Density of States in Transition Rates

In a transition rate, the density of states is the measure of how many distinct final states are available near the energy selected by the drive or interaction. It converts a probability for one normalized final state into a rate summed over a continuum. The result is trustworthy only when the state normalization, channel labels, degeneracy factors, and energy-shell Jacobian are all compatible with the matrix element.

Required background. Fermi’s Golden Rule supplies the rate formula; Plane Waves and Delta Normalization fixes continuum inner products, and Delta Function gives the Jacobian rule. Density of States: First Encounter derives box counting.

From a Final-State Sum to a Rate

Let $\lvert i\rangle$ be the prepared state and let $\lvert\alpha\rangle$ label final eigenstates of the chosen unperturbed Hamiltonian with energies $E_\alpha$. A standard golden-rule expression is

$$\Gamma_i = \frac{2\pi}{\hbar} \sum_\alpha \lvert W_{\alpha i}\rvert^2 \delta(E_\alpha-E_*),$$

where

$$W_{\alpha i} = \langle\alpha\rvert W\lvert i\rangle.$$

The selected energy $E_*$ depends on the process. For a static perturbation,

$$E_*=E_i.$$

For absorption from a monochromatic drive,

$$E_*=E_i+\hbar\omega,$$

while stimulated emission selects $E_i-\hbar\omega$. The delta function is the long-time energy-selection limit; at finite time it is replaced by a kernel of width of order $\hbar/T$.

The exact density of final states is the distribution

$$\rho(E) = \sum_\alpha \delta(E-E_\alpha).$$

It has units of inverse energy when it counts a dimensionless number of states. A density per volume has units

$$[\rho/\mathcal V] = (\text{energy}\times\text{volume})^{-1}.$$

Density of states is not an occupation probability. It says what states exist, not which of them are populated.

The Weighted Density Is the Physical Object

The rate does not usually depend on $\rho(E)$ alone. Define the matrix-element-weighted density

$$\mathcal S_i(E) \equiv \sum_\alpha \lvert W_{\alpha i}\rvert^2 \delta(E-E_\alpha).$$

Then

$$\Gamma_i = \frac{2\pi}{\hbar} \mathcal S_i(E_*).$$

Where $\rho(E)$ is finite and nonzero, define the mean squared matrix element over the energy shell. Then one may write exactly

$$\mathcal S_i(E) = \overline{ \lvert W(E)\rvert^2 } \rho(E).$$

Replacing that shell average by one representative matrix element is an approximation. Pulling the weighted density out of a finite energy-window integral additionally requires slow energy variation. This distinction explains why:

• a large density of states produces no transition if symmetry makes every relevant matrix element vanish;
• two channels with the same density of states can have very different rates;
• a threshold singularity in $\rho(E)$ can be softened or removed by a vanishing matrix element;
• different directions on one energy shell may need explicit angular averaging.

The dimensions also check the formula. Since

$$[\mathcal S_i] = \text{energy},$$

the factor $2\pi/\hbar$ converts it to inverse time.

Channels and Continuum Coordinates

Suppose final states are organized into channels $c$ and continuous coordinates $\mathbf q$. A channel may encode spin, polarization, angular momentum, particle species, propagation direction, band index, or any other discrete label. Write the completeness measure schematically as

$$\sum_\alpha \longrightarrow \sum_c \int d\mu_c(\mathbf q).$$

The channel density of states is

$$\rho_c(E) = \int d\mu_c(\mathbf q)\, \delta\big(E-E_c(\mathbf q)\big),$$

and the total density is

$$\rho(E) = \sum_c\rho_c(E).$$

The rate is

$$\Gamma_i = \frac{2\pi}{\hbar} \sum_c \int d\mu_c(\mathbf q)\, \times \lvert W_{ci}(\mathbf q)\rvert^2 \delta\big(E_c(\mathbf q)-E_*\big).$$

This form is safer than writing $\lvert W\rvert^2\rho$ prematurely because the integration measure and matrix element remain visibly paired.

