

$$\begin{aligned} K_0(x_b,T;x_a,0) &= \int\frac{dp}{2\pi\hbar} \langle x_b\vert p\rangle e^{-ip^2T/(2m\hbar)} \langle p\vert x_a\rangle\\ &= \int\frac{dp}{2\pi\hbar} \exp\left[ \frac{ip(x_b-x_a)}{\hbar} - \frac{ip^2T}{2m\hbar} \right]. \end{aligned}$$

4. Write the schematic propagator for a Hamiltonian with two bound states and a one-channel continuum.

Solution

If the bound states are $\psi_0,\psi_1$ with energies $E_0,E_1$, and continuum states are $\psi_E$ with measure $dE$, then

$$\begin{aligned} K(x_b,T;x_a,0) &= \psi_0(x_b)\psi_0^*(x_a)e^{-iE_0T/\hbar}\\ &\quad+ \psi_1(x_b)\psi_1^*(x_a)e^{-iE_1T/\hbar}\\ &\quad+ \int dE\, \psi_E(x_b)\psi_E^*(x_a)e^{-iET/\hbar}. \end{aligned}$$

The exact continuum measure and normalization must be specified by the problem.

5. Compare the scalar spectral factors for the propagator and resolvent.

Solution

Both are functions of the Hamiltonian. On an energy eigenstate with energy $E$,

$$U(T) \quad\text{multiplies by}\quad e^{-iET/\hbar},$$

whereas

$$R(z)=(zI-H)^{-1} \quad\text{multiplies by}\quad \frac{1}{z-E}.$$

Thus they use the same spectral projectors or generalized eigenstates but attach different functions of $E$.