The Energy-Shell Jacobian

For coordinates $\mathbf q\in\mathbb R^n$ with

$$d\mu_c(\mathbf q) = w_c(\mathbf q)d^nq,$$

the delta function restricts the integral to the constant-energy surface

$$\Sigma_{c,E} = \left\{ \mathbf q: E_c(\mathbf q)=E \right\}.$$

Away from critical points where $\nabla_{\mathbf q}E_c=0$, the coarea formula gives

$$\rho_c(E) = \int_{\Sigma_{c,E}} \frac{ w_c(\mathbf q)d\Sigma }{ \lVert\nabla_{\mathbf q}E_c\rVert }.$$

The factor

$$\frac{1}{ \lVert\nabla_{\mathbf q}E_c\rVert }$$

is the Jacobian that converts a coordinate-space shell thickness into an energy interval. Large constant-energy surfaces increase the density of states; a small energy gradient also increases it because many states fit into a narrow energy window.

At a band extremum, saddle point, flat band, or threshold, the gradient can vanish. The resulting density-of-states singularity may be physical, integrable, regularized by finite size or broadening, or modified by interactions. One should return to the original integral rather than substitute a formally divergent shell expression without analysis.

Box Normalization and Volume Cancellation

Continuum rates are often regulated by a periodic box of volume $\mathcal V=L^d$. The normalized plane waves are

$$\langle\mathbf x\vert\mathbf k\rangle_{\mathcal V} = \frac{1}{\sqrt{\mathcal V}} e^{i\mathbf k\cdot\mathbf x},$$

with

$$\mathbf k = \frac{2\pi}{L} \mathbf n, \qquad \mathbf n\in\mathbb Z^d.$$

The large-box sum becomes

$$\sum_{\mathbf k} \longrightarrow \frac{\mathcal V}{(2\pi)^d} \int d^dk.$$

For a localized initial state and localized interaction $W$, a typical box-normalized matrix element scales as

$$W_{\mathbf k i}^{(\mathcal V)} = \frac{1}{\sqrt{\mathcal V}} \mathcal M_i(\mathbf k),$$

where

$$\mathcal M_i(\mathbf k) = \int d^dx\, e^{-i\mathbf k\cdot\mathbf x} W(\mathbf x)\psi_i(\mathbf x)$$

in this convention. Consequently,

$$\left\lvert W_{\mathbf k i}^{(\mathcal V)} \right\rvert^2 = \frac{1}{\mathcal V} \lvert\mathcal M_i(\mathbf k)\rvert^2.$$

Substitution into the golden-rule sum gives

$$\begin{aligned} \Gamma_i &= \frac{2\pi}{\hbar} \frac{\mathcal V}{(2\pi)^d} \int d^dk\, \frac{ \lvert\mathcal M_i(\mathbf k)\rvert^2 }{\mathcal V} \times \delta\big(E(\mathbf k)-E_*\big)\\ &= \frac{2\pi}{\hbar} \int\frac{d^dk}{(2\pi)^d}\, \lvert\mathcal M_i(\mathbf k)\rvert^2 \delta\big(E(\mathbf k)-E_*\big). \end{aligned}$$

The artificial box volume cancels. If a purported rate for one localized emitter grows with the normalization volume, the continuum normalization has almost certainly been mixed inconsistently.

Momentum Delta Normalization

One may bypass the box by choosing

$$\langle\mathbf x\vert\mathbf k\rangle = e^{i\mathbf k\cdot\mathbf x},$$

so that

$$\langle\mathbf k\vert\mathbf k'\rangle = (2\pi)^d \delta^{(d)}(\mathbf k-\mathbf k')$$

and

$$I = \int\frac{d^dk}{(2\pi)^d} \lvert\mathbf k\rangle \langle\mathbf k\rvert.$$

The continuum rate then has the volume-free form obtained above. Other conventions place factors of $(2\pi)^{d/2}$ in the wavefunction and use $\delta^{(d)}(\mathbf k-\mathbf k')$ without the prefactor. Either convention is valid if completeness, matrix elements, and density-of-states measures are transformed together.

Energy-Normalized States

Sometimes the density-of-states Jacobian is absorbed into the state normalization. Let $q$ be a one-dimensional continuum coordinate within channel $c$, with

$$\langle q,c\vert q',c'\rangle = \delta_{cc'}\delta(q-q').$$

Assume $E_c(q)$ is monotonic on that branch. Define

$$\lvert E,c\rangle = \frac{ \lvert q(E),c\rangle }{ \sqrt{ \left\lvert dE_c/dq\right\rvert } }.$$

Then

$$\langle E,c\vert E',c'\rangle = \delta_{cc'}\delta(E-E')$$

and

$$I_{\mathrm{cont}} = \sum_c \int dE\, \lvert E,c\rangle \langle E,c\rvert.$$

In this convention, the golden-rule rate is

$$\Gamma_i = \frac{2\pi}{\hbar} \sum_c \left\lvert \langle E_*,c\rvert W\lvert i\rangle \right\rvert^2.$$

No explicit $\rho(E_*)$ appears because the Jacobian is already inside $\lvert E,c\rangle$. The energy-normalized matrix element has different units from a box-normalized matrix element. Inserting an additional density of states would double count the same Jacobian.

If a given energy has several roots $q_j(E)$, each monotonic branch must be a separate channel or root label. In one dimension, for example, right- and left-moving states at the same energy account for the familiar factor of two unless the physical boundary condition selects only one direction.

Normalization Dictionary

Convention	Inner product	Completeness measure	Where the density factor lives
Finite box	$\delta_{\mathbf k\mathbf k'}$	$\sum_{\mathbf k}$	in the increasing number of discrete states
Momentum delta	$(2\pi)^d\delta^{(d)}(\mathbf k-\mathbf k')$	$\int d^dk/(2\pi)^d$	in the momentum measure and energy-shell Jacobian
Energy delta	$\delta_{cc'}\delta(E-E')$	$\sum_c\int dE$	absorbed into the definition of $\lvert E,c\rangle$

These are different coordinate descriptions of the same continuum. A calculation may change convention, but it must transform the states, completeness relation, and matrix element as a single package.

Free-Particle Examples in One, Two, and Three Dimensions

For a spinless nonrelativistic particle with threshold energy $E_{\mathrm{th}}$, define

$$\varepsilon \equiv E-E_{\mathrm{th}}, \qquad \varepsilon\gt0,$$

and dispersion

$$\varepsilon = \frac{\hbar^2k^2}{2m}.$$

Per box volume $\mathcal V_d$, the density of states in $d$ spatial dimensions is

$$\frac{\rho_d(\varepsilon)}{\mathcal V_d} = g_{\mathrm{int}} \frac{S_{d-1}}{(2\pi)^d} \frac{m}{\hbar^2} \left( \frac{\sqrt{2m\varepsilon}}{\hbar} \right)^{d-2},$$

where $g_{\mathrm{int}}$ counts included internal degeneracies and

$$S_{d-1} = \frac{2\pi^{d/2}}{ \Gamma(d/2) }$$

is the area of the unit sphere in $d$ dimensions. A factor $\Theta(\varepsilon)$ is understood.

The three most common cases are

$$\begin{array}{c|c} d & \rho_d(\varepsilon)/\mathcal V_d \\ \hline 1 & \displaystyle \frac{g_{\mathrm{int}}}{\pi\hbar} \sqrt{\frac{m}{2\varepsilon}} \\[0.9em] 2 & \displaystyle \frac{g_{\mathrm{int}}m}{2\pi\hbar^2} \\[0.9em] 3 & \displaystyle \frac{g_{\mathrm{int}}}{4\pi^2} \left( \frac{2m}{\hbar^2} \right)^{3/2} \sqrt{\varepsilon} \end{array}$$

Thus

$$\rho_d(\varepsilon) \propto \varepsilon^{d/2-1}.$$

• In one dimension the state density diverges as $\varepsilon^{-1/2}$ at threshold.
• In two dimensions it approaches a constant step.
• In three dimensions it vanishes as $\sqrt{\varepsilon}$.

These are density-of-states statements, not complete rate laws. The matrix element, channel boundary conditions, and any incident-flux factor can introduce additional powers of momentum.

Worked Energy-Shell Integral in Three Dimensions

Consider the volume-independent continuum expression

$$\Gamma_i = \frac{2\pi}{\hbar} \int\frac{d^3k}{(2\pi)^3}\, \times \lvert\mathcal M_i(\mathbf k)\rvert^2 \delta\left( \frac{\hbar^2k^2}{2m}-\varepsilon_* \right).$$

Let

$$k_* = \frac{\sqrt{2m\varepsilon_*}}{\hbar}.$$

For $k\ge0$,

$$\delta\left( \frac{\hbar^2k^2}{2m}-\varepsilon_* \right) = \frac{m}{\hbar^2k_*} \delta(k-k_*).$$

Using $d^3k=k^2dk\,d\Omega_{\mathbf k}$ gives

$$\Gamma_i = \frac{2\pi}{\hbar} \frac{mk_*}{(2\pi)^3\hbar^2} \int d\Omega_{\mathbf k}\, \left\lvert \mathcal M_i(k_*,\Omega_{\mathbf k}) \right\rvert^2.$$

If the on-shell matrix element is isotropic,

$$\Gamma_i = \frac{2\pi}{\hbar} \lvert\mathcal M_i(k_*)\rvert^2 \frac{mk_*}{2\pi^2\hbar^2}.$$

The last factor is precisely the spinless three-dimensional density of states per volume in the momentum-delta convention appropriate to $\mathcal M_i$.

Threshold Scaling and Selection Rules

If the on-shell matrix element approaches a nonzero constant, the free-particle threshold scaling follows the density of states:

$$\Gamma(\varepsilon) \propto \varepsilon^{d/2-1}.$$

But angular momentum and symmetry often force the matrix element to vanish. If a partial-wave amplitude behaves as

$$\mathcal M_\ell(k) \propto k^\ell,$$

then in three dimensions

$$\Gamma_\ell \propto k^{2\ell+1} \propto \varepsilon^{\ell+1/2},$$

for a regular short-range problem without a threshold resonance. Long-range forces, a near-threshold bound or virtual state, and other singular threshold behavior can change the assumed power of the matrix element. The density of states supplies only one part of the threshold law.

Selection Rules in Transition Rates explains how symmetry determines the zeros and allowed tensor structure of the matrix element.

Finite Time and the Continuum Window

A finite observation time $T$ resolves an energy window of order

$$\Delta E_T \sim \frac{\hbar}{T}.$$

Let $\delta E_{\mathrm{level}}$ be the typical spacing of final levels in a large but finite box. A smooth continuum approximation requires many levels inside the finite-time window:

$$\delta E_{\mathrm{level}} \ll \frac{\hbar}{T}.$$

Since

$$\delta E_{\mathrm{level}} \sim \frac{1}{\rho(E_*)},$$

the number of sampled levels is approximately

$$N_{\mathrm{win}} \sim \rho(E_*) \frac{\hbar}{T}.$$

One wants

$$N_{\mathrm{win}} \gg 1.$$

At the same time, the weighted density should vary little across that window. If $E_{\mathrm{var}}$ is its characteristic variation scale, require

$$\frac{\hbar}{T} \ll E_{\mathrm{var}}.$$

Together,

$$\delta E_{\mathrm{level}} \ll \frac{\hbar}{T} \ll E_{\mathrm{var}}.$$

The rate description also requires weak depletion,

$$\Gamma_iT \ll 1,$$

unless the exponential decay or master-equation resummation is performed. These inequalities define a time window; a continuum rate is not an exact all-time law for a finite closed system.

Multiple Channels and Degeneracy Factors

A reliable channel sum states what is included in $c$ and in $\rho_c$:

$$\Gamma_i = \sum_c\Gamma_{i\to c}.$$

Typical discrete factors include

• spin or helicity of the final particles;
• photon polarization;
• valleys, bands, subbands, or transverse modes;
• magnetic quantum numbers;
• right- and left-moving branches;
• particle species and reaction channels.

Average initial-state rates with the actual preparation probabilities for an incoherent mixture. Lack of experimental resolution alone does not turn a coherent superposition into a mixture: amplitudes from its components can interfere before the final-state sum. Final-state quantum numbers are summed when the measurement is inclusive over them. Identical final particles may require a symmetry factor rather than a naive label sum.

A degeneracy factor must appear exactly once. If $g_{\mathrm{int}}$ is already included in $\rho(E)$, summing the same internal states again in $\sum_c$ double counts them.

Relativistic Phase Space

Relativistic decay rates retain the structure “squared amplitude times final-state measure,” but use $d^3p/[(2\pi)^3 2E]$ in natural units, a four-momentum conservation delta function, and an initial normalization factor. Identical-particle counting and initial spin preparation must also be specified. Scattering: QFT Bridge Reference derives that dictionary; nonrelativistic densities cannot be inserted into it without changing the state normalization.

Exercises

Weighted versus unweighted density

Two final channels have the same density of states $\rho(E_*)$. Their on-shell matrix elements are $W_{1i}=w$ and $W_{2i}=0$. Compute the total rate and explain why $2\rho(E_*)\lvert w\rvert^2$ is incorrect.

Solution

The channel sum gives

$$\begin{aligned} \Gamma_i &= \frac{2\pi}{\hbar} \rho(E_*) \left( \lvert w\rvert^2+0 \right) \\ &= \frac{2\pi}{\hbar} \rho(E_*)\lvert w\rvert^2. \end{aligned}$$

The second channel contributes states but no transition strength. Replacing both channel matrix elements by $w$ ignores the selection-rule zero. The physical object is the weighted density, not the unweighted count alone.

Derive the dimensional threshold powers

For $\varepsilon=\hbar^2k^2/(2m)$, derive the scaling of the free-particle density of states in $d$ dimensions and specialize to $d=1,2,3$.

Solution

The number of states in a shell is proportional to

$$k^{d-1}dk.$$

Since

$$\frac{d\varepsilon}{dk} = \frac{\hbar^2k}{m},$$

one has

$$\rho_d(\varepsilon) \propto \frac{k^{d-1}}{d\varepsilon/dk} \propto k^{d-2}.$$

Using $k\propto\sqrt\varepsilon$ gives

$$\rho_d(\varepsilon) \propto \varepsilon^{d/2-1}.$$

Therefore

$$\rho_1\propto\varepsilon^{-1/2}, \qquad \rho_2\propto\varepsilon^0, \qquad \rho_3\propto\varepsilon^{1/2}.$$

Verify energy normalization

Assume $\langle q\vert q'\rangle=\delta(q-q')$ and a monotonic dispersion $E(q)$. Verify that

$$\lvert E\rangle = \frac{\lvert q(E)\rangle}{ \sqrt{\lvert dE/dq\rvert} }$$

satisfies $\langle E\vert E'\rangle=\delta(E-E')$.

Solution

The coordinate delta function transforms as

$$\delta\big(q(E)-q(E')\big) = \left\lvert \frac{dE}{dq} \right\rvert \delta(E-E').$$

Therefore

$$\begin{aligned} \langle E\vert E'\rangle &= \frac{ \delta\big(q(E)-q(E')\big) }{ \sqrt{\lvert dE/dq\rvert_E} \sqrt{\lvert dE/dq\rvert_{E'}} } \\ &= \delta(E-E'). \end{aligned}$$

On the support of the delta function, the two Jacobians are equal. If the dispersion has several roots at the same energy, each branch needs a separate channel label.

Find a golden-rule time window

Near $E_*$, let the mean final-level spacing be $\delta E$, let the weighted density vary on scale $E_{\mathrm{var}}$, and let the predicted rate be $\Gamma$. State inequalities on $T$ for a continuum golden-rule calculation without substantial depletion.

Solution

Many levels must lie inside the finite-time energy window:

$$\delta E \ll \frac{\hbar}{T}.$$

The weighted density must be nearly constant across that window:

$$\frac{\hbar}{T} \ll E_{\mathrm{var}}.$$

Finally, first-order depletion must remain small:

$$\Gamma T \ll 1.$$

Thus a useful interval requires

$$\frac{\hbar}{E_{\mathrm{var}}} \ll T \ll \min\left( \frac{\hbar}{\delta E}, \frac{1}{\Gamma} \right),$$

up to numerical factors set by the precise window convention. If no such interval exists, a constant continuum rate is not controlled.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• E. Fermi, Nuclear Physics, University of Chicago Press, 1950.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• 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.